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Part IV

Appendix

A

Functional Analysis and Set Theory

One should always generalize.(Man muss immer generalisieren)

Carl Jacobi

In order to keep this book as self-contained as possible, in this appendix wecollect without proofs several results from functional analysis that have beenused throughout the book. This part is intended mostly for graduate students.Most proofs will be omitted. Basic references are [AliBo99], [Bre83], [DuSc88],[Ru91], [Yo95]. The reader should be warned that in [DuSc88] and [Ru91] thedefinitions of normal spaces and topological vector spaces are different fromthe standard ones.

A.1 Some Results from Functional Analysis

A.1.1 Topological Spaces

Definition A.1. Let X be a nonempty set. A collection τ ⊆ P (X) is a topol-ogy if

(i) ∅, X ∈ τ ;(ii) if Ui ∈ τ for i = 1, . . . ,M , then U1 ∩ . . . ∩ UM ∈ τ ;(iii) if Uαα∈I is an arbitrary collection of elements of τ then

⋃α∈I Uα ∈ τ .

Example A.2. Given a nonempty set X, the smallest topology consists of∅,X, while the largest topology contains all subsets as open sets, and iscalled the discrete topology .

The pair (X, τ) is called topological space and the elements of τ open sets.For simplicity, we often apply the term topological space only to X. A setC ⊂ X is closed if its complement X \ C is open. The closure E of a set

550 A Functional Analysis and Set Theory

E ⊂ X is the smallest closed set that contains E. The interiorE of a set

E ⊂ X is the union of all its open subsets.A subset E of a topological space X is said to be dense if its closure is

the entire space, i.e., E = X. We say that a topological space is separable ifit contains a countable dense subset.

Given a point x ∈ X, a neighborhood1 of x is any open set U ∈ τ thatcontains x. Given a set E ⊂ X, a neighborhood of E is any open set U ∈ τthat contains E. A topological space is a Hausdorff space if for any x, y ∈ Xwith x = y we may find two disjoint neighborhoods of x and y.

Given a topological space X and a sequence xn, we say that xn con-verges to a point x ∈ X if for every neighborhood U of x we have that xn ∈ Ufor all n sufficiently large. Note that unless the space is Hausdorff, the limitmay not be unique.

A topological space is a normal space if for every pair of disjoint closedsets C1, C2 ⊂ X we may find two disjoint neighborhoods of C1 and C2.

Definition A.3. Let X, Y be two topological spaces and let f : X → Y be afunction from X into Y. We say that f is continuous if f−1 (U) is open forevery open set U ⊂ Y .

The space of all continuous functions f : X → Y is denoted by C (X;Y ).The next two theorems give important characterizations of normal spaces.

Theorem A.4 (Urysohn). A topological space X is normal if and only iffor any two disjoint closed sets C1, C2 ⊂ X there exists a continuous functionf : X → [0, 1] such that f ≡ 1 in C1 and f ≡ 0 in C2.

Theorem A.5 (Tietze extension). A topological space X is normal if andonly for any closed set C ⊂ X and any continuous function f : C → R thereexists a continuous function F : X → R such that F (x) = f (x) for all x ∈ C.Moreover, if f (C) ⊂ [a, b] then F may be constructed so that

F (C) ⊂ [a, b] .

We now introduce the notion of a base for a topology. Let (X, τ) be atopological space. A family β of open sets of X is a base for the topology τif every open set U ∈ τ may be written as union of elements of β. Given apoint x ∈ X, a family βx of neighborhoods of x is a local base at x if everyneighborhood of x contains an element of βx.

Proposition A.6. Let X be a nonempty set and let β ⊂ P (X) be a familyof sets such that

(i) for every x ∈ X there exists B ∈ β such that x ∈ B;

1 The reader should be warned that in some texts (e.g., [DuSc88] and [Ru91]) thedefinition of neighborhood is different.

A.1 Some Results from Functional Analysis 551

(ii) for every B1, B2 ∈ β, with B1 ∩B2 = ∅, and for every x ∈ B1 ∩B2 thereexists B3 ∈ β such that x ∈ B3 and B3 ⊂ B1 ∩B2.

