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Banach and quasi-Banach spaces ofvector-valued sequences with special properties
Vinıcius Vieira Favaro
Workshop on Infinite Dimensional AnalysisBuenos Aires, 22-25 July 2014
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Table of Contents
1 Introduction
2 Main result
3 Applications
4 Bibliography
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Introduction
Introduction
In this work we continue the research initiated in [1, 3] onthe existence of infinite dimensional closed subspaces ofBanach or quasi-Banach sequence spaces formed bysequences with special properties.
Given a Banach space X, in [3] the authors introduce alarge class of Banach or quasi-Banach spaces formed byX-valued sequences, called invariant sequences spaces,which encompasses several classical sequences spaces asparticular cases.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 4: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/4.jpg)
Introduction
Introduction
In this work we continue the research initiated in [1, 3] onthe existence of infinite dimensional closed subspaces ofBanach or quasi-Banach sequence spaces formed bysequences with special properties.
Given a Banach space X, in [3] the authors introduce alarge class of Banach or quasi-Banach spaces formed byX-valued sequences, called invariant sequences spaces,which encompasses several classical sequences spaces asparticular cases.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 5: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/5.jpg)
Introduction
Roughly speaking, the main results of [1, 3] prove that, forevery invariant sequence space E of X-valued sequencesand every subset Γ of (0,∞], there exists a closed infinitedimensional subspace of E formed, up to the null vector,by sequences not belonging to
⋃q∈Γ
`q(X); as well as a closed
infinite dimensional subspace of E formed, up to the nullvector, by sequences not belonging to c0(X).
In other words we can say that E −⋃q∈Γ
`q(X) and
E − c0(X) are spaceable. Remember that a subset A of atopological vector space V is spaceable if A ∪ {0} contains aclosed infinite dimensional subspace of V .
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 6: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/6.jpg)
Introduction
Roughly speaking, the main results of [1, 3] prove that, forevery invariant sequence space E of X-valued sequencesand every subset Γ of (0,∞], there exists a closed infinitedimensional subspace of E formed, up to the null vector,by sequences not belonging to
⋃q∈Γ
`q(X); as well as a closed
infinite dimensional subspace of E formed, up to the nullvector, by sequences not belonging to c0(X).
In other words we can say that E −⋃q∈Γ
`q(X) and
E − c0(X) are spaceable. Remember that a subset A of atopological vector space V is spaceable if A ∪ {0} contains aclosed infinite dimensional subspace of V .
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 7: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/7.jpg)
Introduction
Roughly speaking, the main results of [1, 3] prove that, forevery invariant sequence space E of X-valued sequencesand every subset Γ of (0,∞], there exists a closed infinitedimensional subspace of E formed, up to the null vector,by sequences not belonging to
⋃q∈Γ
`q(X); as well as a closed
infinite dimensional subspace of E formed, up to the nullvector, by sequences not belonging to c0(X).
In other words we can say that E −⋃q∈Γ
`q(X) and
E − c0(X) are spaceable. Remember that a subset A of atopological vector space V is spaceable if A ∪ {0} contains aclosed infinite dimensional subspace of V .
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 8: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/8.jpg)
Introduction
Roughly speaking, the main results of [1, 3] prove that, forevery invariant sequence space E of X-valued sequencesand every subset Γ of (0,∞], there exists a closed infinitedimensional subspace of E formed, up to the null vector,by sequences not belonging to
⋃q∈Γ
`q(X); as well as a closed
infinite dimensional subspace of E formed, up to the nullvector, by sequences not belonging to c0(X).
In other words we can say that E −⋃q∈Γ
`q(X) and
E − c0(X) are spaceable. Remember that a subset A of atopological vector space V is spaceable if A ∪ {0} contains aclosed infinite dimensional subspace of V .
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 9: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/9.jpg)
Introduction
In this talk we consider the following much more generalsituation: given Banach spaces X and Y , a mapf : X −→ Y , a set Γ ⊆ (0,+∞] and an invariant sequencespace E of X-valued sequences, we investigate the existenceof closed infinite dimensional subspaces of E formed, up tothe origin, by sequences (xj)
∞j=1 ∈ E such that either
(f(xj))∞j=1 /∈
⋃q∈Γ
`q(Y ) or
(f(xj))∞j=1 /∈
⋃q∈Γ
`wq (Y ) or
(f(xj))∞j=1 /∈ c0(Y ).
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 10: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/10.jpg)
Introduction
In this talk we consider the following much more generalsituation: given Banach spaces X and Y , a mapf : X −→ Y , a set Γ ⊆ (0,+∞] and an invariant sequencespace E of X-valued sequences, we investigate the existenceof closed infinite dimensional subspaces of E formed, up tothe origin, by sequences (xj)
∞j=1 ∈ E such that either
(f(xj))∞j=1 /∈
⋃q∈Γ
`q(Y ) or
(f(xj))∞j=1 /∈
⋃q∈Γ
`wq (Y ) or
(f(xj))∞j=1 /∈ c0(Y ).
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 11: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/11.jpg)
Introduction
In this talk we consider the following much more generalsituation: given Banach spaces X and Y , a mapf : X −→ Y , a set Γ ⊆ (0,+∞] and an invariant sequencespace E of X-valued sequences, we investigate the existenceof closed infinite dimensional subspaces of E formed, up tothe origin, by sequences (xj)
∞j=1 ∈ E such that either
(f(xj))∞j=1 /∈
⋃q∈Γ
`q(Y ) or
(f(xj))∞j=1 /∈
⋃q∈Γ
`wq (Y ) or
(f(xj))∞j=1 /∈ c0(Y ).
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 12: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/12.jpg)
Introduction
In this talk we consider the following much more generalsituation: given Banach spaces X and Y , a mapf : X −→ Y , a set Γ ⊆ (0,+∞] and an invariant sequencespace E of X-valued sequences, we investigate the existenceof closed infinite dimensional subspaces of E formed, up tothe origin, by sequences (xj)
∞j=1 ∈ E such that either
(f(xj))∞j=1 /∈
⋃q∈Γ
`q(Y ) or
(f(xj))∞j=1 /∈
⋃q∈Γ
`wq (Y ) or
(f(xj))∞j=1 /∈ c0(Y ).
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 13: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/13.jpg)
Introduction
As usual, `p(X) and `wp (X) are the Banach spaces(p-Banach spaces if 0 < p < 1) of p-summable and weaklyp-summable X-valued sequences, respectively, and c0(X) isthe Banach space of norm null X-valued sequences. Lettingf be the identity on X, the cases of sequences (xj)
∞j=1 ∈ E
such that (f(xj))∞j=1 /∈
⋃q∈Γ
`q(Y ) or (f(xj))∞j=1 /∈ c0(Y )
recover the situation investigated in [1, 3]. So, the resultsproved here generalize the previous results in twodirections: we consider f belonging to a large class offunctions and we consider spaces formed by sequences(xj)
∞j=1 ∈ E such that (f(xj))
∞j=1 does not belong to⋃
q∈Γ
`wq (Y ), a condition much more restrictive than not to
belong to⋃q∈Γ
`q(Y ).
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Introduction
As usual, `p(X) and `wp (X) are the Banach spaces(p-Banach spaces if 0 < p < 1) of p-summable and weaklyp-summable X-valued sequences, respectively, and c0(X) isthe Banach space of norm null X-valued sequences. Lettingf be the identity on X, the cases of sequences (xj)
∞j=1 ∈ E
such that (f(xj))∞j=1 /∈
⋃q∈Γ
`q(Y ) or (f(xj))∞j=1 /∈ c0(Y )
recover the situation investigated in [1, 3]. So, the resultsproved here generalize the previous results in twodirections: we consider f belonging to a large class offunctions and we consider spaces formed by sequences(xj)
∞j=1 ∈ E such that (f(xj))
∞j=1 does not belong to⋃
q∈Γ
`wq (Y ), a condition much more restrictive than not to
belong to⋃q∈Γ
`q(Y ).
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 15: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/15.jpg)
Introduction
As usual, `p(X) and `wp (X) are the Banach spaces(p-Banach spaces if 0 < p < 1) of p-summable and weaklyp-summable X-valued sequences, respectively, and c0(X) isthe Banach space of norm null X-valued sequences. Lettingf be the identity on X, the cases of sequences (xj)
∞j=1 ∈ E
such that (f(xj))∞j=1 /∈
⋃q∈Γ
`q(Y ) or (f(xj))∞j=1 /∈ c0(Y )
recover the situation investigated in [1, 3]. So, the resultsproved here generalize the previous results in twodirections: we consider f belonging to a large class offunctions and we consider spaces formed by sequences(xj)
∞j=1 ∈ E such that (f(xj))
∞j=1 does not belong to⋃
q∈Γ
`wq (Y ), a condition much more restrictive than not to
belong to⋃q∈Γ
`q(Y ).
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 16: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/16.jpg)
Introduction
As usual, `p(X) and `wp (X) are the Banach spaces(p-Banach spaces if 0 < p < 1) of p-summable and weaklyp-summable X-valued sequences, respectively, and c0(X) isthe Banach space of norm null X-valued sequences. Lettingf be the identity on X, the cases of sequences (xj)
∞j=1 ∈ E
such that (f(xj))∞j=1 /∈
⋃q∈Γ
`q(Y ) or (f(xj))∞j=1 /∈ c0(Y )
recover the situation investigated in [1, 3]. So, the resultsproved here generalize the previous results in twodirections: we consider f belonging to a large class offunctions and we consider spaces formed by sequences(xj)
∞j=1 ∈ E such that (f(xj))
∞j=1 does not belong to⋃
q∈Γ
`wq (Y ), a condition much more restrictive than not to
belong to⋃q∈Γ
`q(Y ).
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 17: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/17.jpg)
Introduction
As usual, `p(X) and `wp (X) are the Banach spaces(p-Banach spaces if 0 < p < 1) of p-summable and weaklyp-summable X-valued sequences, respectively, and c0(X) isthe Banach space of norm null X-valued sequences. Lettingf be the identity on X, the cases of sequences (xj)
∞j=1 ∈ E
such that (f(xj))∞j=1 /∈
⋃q∈Γ
`q(Y ) or (f(xj))∞j=1 /∈ c0(Y )
recover the situation investigated in [1, 3]. So, the resultsproved here generalize the previous results in twodirections: we consider f belonging to a large class offunctions and we consider spaces formed by sequences(xj)
∞j=1 ∈ E such that (f(xj))
∞j=1 does not belong to⋃
q∈Γ
`wq (Y ), a condition much more restrictive than not to
belong to⋃q∈Γ
`q(Y ).
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 18: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/18.jpg)
Main result
Main result
Definition
Let X 6= {0} be a Banach space.(a) Given x ∈ XN, by x0 we mean the zerofree version of x, thatis: if x has only finitely many non-zero coordinates, thenx0 = 0; otherwise, x0 = (xj)
∞j=1 where xj is the j-th non-zero
coordinate of x.(b) By an invariant sequence space over X we mean aninfinite-dimensional Banach or quasi-Banach space E ofX-valued sequences enjoying the following conditions:(b1) For x ∈ XN such that x0 6= 0, x ∈ E if and only if x0 ∈ E,and in this case ‖x‖E ≤ K‖x0‖E for some constant Kdepending only on E.(b2) ‖xj‖X ≤ ‖x‖E for every x = (xj)
∞j=1 ∈ E and every j ∈ N.
An invariant sequence space is an invariant sequence space oversome Banach space X.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Main result
Definition
Let X 6= {0} be a Banach space.(a) Given x ∈ XN, by x0 we mean the zerofree version of x, thatis: if x has only finitely many non-zero coordinates, thenx0 = 0; otherwise, x0 = (xj)
∞j=1 where xj is the j-th non-zero
coordinate of x.(b) By an invariant sequence space over X we mean aninfinite-dimensional Banach or quasi-Banach space E ofX-valued sequences enjoying the following conditions:(b1) For x ∈ XN such that x0 6= 0, x ∈ E if and only if x0 ∈ E,and in this case ‖x‖E ≤ K‖x0‖E for some constant Kdepending only on E.(b2) ‖xj‖X ≤ ‖x‖E for every x = (xj)
∞j=1 ∈ E and every j ∈ N.
An invariant sequence space is an invariant sequence space oversome Banach space X.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 20: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/20.jpg)
Main result
Main result
Definition
Let X 6= {0} be a Banach space.(a) Given x ∈ XN, by x0 we mean the zerofree version of x, thatis: if x has only finitely many non-zero coordinates, thenx0 = 0; otherwise, x0 = (xj)
∞j=1 where xj is the j-th non-zero
coordinate of x.(b) By an invariant sequence space over X we mean aninfinite-dimensional Banach or quasi-Banach space E ofX-valued sequences enjoying the following conditions:(b1) For x ∈ XN such that x0 6= 0, x ∈ E if and only if x0 ∈ E,and in this case ‖x‖E ≤ K‖x0‖E for some constant Kdepending only on E.(b2) ‖xj‖X ≤ ‖x‖E for every x = (xj)
∞j=1 ∈ E and every j ∈ N.
An invariant sequence space is an invariant sequence space oversome Banach space X.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Main result
Definition
Let X 6= {0} be a Banach space.(a) Given x ∈ XN, by x0 we mean the zerofree version of x, thatis: if x has only finitely many non-zero coordinates, thenx0 = 0; otherwise, x0 = (xj)
∞j=1 where xj is the j-th non-zero
coordinate of x.(b) By an invariant sequence space over X we mean aninfinite-dimensional Banach or quasi-Banach space E ofX-valued sequences enjoying the following conditions:(b1) For x ∈ XN such that x0 6= 0, x ∈ E if and only if x0 ∈ E,and in this case ‖x‖E ≤ K‖x0‖E for some constant Kdepending only on E.(b2) ‖xj‖X ≤ ‖x‖E for every x = (xj)
∞j=1 ∈ E and every j ∈ N.
An invariant sequence space is an invariant sequence space oversome Banach space X.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 22: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/22.jpg)
Main result
Main result
Definition
Let X 6= {0} be a Banach space.(a) Given x ∈ XN, by x0 we mean the zerofree version of x, thatis: if x has only finitely many non-zero coordinates, thenx0 = 0; otherwise, x0 = (xj)
∞j=1 where xj is the j-th non-zero
coordinate of x.(b) By an invariant sequence space over X we mean aninfinite-dimensional Banach or quasi-Banach space E ofX-valued sequences enjoying the following conditions:(b1) For x ∈ XN such that x0 6= 0, x ∈ E if and only if x0 ∈ E,and in this case ‖x‖E ≤ K‖x0‖E for some constant Kdepending only on E.(b2) ‖xj‖X ≤ ‖x‖E for every x = (xj)
∞j=1 ∈ E and every j ∈ N.
An invariant sequence space is an invariant sequence space oversome Banach space X.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 23: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/23.jpg)
Main result
Main result
Definition
Let X 6= {0} be a Banach space.(a) Given x ∈ XN, by x0 we mean the zerofree version of x, thatis: if x has only finitely many non-zero coordinates, thenx0 = 0; otherwise, x0 = (xj)
∞j=1 where xj is the j-th non-zero
coordinate of x.(b) By an invariant sequence space over X we mean aninfinite-dimensional Banach or quasi-Banach space E ofX-valued sequences enjoying the following conditions:(b1) For x ∈ XN such that x0 6= 0, x ∈ E if and only if x0 ∈ E,and in this case ‖x‖E ≤ K‖x0‖E for some constant Kdepending only on E.(b2) ‖xj‖X ≤ ‖x‖E for every x = (xj)
∞j=1 ∈ E and every j ∈ N.
An invariant sequence space is an invariant sequence space oversome Banach space X.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 24: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/24.jpg)
Main result
Main result
Definition
Let X 6= {0} be a Banach space.(a) Given x ∈ XN, by x0 we mean the zerofree version of x, thatis: if x has only finitely many non-zero coordinates, thenx0 = 0; otherwise, x0 = (xj)
∞j=1 where xj is the j-th non-zero
coordinate of x.(b) By an invariant sequence space over X we mean aninfinite-dimensional Banach or quasi-Banach space E ofX-valued sequences enjoying the following conditions:(b1) For x ∈ XN such that x0 6= 0, x ∈ E if and only if x0 ∈ E,and in this case ‖x‖E ≤ K‖x0‖E for some constant Kdepending only on E.(b2) ‖xj‖X ≤ ‖x‖E for every x = (xj)
∞j=1 ∈ E and every j ∈ N.
An invariant sequence space is an invariant sequence space oversome Banach space X.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 25: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/25.jpg)
Main result
Example
(a) For 0 < p ≤ ∞, `p(X), `wp (X), `up(X) (unconditionallyp-summable X-valued sequences) and `m(s;p) (X) (mixedsequence space) are invariant sequence spaces over X with theirrespective usual norms (p-norms if 0 < p < 1).(b) The Lorentz sequence spaces, Orlicz sequence space, . . . .
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Example
(a) For 0 < p ≤ ∞, `p(X), `wp (X), `up(X) (unconditionallyp-summable X-valued sequences) and `m(s;p) (X) (mixedsequence space) are invariant sequence spaces over X with theirrespective usual norms (p-norms if 0 < p < 1).(b) The Lorentz sequence spaces, Orlicz sequence space, . . . .
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Example
(a) For 0 < p ≤ ∞, `p(X), `wp (X), `up(X) (unconditionallyp-summable X-valued sequences) and `m(s;p) (X) (mixedsequence space) are invariant sequence spaces over X with theirrespective usual norms (p-norms if 0 < p < 1).(b) The Lorentz sequence spaces, Orlicz sequence space, . . . .