Then the collection τ ⊂ P (X) of arbitrary unions of members of β is atopology for which β is a base.

As we will see later on, the metrizability and the normability of a giventopology depend on the properties of a base (see Theorems A.17 and A.40below).

Definition A.7. Let (X, τ) be a topological space.

(i) The space X satisfies the first axiom of countability if every x ∈ X admitsa countable base of open sets.

(ii) The space X satisfies the second axiom of countability if it has a countablebase.

The following result is used in Chapter 4.

Theorem A.8 (Lindelof). Let (X, τ) be a topological space satisfying thesecond axiom of countability. Then every family F ⊂ τ contains a countablesubfamily Un ⊂ F such that

⋃U∈F

U =⋃n

Un.

In the text we use several notions of compactness.

Definition A.9. Let X be a topological space.

(i) A set K ⊂ X is compact if given any open cover of K, i.e., any collectionUα of elements of τ such that

⋃α Uα ⊃ K, then we may find a finite

subcover (i.e., a finite subcollection of Uα whose union still contains K);(ii) a set E ⊂ X is relatively compact (or precompact) if its closure E is

compact;(iii) a set E ⊂ X is σ-compact if it can be written as a countable union of

compact sets;(iv) the topological space X is locally compact if every point x ∈ X has a

neighborhood whose closure is compact.

Remark A.10. A closed subset of a compact topological space is compact. Onthe other hand, a compact set of a Hausdorff space is closed.

Note that if the topology is not Hausdorff then compact sets may not beclosed.

Example A.11. Given a nonempty set X endowed with the smallest topology,any nonempty set strictly contained in X is compact but not closed.


The next theorem is used to construct cutoff functions and partitions ofunity. These two notions will play a central role in [FoLe10].

Theorem A.12. If X is a locally compact Hausdorff space and K ⊂ U ⊂ X,with K compact and U open, then there exists W open such that W is compactand K ⊂W ⊂W ⊂ U .

Corollary A.13. If X is a locally compact Hausdorff space and K ⊂ U ⊂ X,with K compact and U open, then there exists a function ϕ ∈ Cc (X) suchthat 0 ≤ ϕ ≤ 1, ϕ ≡ 1 on K, and ϕ ≡ 0 on X \ U .

The function ϕ is usually referred to as a cutoff function. To introducepartitions of unity we need the notion of a locally finite family.

Definition A.14. Let X be a topological space and let F be a collection ofsubsets of X. Then

(i) F is locally finite if every x ∈ X has a neighborhood meeting only finitelymany U ∈ F .

(ii) F is σ-locally finite if

F =∞⋃

n=1

Fn,

where each Fn is a locally finite collection in X.

We are now ready to introduce partitions of unity. Their existence is provedin Theorems 2.77 and 6.18.

Definition A.15. If X is a topological space, a partition of unity on X is afamily ϕαα∈I of continuous functions ϕα : X → [0, 1] such that

∑α∈I

ϕα (x) = 1

for all x ∈ X. A partition of unity is locally finite if for every x ∈ X thereexists a neighborhood U of x such that the set α ∈ I : U ∩ suppϕα = ∅ isfinite. If Uαα∈I is an open cover of X, a partition of unity subordinated tothe cover Uαα∈I is a partition of unity ϕαα∈I such that suppϕα ⊂ Uα

for each α ∈ I.

A.1.2 Metric Spaces

Definition A.16. A metric on a set X is a map d : X × X → [0,∞) suchthat

(i) d (x, y) ≤ d (x, z) + d (z, y) for all x, y, z ∈ X;(ii) d (x, y) = d (y, x) for all x, y ∈ X;(iii) d (x, y) = 0 if and only if x = y.


A metric space (X, d) is a set X endowed with a metric d. When there isno possibility of confusion we abbreviate by saying that X is a metric space.If r > 0, the (open) ball of center x0 ∈ X and radius r is the set

B (x0, r) := x ∈ X : d (x0, x) < r .

If x ∈ X and E ⊂ X, the distance of x from the set E is defined by

dist (x,E) := inf d (x, y) : y ∈ E ,

while the distance between two sets E1, E2 ⊂ X is defined by

dist (E1, E2) := inf d (x, y) : x ∈ E1, y ∈ E2 .