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 28: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/28.jpg)
Main result
Example
(a) For 0 < p ≤ ∞, `p(X), `wp (X), `up(X) (unconditionallyp-summable X-valued sequences) and `m(s;p) (X) (mixedsequence space) are invariant sequence spaces over X with theirrespective usual norms (p-norms if 0 < p < 1).(b) The Lorentz sequence spaces, Orlicz sequence space, . . . .
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 29: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/29.jpg)
Main result
Example
(a) For 0 < p ≤ ∞, `p(X), `wp (X), `up(X) (unconditionallyp-summable X-valued sequences) and `m(s;p) (X) (mixedsequence space) are invariant sequence spaces over X with theirrespective usual norms (p-norms if 0 < p < 1).(b) The Lorentz sequence spaces, Orlicz sequence space, . . . .
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 30: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/30.jpg)
Main result
Example
(a) For 0 < p ≤ ∞, `p(X), `wp (X), `up(X) (unconditionallyp-summable X-valued sequences) and `m(s;p) (X) (mixedsequence space) are invariant sequence spaces over X with theirrespective usual norms (p-norms if 0 < p < 1).(b) The Lorentz sequence spaces, Orlicz sequence space, . . . .
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 31: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/31.jpg)
Main result
Example
(a) For 0 < p ≤ ∞, `p(X), `wp (X), `up(X) (unconditionallyp-summable X-valued sequences) and `m(s;p) (X) (mixedsequence space) are invariant sequence spaces over X with theirrespective usual norms (p-norms if 0 < p < 1).(b) The Lorentz sequence spaces, Orlicz sequence space, . . . .
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 32: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/32.jpg)
Main result
Definition
Let X and Y be Banach spaces, E be an invariant sequencespace over X, Γ ⊆ (0,+∞] and f : X −→ Y be a function. Wedefine the sets:
C(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`q(Y )
},
Cw(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`wq (Y )
}and
C(E, f, 0) ={
(xj)∞j=1 ∈ E : (f(xj))
∞j=1 /∈ c0(Y )
}.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Definition
Let X and Y be Banach spaces, E be an invariant sequencespace over X, Γ ⊆ (0,+∞] and f : X −→ Y be a function. Wedefine the sets:
C(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`q(Y )
},
Cw(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`wq (Y )
}and
C(E, f, 0) ={
(xj)∞j=1 ∈ E : (f(xj))
∞j=1 /∈ c0(Y )
}.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Definition
Let X and Y be Banach spaces, E be an invariant sequencespace over X, Γ ⊆ (0,+∞] and f : X −→ Y be a function. Wedefine the sets:
C(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`q(Y )
},
Cw(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`wq (Y )
}and
C(E, f, 0) ={
(xj)∞j=1 ∈ E : (f(xj))
∞j=1 /∈ c0(Y )
}.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Definition
Let X and Y be Banach spaces, E be an invariant sequencespace over X, Γ ⊆ (0,+∞] and f : X −→ Y be a function. Wedefine the sets:
C(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`q(Y )
},
Cw(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`wq (Y )
}and
C(E, f, 0) ={
(xj)∞j=1 ∈ E : (f(xj))
∞j=1 /∈ c0(Y )
}.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Definition
A map f : X −→ Y between normed spaces is said to be:(a) Non-contractive if f(0) = 0 and for every scalar α 6= 0 thereis a constant K(α) > 0 such that
‖f(αx)‖Y ≥ K(α) · ‖f(x)‖Y
for every x ∈ X.(b) Strongly non-contractive if f(0) = 0 and for every scalarα 6= 0 there is a constant K(α) > 0 such that
|ϕ(f(αx))| ≥ K(α) · |ϕ(f(x))|
for all x ∈ X and ϕ ∈ Y ′.
By the Hahn–Banach theorem, strongly non-contractivefunctions are non-contractive.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Definition
A map f : X −→ Y between normed spaces is said to be:(a) Non-contractive if f(0) = 0 and for every scalar α 6= 0 thereis a constant K(α) > 0 such that
‖f(αx)‖Y ≥ K(α) · ‖f(x)‖Y
for every x ∈ X.(b) Strongly non-contractive if f(0) = 0 and for every scalarα 6= 0 there is a constant K(α) > 0 such that
|ϕ(f(αx))| ≥ K(α) · |ϕ(f(x))|
for all x ∈ X and ϕ ∈ Y ′.
By the Hahn–Banach theorem, strongly non-contractivefunctions are non-contractive.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Definition
A map f : X −→ Y between normed spaces is said to be:(a) Non-contractive if f(0) = 0 and for every scalar α 6= 0 thereis a constant K(α) > 0 such that
‖f(αx)‖Y ≥ K(α) · ‖f(x)‖Y
for every x ∈ X.(b) Strongly non-contractive if f(0) = 0 and for every scalarα 6= 0 there is a constant K(α) > 0 such that
|ϕ(f(αx))| ≥ K(α) · |ϕ(f(x))|
for all x ∈ X and ϕ ∈ Y ′.
By the Hahn–Banach theorem, strongly non-contractivefunctions are non-contractive.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Definition
A map f : X −→ Y between normed spaces is said to be:(a) Non-contractive if f(0) = 0 and for every scalar α 6= 0 thereis a constant K(α) > 0 such that
‖f(αx)‖Y ≥ K(α) · ‖f(x)‖Y
for every x ∈ X.(b) Strongly non-contractive if f(0) = 0 and for every scalarα 6= 0 there is a constant K(α) > 0 such that
|ϕ(f(αx))| ≥ K(α) · |ϕ(f(x))|
for all x ∈ X and ϕ ∈ Y ′.
By the Hahn–Banach theorem, strongly non-contractivefunctions are non-contractive.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Example
Subhomogeneous functions (with f(0) = 0) arenon-contractive;
bounded and unbounded linear operators are stronglynon-contractive (hence non-contractive); and
homogeneous polynomials (continuous or not) are stronglynon-contractive (hence non-contractive).
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Example
Subhomogeneous functions (with f(0) = 0) arenon-contractive;
bounded and unbounded linear operators are stronglynon-contractive (hence non-contractive); and
homogeneous polynomials (continuous or not) are stronglynon-contractive (hence non-contractive).
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Example
Subhomogeneous functions (with f(0) = 0) arenon-contractive;
bounded and unbounded linear operators are stronglynon-contractive (hence non-contractive); and
homogeneous polynomials (continuous or not) are stronglynon-contractive (hence non-contractive).
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Theorem
Let X and Y be Banach spaces, E be an invariant sequencespace over X, f : X −→ Y be a function and Γ ⊆ (0,+∞].(a) If f is non-contractive, then C(E, f,Γ) and C(E, f, 0) areeither empty or spaceable.(b) If f is strongly non-contractive, then Cw(E, f,Γ) is eitherempty or spaceable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 44: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/44.jpg)
Main result
Theorem
Let X and Y be Banach spaces, E be an invariant sequencespace over X, f : X −→ Y be a function and Γ ⊆ (0,+∞].(a) If f is non-contractive, then C(E, f,Γ) and C(E, f, 0) areeither empty or spaceable.(b) If f is strongly non-contractive, then Cw(E, f,Γ) is eitherempty or spaceable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 45: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/45.jpg)
Main result
Theorem
Let X and Y be Banach spaces, E be an invariant sequencespace over X, f : X −→ Y be a function and Γ ⊆ (0,+∞].(a) If f is non-contractive, then C(E, f,Γ) and C(E, f, 0) areeither empty or spaceable.(b) If f is strongly non-contractive, then Cw(E, f,Γ) is eitherempty or spaceable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 46: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/46.jpg)
Main result
Theorem
Let X and Y be Banach spaces, E be an invariant sequencespace over X, f : X −→ Y be a function and Γ ⊆ (0,+∞].(a) If f is non-contractive, then C(E, f,Γ) and C(E, f, 0) areeither empty or spaceable.(b) If f is strongly non-contractive, then Cw(E, f,Γ) is eitherempty or spaceable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 47: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/47.jpg)
Main result
Theorem
Let X and Y be Banach spaces, E be an invariant sequencespace over X, f : X −→ Y be a function and Γ ⊆ (0,+∞].(a) If f is non-contractive, then C(E, f,Γ) and C(E, f, 0) areeither empty or spaceable.(b) If f is strongly non-contractive, then Cw(E, f,Γ) is eitherempty or spaceable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 48: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/48.jpg)
Main result
Theorem
Let X and Y be Banach spaces, E be an invariant sequencespace over X, f : X −→ Y be a function and Γ ⊆ (0,+∞].(a) If f is non-contractive, then C(E, f,Γ) and C(E, f, 0) areeither empty or spaceable.(b) If f is strongly non-contractive, then Cw(E, f,Γ) is eitherempty or spaceable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 49: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/49.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
Remember that
C(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`q(Y )
}.
Let us fix a notation. For α = (αn)∞n=1 ∈ KN and w ∈ X wedenote
w ⊗ α = α⊗ w := (αnw)∞n=1 ∈ XN.
Assume that C(E, f,Γ) is non-empty and choose x ∈ C(E, f,Γ).Since E is an invariant sequence space, then x0 ∈ E and thecondition f(0) = 0 guarantees that x0 ∈ C(E, f,Γ). Writingx0 = (xj)
∞j=1 we have that xj 6= 0 for every j.
Step 1: Split N into countably many infinite pairwise disjointsubsets (Ni)
∞i=1. For every i ∈ N set Ni = {i1 < i2 < . . .} and
define
yi =
∞∑j=1
xj ⊗ eij ∈ XN.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Sketch of the proof for the case C(E, f,Γ).
Remember that
C(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`q(Y )
}.
Let us fix a notation. For α = (αn)∞n=1 ∈ KN and w ∈ X wedenote
w ⊗ α = α⊗ w := (αnw)∞n=1 ∈ XN.
Assume that C(E, f,Γ) is non-empty and choose x ∈ C(E, f,Γ).Since E is an invariant sequence space, then x0 ∈ E and thecondition f(0) = 0 guarantees that x0 ∈ C(E, f,Γ). Writingx0 = (xj)
∞j=1 we have that xj 6= 0 for every j.
Step 1: Split N into countably many infinite pairwise disjointsubsets (Ni)
∞i=1. For every i ∈ N set Ni = {i1 < i2 < . . .} and
define
yi =
∞∑j=1
xj ⊗ eij ∈ XN.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 51: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/51.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
Remember that
C(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`q(Y )
}.
Let us fix a notation. For α = (αn)∞n=1 ∈ KN and w ∈ X wedenote
w ⊗ α = α⊗ w := (αnw)∞n=1 ∈ XN.
Assume that C(E, f,Γ) is non-empty and choose x ∈ C(E, f,Γ).Since E is an invariant sequence space, then x0 ∈ E and thecondition f(0) = 0 guarantees that x0 ∈ C(E, f,Γ). Writingx0 = (xj)
∞j=1 we have that xj 6= 0 for every j.
Step 1: Split N into countably many infinite pairwise disjointsubsets (Ni)
∞i=1. For every i ∈ N set Ni = {i1 < i2 < . . .} and
define
yi =
∞∑j=1
xj ⊗ eij ∈ XN.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 52: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/52.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
Remember that
C(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`q(Y )
}.
Let us fix a notation. For α = (αn)∞n=1 ∈ KN and w ∈ X wedenote
w ⊗ α = α⊗ w := (αnw)∞n=1 ∈ XN.
Assume that C(E, f,Γ) is non-empty and choose x ∈ C(E, f,Γ).Since E is an invariant sequence space, then x0 ∈ E and thecondition f(0) = 0 guarantees that x0 ∈ C(E, f,Γ). Writingx0 = (xj)
∞j=1 we have that xj 6= 0 for every j.
Step 1: Split N into countably many infinite pairwise disjointsubsets (Ni)
∞i=1. For every i ∈ N set Ni = {i1 < i2 < . . .} and
define
yi =
∞∑j=1
xj ⊗ eij ∈ XN.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Sketch of the proof for the case C(E, f,Γ).
Remember that
C(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`q(Y )
}.
Let us fix a notation. For α = (αn)∞n=1 ∈ KN and w ∈ X wedenote
w ⊗ α = α⊗ w := (αnw)∞n=1 ∈ XN.
Assume that C(E, f,Γ) is non-empty and choose x ∈ C(E, f,Γ).Since E is an invariant sequence space, then x0 ∈ E and thecondition f(0) = 0 guarantees that x0 ∈ C(E, f,Γ). Writingx0 = (xj)
∞j=1 we have that xj 6= 0 for every j.
Step 1: Split N into countably many infinite pairwise disjointsubsets (Ni)
∞i=1. For every i ∈ N set Ni = {i1 < i2 < . . .} and
define
yi =
∞∑j=1
xj ⊗ eij ∈ XN.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 54: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/54.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
Remember that
C(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`q(Y )
}.
Let us fix a notation. For α = (αn)∞n=1 ∈ KN and w ∈ X wedenote
w ⊗ α = α⊗ w := (αnw)∞n=1 ∈ XN.
Assume that C(E, f,Γ) is non-empty and choose x ∈ C(E, f,Γ).Since E is an invariant sequence space, then x0 ∈ E and thecondition f(0) = 0 guarantees that x0 ∈ C(E, f,Γ). Writingx0 = (xj)
∞j=1 we have that xj 6= 0 for every j.
Step 1: Split N into countably many infinite pairwise disjointsubsets (Ni)
∞i=1. For every i ∈ N set Ni = {i1 < i2 < . . .} and
define
yi =
∞∑j=1
xj ⊗ eij ∈ XN.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 55: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/55.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
Remember that
C(E, f,Γ) =
{(xj)
∞j=1 ∈ E : (f(xj))
∞j=1 /∈
⋃q∈Γ
`q(Y )
}.
Let us fix a notation. For α = (αn)∞n=1 ∈ KN and w ∈ X wedenote
w ⊗ α = α⊗ w := (αnw)∞n=1 ∈ XN.
Assume that C(E, f,Γ) is non-empty and choose x ∈ C(E, f,Γ).Since E is an invariant sequence space, then x0 ∈ E and thecondition f(0) = 0 guarantees that x0 ∈ C(E, f,Γ). Writingx0 = (xj)
∞j=1 we have that xj 6= 0 for every j.
Step 1: Split N into countably many infinite pairwise disjointsubsets (Ni)
∞i=1. For every i ∈ N set Ni = {i1 < i2 < . . .} and
define
yi =
∞∑j=1
xj ⊗ eij ∈ XN.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 56: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/56.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
Observe that y0i = x0, so 0 6= y0
i ∈ E, hence yi ∈ E for every ibecause E is an invariant sequence space. For q ∈ Γ, q < +∞,
we have∞∑j=1‖f(xj)‖qY = +∞ because x0 ∈ C(E, f,Γ). If
+∞ ∈ Γ, by the same reason we have supi‖f(xi)‖Y = +∞. It
follows that each yi ∈ C(E, f,Γ).
Step 2: Define s = 1 if E is a Banach space and s = s if E is ans-Banach space, 0 < s < 1. We need to prove that the operator
T : `s −→ E , T ((ai)∞i=1) =
∞∑i=1
aiyi,
is well defined. It is possible to prove that∞∑i=1‖aiyi‖E < +∞ if
E is a Banach space and∞∑i=1‖aiyi‖sE < +∞ if E is an s-Banach
space, 0 < s < 1.G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 57: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/57.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
Observe that y0i = x0, so 0 6= y0
i ∈ E, hence yi ∈ E for every ibecause E is an invariant sequence space. For q ∈ Γ, q < +∞,
we have∞∑j=1‖f(xj)‖qY = +∞ because x0 ∈ C(E, f,Γ). If
+∞ ∈ Γ, by the same reason we have supi‖f(xi)‖Y = +∞. It
follows that each yi ∈ C(E, f,Γ).
Step 2: Define s = 1 if E is a Banach space and s = s if E is ans-Banach space, 0 < s < 1. We need to prove that the operator
T : `s −→ E , T ((ai)∞i=1) =
∞∑i=1
aiyi,
is well defined. It is possible to prove that∞∑i=1‖aiyi‖E < +∞ if
E is a Banach space and∞∑i=1‖aiyi‖sE < +∞ if E is an s-Banach
space, 0 < s < 1.G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 58: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/58.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
Observe that y0i = x0, so 0 6= y0
i ∈ E, hence yi ∈ E for every ibecause E is an invariant sequence space. For q ∈ Γ, q < +∞,
we have∞∑j=1‖f(xj)‖qY = +∞ because x0 ∈ C(E, f,Γ). If
+∞ ∈ Γ, by the same reason we have supi‖f(xi)‖Y = +∞. It
follows that each yi ∈ C(E, f,Γ).
Step 2: Define s = 1 if E is a Banach space and s = s if E is ans-Banach space, 0 < s < 1. We need to prove that the operator
T : `s −→ E , T ((ai)∞i=1) =
∞∑i=1
aiyi,
is well defined. It is possible to prove that∞∑i=1‖aiyi‖E < +∞ if
E is a Banach space and∞∑i=1‖aiyi‖sE < +∞ if E is an s-Banach
space, 0 < s < 1.G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 59: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/59.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
Observe that y0i = x0, so 0 6= y0
i ∈ E, hence yi ∈ E for every ibecause E is an invariant sequence space. For q ∈ Γ, q < +∞,
we have∞∑j=1‖f(xj)‖qY = +∞ because x0 ∈ C(E, f,Γ). If
+∞ ∈ Γ, by the same reason we have supi‖f(xi)‖Y = +∞. It
follows that each yi ∈ C(E, f,Γ).
Step 2: Define s = 1 if E is a Banach space and s = s if E is ans-Banach space, 0 < s < 1. We need to prove that the operator
T : `s −→ E , T ((ai)∞i=1) =
∞∑i=1
aiyi,
is well defined. It is possible to prove that∞∑i=1‖aiyi‖E < +∞ if
E is a Banach space and∞∑i=1‖aiyi‖sE < +∞ if E is an s-Banach
space, 0 < s < 1.G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 60: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/60.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
Observe that y0i = x0, so 0 6= y0
i ∈ E, hence yi ∈ E for every ibecause E is an invariant sequence space. For q ∈ Γ, q < +∞,
we have∞∑j=1‖f(xj)‖qY = +∞ because x0 ∈ C(E, f,Γ). If
+∞ ∈ Γ, by the same reason we have supi‖f(xi)‖Y = +∞. It
follows that each yi ∈ C(E, f,Γ).
Step 2: Define s = 1 if E is a Banach space and s = s if E is ans-Banach space, 0 < s < 1. We need to prove that the operator
T : `s −→ E , T ((ai)∞i=1) =
∞∑i=1
aiyi,
is well defined. It is possible to prove that∞∑i=1‖aiyi‖E < +∞ if
E is a Banach space and∞∑i=1‖aiyi‖sE < +∞ if E is an s-Banach
space, 0 < s < 1.G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 61: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/61.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
Observe that y0i = x0, so 0 6= y0
i ∈ E, hence yi ∈ E for every ibecause E is an invariant sequence space. For q ∈ Γ, q < +∞,
we have∞∑j=1‖f(xj)‖qY = +∞ because x0 ∈ C(E, f,Γ). If
+∞ ∈ Γ, by the same reason we have supi‖f(xi)‖Y = +∞. It
follows that each yi ∈ C(E, f,Γ).
Step 2: Define s = 1 if E is a Banach space and s = s if E is ans-Banach space, 0 < s < 1. We need to prove that the operator
T : `s −→ E , T ((ai)∞i=1) =
∞∑i=1
aiyi,
is well defined. It is possible to prove that∞∑i=1‖aiyi‖E < +∞ if
E is a Banach space and∞∑i=1‖aiyi‖sE < +∞ if E is an s-Banach
space, 0 < s < 1.G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 62: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/62.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
Observe that y0i = x0, so 0 6= y0
i ∈ E, hence yi ∈ E for every ibecause E is an invariant sequence space. For q ∈ Γ, q < +∞,
we have∞∑j=1‖f(xj)‖qY = +∞ because x0 ∈ C(E, f,Γ). If
+∞ ∈ Γ, by the same reason we have supi‖f(xi)‖Y = +∞. It
follows that each yi ∈ C(E, f,Γ).
Step 2: Define s = 1 if E is a Banach space and s = s if E is ans-Banach space, 0 < s < 1. We need to prove that the operator
T : `s −→ E , T ((ai)∞i=1) =
∞∑i=1
aiyi,
is well defined. It is possible to prove that∞∑i=1‖aiyi‖E < +∞ if
E is a Banach space and∞∑i=1‖aiyi‖sE < +∞ if E is an s-Banach
space, 0 < s < 1.G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 63: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/63.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
In both cases the series∞∑i=1
aiyi converges in E, hence the
operator is well defined. It is easy to see that T is linear andinjective. Thus T (`s) is a closed infinite-dimensional subspaceof E.
Step 3: We have to show that if z = (zn)∞n=1 ∈ T (`s), z 6= 0,then (f(zn))∞n=1 /∈
⋃q∈Γ
`q(Y ). Given such a z, there are
sequences(a
(k)i
)∞i=1∈ `s, k ∈ N, such that
z = limk→∞ T((a
(k)i
)∞i=1
)in E.
Using the convergence below and the fact that f isnon-contractive, it is possible to prove (hardwork) that there isa subsequence
(zmj
)∞j=1
of z = (zn)∞n=1 satisfying:
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Main result
Sketch of the proof for the case C(E, f,Γ).
In both cases the series∞∑i=1
aiyi converges in E, hence the
operator is well defined. It is easy to see that T is linear andinjective. Thus T (`s) is a closed infinite-dimensional subspaceof E.
Step 3: We have to show that if z = (zn)∞n=1 ∈ T (`s), z 6= 0,then (f(zn))∞n=1 /∈
⋃q∈Γ
`q(Y ). Given such a z, there are
sequences(a
(k)i
)∞i=1∈ `s, k ∈ N, such that
z = limk→∞ T((a
(k)i
)∞i=1
)in E.
Using the convergence below and the fact that f isnon-contractive, it is possible to prove (hardwork) that there isa subsequence
(zmj
)∞j=1
of z = (zn)∞n=1 satisfying:
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 65: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/65.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
In both cases the series∞∑i=1
aiyi converges in E, hence the
operator is well defined. It is easy to see that T is linear andinjective. Thus T (`s) is a closed infinite-dimensional subspaceof E.
Step 3: We have to show that if z = (zn)∞n=1 ∈ T (`s), z 6= 0,then (f(zn))∞n=1 /∈
⋃q∈Γ
`q(Y ). Given such a z, there are
sequences(a
(k)i
)∞i=1∈ `s, k ∈ N, such that
z = limk→∞ T((a
(k)i
)∞i=1
)in E.
Using the convergence below and the fact that f isnon-contractive, it is possible to prove (hardwork) that there isa subsequence
(zmj
)∞j=1
of z = (zn)∞n=1 satisfying:
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 66: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/66.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
In both cases the series∞∑i=1
aiyi converges in E, hence the
operator is well defined. It is easy to see that T is linear andinjective. Thus T (`s) is a closed infinite-dimensional subspaceof E.
Step 3: We have to show that if z = (zn)∞n=1 ∈ T (`s), z 6= 0,then (f(zn))∞n=1 /∈
⋃q∈Γ
`q(Y ). Given such a z, there are
sequences(a
(k)i
)∞i=1∈ `s, k ∈ N, such that
z = limk→∞ T((a
(k)i
)∞i=1
)in E.
Using the convergence below and the fact that f isnon-contractive, it is possible to prove (hardwork) that there isa subsequence
(zmj
)∞j=1
of z = (zn)∞n=1 satisfying:
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 67: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/67.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
In both cases the series∞∑i=1
aiyi converges in E, hence the
operator is well defined. It is easy to see that T is linear andinjective. Thus T (`s) is a closed infinite-dimensional subspaceof E.
Step 3: We have to show that if z = (zn)∞n=1 ∈ T (`s), z 6= 0,then (f(zn))∞n=1 /∈
⋃q∈Γ
`q(Y ). Given such a z, there are
sequences(a
(k)i
)∞i=1∈ `s, k ∈ N, such that
z = limk→∞ T((a
(k)i
)∞i=1
)in E.
Using the convergence below and the fact that f isnon-contractive, it is possible to prove (hardwork) that there isa subsequence
(zmj
)∞j=1
of z = (zn)∞n=1 satisfying:
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 68: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/68.jpg)
Main result
Sketch of the proof for the case C(E, f,Γ).
In both cases the series∞∑i=1
aiyi converges in E, hence the
operator is well defined. It is easy to see that T is linear andinjective. Thus T (`s) is a closed infinite-dimensional subspaceof E.
Step 3: We have to show that if z = (zn)∞n=1 ∈ T (`s), z 6= 0,then (f(zn))∞n=1 /∈
⋃q∈Γ
`q(Y ). Given such a z, there are
sequences(a
(k)i
)∞i=1∈ `s, k ∈ N, such that
z = limk→∞ T((a
(k)i
)∞i=1
)in E.
Using the convergence below and the fact that f isnon-contractive, it is possible to prove (hardwork) that there isa subsequence
(zmj
)∞j=1
of z = (zn)∞n=1 satisfying:
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 69: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/69.jpg)
Main result
∞∑j=1
∥∥∥f(zmj)
∥∥∥q =∞, for all q ∈ Γ, if ∞ /∈ Γ, and
supj ‖f(zmj )‖ =∞, if ∞ ∈ Γ. This shows that(f(zn))∞n=1 /∈
⋃q∈Γ
`q(X) and completes the proof that
z ∈ C(E, f,Γ).�
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 70: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/70.jpg)
Main result
∞∑j=1
∥∥∥f(zmj)
∥∥∥q =∞, for all q ∈ Γ, if ∞ /∈ Γ, and
supj ‖f(zmj )‖ =∞, if ∞ ∈ Γ. This shows that(f(zn))∞n=1 /∈
⋃q∈Γ
`q(X) and completes the proof that
z ∈ C(E, f,Γ).�
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 71: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/71.jpg)
Main result
∞∑j=1
∥∥∥f(zmj)
∥∥∥q =∞, for all q ∈ Γ, if ∞ /∈ Γ, and
supj ‖f(zmj )‖ =∞, if ∞ ∈ Γ. This shows that(f(zn))∞n=1 /∈
⋃q∈Γ
`q(X) and completes the proof that
z ∈ C(E, f,Γ).�
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
![Page 72: Banach and quasi-Banach spaces of vector-valued sequences ...mate.dm.uba.ar/~widaba14/charlas/widaba14_Favaro.pdf · Banach and quasi-Banach spaces of vector-valued sequences with](https://reader033.vdocuments.net/reader033/viewer/2022050219/5f65451c2e0ce4590d5210e0/html5/thumbnails/72.jpg)
Applications
Applications
Definition
Let X and Y be Banach spaces and 0 < p ≤ q < +∞. We saythat
a linear operator u : X −→ Y is absolutely (q; p)-summing if(u(xj))
∞j=1 ∈ `q(Y ) for each (xj)
∞j=1 ∈ `wp (X).
an n-homogeneous polynomial P : X −→ Y is p-dominatedif (P (xj))
∞j=1 ∈ `p/n(Y ) for each (xj)
∞j=1 ∈ `wp (X).
Corollary
Let X and Y be Banach spaces.(a) Let 1 ≤ p ≤ q < +∞ and let u : X −→ Y be a non-absolutely(q, p)-summing linear operator. Then the set{
(xj)∞j=1 ∈ `wp (X) : (u(xj))
∞j=1 /∈ `q(Y )
}is spaceable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Applications
Definition
Let X and Y be Banach spaces and 0 < p ≤ q < +∞. We saythat
a linear operator u : X −→ Y is absolutely (q; p)-summing if(u(xj))
∞j=1 ∈ `q(Y ) for each (xj)
∞j=1 ∈ `wp (X).
an n-homogeneous polynomial P : X −→ Y is p-dominatedif (P (xj))
∞j=1 ∈ `p/n(Y ) for each (xj)
∞j=1 ∈ `wp (X).
Corollary
Let X and Y be Banach spaces.(a) Let 1 ≤ p ≤ q < +∞ and let u : X −→ Y be a non-absolutely(q, p)-summing linear operator. Then the set{
(xj)∞j=1 ∈ `wp (X) : (u(xj))
∞j=1 /∈ `q(Y )
}is spaceable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Applications
Definition
Let X and Y be Banach spaces and 0 < p ≤ q < +∞. We saythat
a linear operator u : X −→ Y is absolutely (q; p)-summing if(u(xj))
∞j=1 ∈ `q(Y ) for each (xj)
∞j=1 ∈ `wp (X).
an n-homogeneous polynomial P : X −→ Y is p-dominatedif (P (xj))
∞j=1 ∈ `p/n(Y ) for each (xj)
∞j=1 ∈ `wp (X).
Corollary
Let X and Y be Banach spaces.(a) Let 1 ≤ p ≤ q < +∞ and let u : X −→ Y be a non-absolutely(q, p)-summing linear operator. Then the set{
(xj)∞j=1 ∈ `wp (X) : (u(xj))
∞j=1 /∈ `q(Y )
}is spaceable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Applications
Definition
Let X and Y be Banach spaces and 0 < p ≤ q < +∞. We saythat
a linear operator u : X −→ Y is absolutely (q; p)-summing if(u(xj))
∞j=1 ∈ `q(Y ) for each (xj)
∞j=1 ∈ `wp (X).
an n-homogeneous polynomial P : X −→ Y is p-dominatedif (P (xj))
∞j=1 ∈ `p/n(Y ) for each (xj)
∞j=1 ∈ `wp (X).
Corollary
Let X and Y be Banach spaces.(a) Let 1 ≤ p ≤ q < +∞ and let u : X −→ Y be a non-absolutely(q, p)-summing linear operator. Then the set{
(xj)∞j=1 ∈ `wp (X) : (u(xj))
∞j=1 /∈ `q(Y )
}is spaceable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Applications
Definition
Let X and Y be Banach spaces and 0 < p ≤ q < +∞. We saythat
a linear operator u : X −→ Y is absolutely (q; p)-summing if(u(xj))
∞j=1 ∈ `q(Y ) for each (xj)
∞j=1 ∈ `wp (X).
an n-homogeneous polynomial P : X −→ Y is p-dominatedif (P (xj))
∞j=1 ∈ `p/n(Y ) for each (xj)
∞j=1 ∈ `wp (X).
Corollary
Let X and Y be Banach spaces.(a) Let 1 ≤ p ≤ q < +∞ and let u : X −→ Y be a non-absolutely(q, p)-summing linear operator. Then the set{
(xj)∞j=1 ∈ `wp (X) : (u(xj))
∞j=1 /∈ `q(Y )
}is spaceable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Corollary
(b) Let 0 < p < +∞ and let P : X −→ Y be a non-p-dominatedn-homogeneous polynomial. Then the set{
(xj)∞j=1 ∈ `wp (X) : (P (xj))
∞j=1 /∈ `p/n(Y )
}is spaceable.
Proof. It is enough to note that linear operators andhomogeneous polynomials are (strongly) non-contractive mapsand that `wp (X) is an invariant sequence space over X.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Corollary
(b) Let 0 < p < +∞ and let P : X −→ Y be a non-p-dominatedn-homogeneous polynomial. Then the set{
(xj)∞j=1 ∈ `wp (X) : (P (xj))
∞j=1 /∈ `p/n(Y )
}is spaceable.
Proof. It is enough to note that linear operators andhomogeneous polynomials are (strongly) non-contractive mapsand that `wp (X) is an invariant sequence space over X.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Corollary
(b) Let 0 < p < +∞ and let P : X −→ Y be a non-p-dominatedn-homogeneous polynomial. Then the set{
(xj)∞j=1 ∈ `wp (X) : (P (xj))
∞j=1 /∈ `p/n(Y )
}is spaceable.
Proof. It is enough to note that linear operators andhomogeneous polynomials are (strongly) non-contractive mapsand that `wp (X) is an invariant sequence space over X.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Corollary
(b) Let 0 < p < +∞ and let P : X −→ Y be a non-p-dominatedn-homogeneous polynomial. Then the set{
(xj)∞j=1 ∈ `wp (X) : (P (xj))
∞j=1 /∈ `p/n(Y )
}is spaceable.
Proof. It is enough to note that linear operators andhomogeneous polynomials are (strongly) non-contractive mapsand that `wp (X) is an invariant sequence space over X.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Definition
Let X and Y be Banach spaces. We say that
a linear operator u : X −→ Y is completely continuous ifu(xj) −→ u(x) in Y whenever xj
w−→ x in X.
an n-homogeneous polynomial P : X −→ Y is weaklysequentially continuous at the origin if P (xj) −→ 0 in Y
whenever xjw−→ 0 in X.
By cw0 (X) we denote the closed subspace of `∞(X) formed byweakly null X-valued sequences. It is easy to check that cw0 (X)is an invariant sequence space over X, so by last Theorem wehave:
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Definition
Let X and Y be Banach spaces. We say that
a linear operator u : X −→ Y is completely continuous ifu(xj) −→ u(x) in Y whenever xj
w−→ x in X.
an n-homogeneous polynomial P : X −→ Y is weaklysequentially continuous at the origin if P (xj) −→ 0 in Y
whenever xjw−→ 0 in X.
By cw0 (X) we denote the closed subspace of `∞(X) formed byweakly null X-valued sequences. It is easy to check that cw0 (X)is an invariant sequence space over X, so by last Theorem wehave:
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Definition
Let X and Y be Banach spaces. We say that
a linear operator u : X −→ Y is completely continuous ifu(xj) −→ u(x) in Y whenever xj
w−→ x in X.
an n-homogeneous polynomial P : X −→ Y is weaklysequentially continuous at the origin if P (xj) −→ 0 in Y
whenever xjw−→ 0 in X.
By cw0 (X) we denote the closed subspace of `∞(X) formed byweakly null X-valued sequences. It is easy to check that cw0 (X)is an invariant sequence space over X, so by last Theorem wehave:
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Definition
Let X and Y be Banach spaces. We say that
a linear operator u : X −→ Y is completely continuous ifu(xj) −→ u(x) in Y whenever xj
w−→ x in X.
an n-homogeneous polynomial P : X −→ Y is weaklysequentially continuous at the origin if P (xj) −→ 0 in Y
whenever xjw−→ 0 in X.
By cw0 (X) we denote the closed subspace of `∞(X) formed byweakly null X-valued sequences. It is easy to check that cw0 (X)is an invariant sequence space over X, so by last Theorem wehave:
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Corollary
Let X and Y be Banach spaces.(a) Let u : X −→ Y be a non-completely continuous linearoperator. Then the set{
(xj)∞j=1 ∈ cw0 (X) : (u(xj))
∞j=1 /∈ c0(Y )
}is spaceable. In particular, if X lacks the Schur property, thenthere exists an infinite dimensional Banach space formed, up tothe origin, by weakly null but non-norm null X-valuedsequences.(b) Let P : X −→ Y be an n-homogeneous polynomial that failsto be weakly sequentially continuous at the origin. Then the set{
(xj)∞j=1 ∈ cw0 (X) : (P (xj))
∞j=1 /∈ c0(Y )
}is spaceable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Corollary
Let X and Y be Banach spaces.(a) Let u : X −→ Y be a non-completely continuous linearoperator. Then the set{
(xj)∞j=1 ∈ cw0 (X) : (u(xj))
∞j=1 /∈ c0(Y )
}is spaceable. In particular, if X lacks the Schur property, thenthere exists an infinite dimensional Banach space formed, up tothe origin, by weakly null but non-norm null X-valuedsequences.(b) Let P : X −→ Y be an n-homogeneous polynomial that failsto be weakly sequentially continuous at the origin. Then the set{
(xj)∞j=1 ∈ cw0 (X) : (P (xj))
∞j=1 /∈ c0(Y )
}is spaceable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Corollary
Let X and Y be Banach spaces.(a) Let u : X −→ Y be a non-completely continuous linearoperator. Then the set{
(xj)∞j=1 ∈ cw0 (X) : (u(xj))
∞j=1 /∈ c0(Y )
}is spaceable. In particular, if X lacks the Schur property, thenthere exists an infinite dimensional Banach space formed, up tothe origin, by weakly null but non-norm null X-valuedsequences.(b) Let P : X −→ Y be an n-homogeneous polynomial that failsto be weakly sequentially continuous at the origin. Then the set{
(xj)∞j=1 ∈ cw0 (X) : (P (xj))
∞j=1 /∈ c0(Y )
}is spaceable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Recall that a subset A of a topological vector space E isα-lineable if A ∪ {0} contains an α-dimensional linear subspaceof E. And A is maximal dense-lineable if A ∪ {0} contains adense linear subspace V of E with dim(V ) = dim(E).
Proposition
Let X and Y be Banach spaces and p > 0. Then the sets{(xj)
∞j=1 ∈ c0(X) : (u(xj))
∞j=1 /∈ c0(Y )
}and{
(xj)∞j=1 ∈ `p(X) : (u(xj))
∞j=1 /∈ `wp (Y )
}are spaceable for every unbounded linear operator u : X −→ Y .Moreover, if X is separable and p < +∞, then these subsets arealso maximal dense-lineable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Recall that a subset A of a topological vector space E isα-lineable if A ∪ {0} contains an α-dimensional linear subspaceof E. And A is maximal dense-lineable if A ∪ {0} contains adense linear subspace V of E with dim(V ) = dim(E).
Proposition
Let X and Y be Banach spaces and p > 0. Then the sets{(xj)
∞j=1 ∈ c0(X) : (u(xj))
∞j=1 /∈ c0(Y )
}and{
(xj)∞j=1 ∈ `p(X) : (u(xj))
∞j=1 /∈ `wp (Y )
}are spaceable for every unbounded linear operator u : X −→ Y .Moreover, if X is separable and p < +∞, then these subsets arealso maximal dense-lineable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Recall that a subset A of a topological vector space E isα-lineable if A ∪ {0} contains an α-dimensional linear subspaceof E. And A is maximal dense-lineable if A ∪ {0} contains adense linear subspace V of E with dim(V ) = dim(E).
Proposition
Let X and Y be Banach spaces and p > 0. Then the sets{(xj)
∞j=1 ∈ c0(X) : (u(xj))
∞j=1 /∈ c0(Y )
}and{
(xj)∞j=1 ∈ `p(X) : (u(xj))
∞j=1 /∈ `wp (Y )
}are spaceable for every unbounded linear operator u : X −→ Y .Moreover, if X is separable and p < +∞, then these subsets arealso maximal dense-lineable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Recall that a subset A of a topological vector space E isα-lineable if A ∪ {0} contains an α-dimensional linear subspaceof E. And A is maximal dense-lineable if A ∪ {0} contains adense linear subspace V of E with dim(V ) = dim(E).
Proposition
Let X and Y be Banach spaces and p > 0. Then the sets{(xj)
∞j=1 ∈ c0(X) : (u(xj))
∞j=1 /∈ c0(Y )
}and{
(xj)∞j=1 ∈ `p(X) : (u(xj))
∞j=1 /∈ `wp (Y )
}are spaceable for every unbounded linear operator u : X −→ Y .Moreover, if X is separable and p < +∞, then these subsets arealso maximal dense-lineable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Theorem (L. Bernal-Gonzalez and M. Cabrera - JFA 2014)
Assume that X is a metrizable and separable topological vectorspace. Let A ⊂ X and α be an infinite cardinal number suchthat A is α-lineable. If there exists a subset B ⊂ X such thatA+B ⊂ A, A ∩B = ∅ and B is dense-lineable, then A ∪ {0}contains a dense vector space D with dim(D) = α.
Proposition
Let X and Y be Banach spaces and p > 0. Then the sets{(xj)
∞j=1 ∈ c0(X) : (u(xj))
∞j=1 /∈ c0(Y )
}and{
(xj)∞j=1 ∈ `p(X) : (u(xj))
∞j=1 /∈ `wp (Y )
}are spaceable for every unbounded linear operator u : X −→ Y .Moreover, if X is separable and p < +∞, then these subsets arealso maximal dense-lineable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Theorem (L. Bernal-Gonzalez and M. Cabrera - JFA 2014)
Assume that X is a metrizable and separable topological vectorspace. Let A ⊂ X and α be an infinite cardinal number suchthat A is α-lineable. If there exists a subset B ⊂ X such thatA+B ⊂ A, A ∩B = ∅ and B is dense-lineable, then A ∪ {0}contains a dense vector space D with dim(D) = α.
Proposition
Let X and Y be Banach spaces and p > 0. Then the sets{(xj)
∞j=1 ∈ c0(X) : (u(xj))
∞j=1 /∈ c0(Y )
}and{
(xj)∞j=1 ∈ `p(X) : (u(xj))
∞j=1 /∈ `wp (Y )
}are spaceable for every unbounded linear operator u : X −→ Y .Moreover, if X is separable and p < +∞, then these subsets arealso maximal dense-lineable.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Proof. It is not difficult to see that the spaceability of both setsfollows from the main theorem. We shall apply theBernal-Cabrera Theorem to prove the second assertion. Assumethat X is separable and p < +∞. It is clear that c0(X) and`p(X) are separable as well. Let A be either C(c0(X), u, 0) orCw(`p(X), u, {p}). By the spaceability of A we have thatA ∪ {0} contains a c-dimensional subspace, where c is thecardinality of the continuum. Let c00(X) denote the space ofeventually null X-valued sequences. It is clear thatA+ c00(X) ⊆ A, A ∩ c00(X) = ∅ and c00(X) is a dense infinitedimensional subspace of c0(X) and `p(X). By theBernal-Cabrera Theorem, A ∪ {0} contains a c-dimensionaldense subspace, and the result follows because c0(X) and `p(X)are c-dimensional (remember that they are separable infinitedimensional Banach or quasi-Banach spaces). �
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Proof. It is not difficult to see that the spaceability of both setsfollows from the main theorem. We shall apply theBernal-Cabrera Theorem to prove the second assertion. Assumethat X is separable and p < +∞. It is clear that c0(X) and`p(X) are separable as well. Let A be either C(c0(X), u, 0) orCw(`p(X), u, {p}). By the spaceability of A we have thatA ∪ {0} contains a c-dimensional subspace, where c is thecardinality of the continuum. Let c00(X) denote the space ofeventually null X-valued sequences. It is clear thatA+ c00(X) ⊆ A, A ∩ c00(X) = ∅ and c00(X) is a dense infinitedimensional subspace of c0(X) and `p(X). By theBernal-Cabrera Theorem, A ∪ {0} contains a c-dimensionaldense subspace, and the result follows because c0(X) and `p(X)are c-dimensional (remember that they are separable infinitedimensional Banach or quasi-Banach spaces). �
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Proof. It is not difficult to see that the spaceability of both setsfollows from the main theorem. We shall apply theBernal-Cabrera Theorem to prove the second assertion. Assumethat X is separable and p < +∞. It is clear that c0(X) and`p(X) are separable as well. Let A be either C(c0(X), u, 0) orCw(`p(X), u, {p}). By the spaceability of A we have thatA ∪ {0} contains a c-dimensional subspace, where c is thecardinality of the continuum. Let c00(X) denote the space ofeventually null X-valued sequences. It is clear thatA+ c00(X) ⊆ A, A ∩ c00(X) = ∅ and c00(X) is a dense infinitedimensional subspace of c0(X) and `p(X). By theBernal-Cabrera Theorem, A ∪ {0} contains a c-dimensionaldense subspace, and the result follows because c0(X) and `p(X)are c-dimensional (remember that they are separable infinitedimensional Banach or quasi-Banach spaces). �
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Proof. It is not difficult to see that the spaceability of both setsfollows from the main theorem. We shall apply theBernal-Cabrera Theorem to prove the second assertion. Assumethat X is separable and p < +∞. It is clear that c0(X) and`p(X) are separable as well. Let A be either C(c0(X), u, 0) orCw(`p(X), u, {p}). By the spaceability of A we have thatA ∪ {0} contains a c-dimensional subspace, where c is thecardinality of the continuum. Let c00(X) denote the space ofeventually null X-valued sequences. It is clear thatA+ c00(X) ⊆ A, A ∩ c00(X) = ∅ and c00(X) is a dense infinitedimensional subspace of c0(X) and `p(X). By theBernal-Cabrera Theorem, A ∪ {0} contains a c-dimensionaldense subspace, and the result follows because c0(X) and `p(X)are c-dimensional (remember that they are separable infinitedimensional Banach or quasi-Banach spaces). �
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Proof. It is not difficult to see that the spaceability of both setsfollows from the main theorem. We shall apply theBernal-Cabrera Theorem to prove the second assertion. Assumethat X is separable and p < +∞. It is clear that c0(X) and`p(X) are separable as well. Let A be either C(c0(X), u, 0) orCw(`p(X), u, {p}). By the spaceability of A we have thatA ∪ {0} contains a c-dimensional subspace, where c is thecardinality of the continuum. Let c00(X) denote the space ofeventually null X-valued sequences. It is clear thatA+ c00(X) ⊆ A, A ∩ c00(X) = ∅ and c00(X) is a dense infinitedimensional subspace of c0(X) and `p(X). By theBernal-Cabrera Theorem, A ∪ {0} contains a c-dimensionaldense subspace, and the result follows because c0(X) and `p(X)are c-dimensional (remember that they are separable infinitedimensional Banach or quasi-Banach spaces). �
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Proof. It is not difficult to see that the spaceability of both setsfollows from the main theorem. We shall apply theBernal-Cabrera Theorem to prove the second assertion. Assumethat X is separable and p < +∞. It is clear that c0(X) and`p(X) are separable as well. Let A be either C(c0(X), u, 0) orCw(`p(X), u, {p}). By the spaceability of A we have thatA ∪ {0} contains a c-dimensional subspace, where c is thecardinality of the continuum. Let c00(X) denote the space ofeventually null X-valued sequences. It is clear thatA+ c00(X) ⊆ A, A ∩ c00(X) = ∅ and c00(X) is a dense infinitedimensional subspace of c0(X) and `p(X). By theBernal-Cabrera Theorem, A ∪ {0} contains a c-dimensionaldense subspace, and the result follows because c0(X) and `p(X)are c-dimensional (remember that they are separable infinitedimensional Banach or quasi-Banach spaces). �
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Proof. It is not difficult to see that the spaceability of both setsfollows from the main theorem. We shall apply theBernal-Cabrera Theorem to prove the second assertion. Assumethat X is separable and p < +∞. It is clear that c0(X) and`p(X) are separable as well. Let A be either C(c0(X), u, 0) orCw(`p(X), u, {p}). By the spaceability of A we have thatA ∪ {0} contains a c-dimensional subspace, where c is thecardinality of the continuum. Let c00(X) denote the space ofeventually null X-valued sequences. It is clear thatA+ c00(X) ⊆ A, A ∩ c00(X) = ∅ and c00(X) is a dense infinitedimensional subspace of c0(X) and `p(X). By theBernal-Cabrera Theorem, A ∪ {0} contains a c-dimensionaldense subspace, and the result follows because c0(X) and `p(X)are c-dimensional (remember that they are separable infinitedimensional Banach or quasi-Banach spaces). �
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Proof. It is not difficult to see that the spaceability of both setsfollows from the main theorem. We shall apply theBernal-Cabrera Theorem to prove the second assertion. Assumethat X is separable and p < +∞. It is clear that c0(X) and`p(X) are separable as well. Let A be either C(c0(X), u, 0) orCw(`p(X), u, {p}). By the spaceability of A we have thatA ∪ {0} contains a c-dimensional subspace, where c is thecardinality of the continuum. Let c00(X) denote the space ofeventually null X-valued sequences. It is clear thatA+ c00(X) ⊆ A, A ∩ c00(X) = ∅ and c00(X) is a dense infinitedimensional subspace of c0(X) and `p(X). By theBernal-Cabrera Theorem, A ∪ {0} contains a c-dimensionaldense subspace, and the result follows because c0(X) and `p(X)are c-dimensional (remember that they are separable infinitedimensional Banach or quasi-Banach spaces). �
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Example
Let X be an infinite dimensional Banach space and0 < p < +∞. We know that `wp (X)− `p(X) is spaceable, thatis, there exists an infinite dimensional Banach/quasi-Banachspace formed, up to the origin, by X-valued sequences (xj)
∞j=1
such that∞∑j=1|ϕ(xj)|p < +∞ for every bounded linear functional
ϕ ∈ X ′ and∞∑j=1‖xj‖p = +∞. Considering an unbounded linear
functional ϕ on X, the last Proposition yields the following dualresult: there exists an infinite dimensionalBanach/quasi-Banach space formed, up to the origin, by
X-valued sequences (xj)∞j=1 such that
∞∑j=1‖xj‖p < +∞ and
∞∑j=1|ϕ(xj)|p = +∞.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Example
Let X be an infinite dimensional Banach space and0 < p < +∞. We know that `wp (X)− `p(X) is spaceable, thatis, there exists an infinite dimensional Banach/quasi-Banachspace formed, up to the origin, by X-valued sequences (xj)
∞j=1
such that∞∑j=1|ϕ(xj)|p < +∞ for every bounded linear functional
ϕ ∈ X ′ and∞∑j=1‖xj‖p = +∞. Considering an unbounded linear
functional ϕ on X, the last Proposition yields the following dualresult: there exists an infinite dimensionalBanach/quasi-Banach space formed, up to the origin, by
X-valued sequences (xj)∞j=1 such that
∞∑j=1‖xj‖p < +∞ and
∞∑j=1|ϕ(xj)|p = +∞.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Example
Let X be an infinite dimensional Banach space and0 < p < +∞. We know that `wp (X)− `p(X) is spaceable, thatis, there exists an infinite dimensional Banach/quasi-Banachspace formed, up to the origin, by X-valued sequences (xj)
∞j=1
such that∞∑j=1|ϕ(xj)|p < +∞ for every bounded linear functional
ϕ ∈ X ′ and∞∑j=1‖xj‖p = +∞. Considering an unbounded linear
functional ϕ on X, the last Proposition yields the following dualresult: there exists an infinite dimensionalBanach/quasi-Banach space formed, up to the origin, by
X-valued sequences (xj)∞j=1 such that
∞∑j=1‖xj‖p < +∞ and
∞∑j=1|ϕ(xj)|p = +∞.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Applications
Example
Let X be an infinite dimensional Banach space and0 < p < +∞. We know that `wp (X)− `p(X) is spaceable, thatis, there exists an infinite dimensional Banach/quasi-Banachspace formed, up to the origin, by X-valued sequences (xj)
∞j=1
such that∞∑j=1|ϕ(xj)|p < +∞ for every bounded linear functional
ϕ ∈ X ′ and∞∑j=1‖xj‖p = +∞. Considering an unbounded linear
functional ϕ on X, the last Proposition yields the following dualresult: there exists an infinite dimensionalBanach/quasi-Banach space formed, up to the origin, by
X-valued sequences (xj)∞j=1 such that
∞∑j=1‖xj‖p < +∞ and
∞∑j=1|ϕ(xj)|p = +∞.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Bibliography
[1] C. S. Barroso, G. Botelho, V. V. F. and D. Pellegrino,Lineability and spaceability for the weak form of Peano’stheorem and vector-valued sequence spaces,Proc. Amer. Math. Soc. 141 (2013), 1913–1923.
[2] L. Bernal-Gonzalez and M. Ordonez Cabrera,Lineability criteria, with applications,J. Funct. Anal. 266 (2014), 3997–4025.
[3] G. Botelho, D. Diniz, V. V. F. and D. Pellegrino,Spaceability in Banach and quasi-Banach sequence spaces,Linear Algebra Appl. 434 (2011), 1255–1260.
[4] G. Botelho, D. Pellegrino and P. Rueda,Dominated polynomials on infinite dimensional spaces,Proc. Amer. Math. Soc. 138 (2010), 209–216.
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014
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Bibliography
Thank You very much!!!
G. Botelho, V. V. Favaro Buenos Aires 07/25/2014