A metric space (X, d) can always be rendered into a topological space(X, τ) by taking as a base for the topology τ the family of all open balls. Wethen say that τ is determined by d. Note that (X, τ) is a Hausdorff normalspace. Indeed, if x = y then B

(x, d(x,y)

2

)and B

(y, d(x,y)

2

)are disjoint neigh-

borhoods of x and y, respectively, while if C1, C2 ⊂ X are disjoint and closed,then the open sets

U1 := x ∈ X : dist (x,C1) < dist (x,C2) ,U2 := x ∈ X : dist (x,C1) > dist (x,C2) ,

are two disjoint neighborhoods of C1 and C2, respectively.A topological space X is metrizable if its topology can be determined by

a metric.

Theorem A.17. A topological space is metrizable if and only if

(i) singletons are closed;(ii) for any closed set C ⊂ X and for any x /∈ C there exist disjoint open

neighborhoods of C and x;(iii) it has a σ-locally finite base.

A sequence xn ⊂ X converges (strongly) to x ∈ X if

limn→∞

d (xn, x0) = 0.

A Cauchy sequence in a metric space is a sequence xn ⊂ X such that

limn, m→∞

d (xn, xm) = 0.

A metric space X is said to be complete if every Cauchy sequence is conver-gent.

Theorem A.18 (Baire category theorem). Let (X, d) be a complete met-ric space. Then the intersection of a countable family of open dense sets in Xis still dense in X.


A.1.3 Topological Vector Spaces

Let X be a vector space over R and let E ⊂ X. The set E is said to be balancedif tx ∈ E for all x ∈ E and t ∈ [−1, 1]. We say that E ⊂ X is absorbing if forevery x ∈ X there exists t > 0 such that sx ∈ E for all 0 ≤ s ≤ t.

Definition A.19. Given a vector space X over R endowed with a topology τ ,the pair (X, τ) is called a topological vector space if the functions

X ×X → X,(x, y) → x+ y, and

R×X → X,(t, x) → tx,

are continuous with respect to τ .

Remark A.20. (i) In a topological vector space a set U is open if and only ifx + U is open for all x ∈ X. Hence to give a base it is enough to give alocal base at the origin.

(ii) Using the continuity of addition and scalar multiplication it is possible toshow that each neighborhood U of the origin is absorbing and it containsa neighborhood of zero W such that W +W ⊂ U and W ⊂ U , as well asa balanced neighborhood of zero.

As a corollary of Theorem A.17 we have the following:

Corollary A.21. A topological vector space is metrizable if and only if

(i) singletons are closed;(ii) it has a countable base.

Definition A.22. Let X be a topological vector space. A set E ⊂ X is saidto be topologically bounded if for each neighborhood U of 0 there exists t > 0such that E ⊂ tU .

Note that when the topology τ is generated by a metric d, sets boundedin the topological sense and in the metric sense may be different. To see this,it suffices to observe that the metric d1 := d

d+1 generates the same topologyas d, but since d1 ≤ 1, every set in X is bounded with respect to d1.

We now define Cauchy sequences in a topological vector space.

Definition A.23. Let X be a topological vector space. A sequence xn ⊂ Xis called a Cauchy sequence if for every neighborhood U of the origin thereexists an integer n ∈ N such that

xn − xk ∈ U

for all k, n ≥ n. The space X is complete if every Cauchy sequence is conver-gent.

Note that Cauchy (and hence convergent) sequences are bounded in thetopological sense.


Proposition A.24. Let X be a topological vector space and let xn ⊂ X bea Cauchy sequence. Then the set xn : n ∈ N is topologically bounded.

Proof. Let U be a neighborhood of the origin. Using the continuity of addition,construct a balanced neighborhood of the origin W such that W + W ⊂ U .Since xn ⊂ X is a Cauchy sequence there exists n ∈ N such that xn−xk ∈Wfor all k, n ≥ n. In particular, xn ∈ xn+W for all n ≥ n. Since W is absorbingwe may find t > 1 so large that xk ∈ tW ⊂ tU for all k = 1, . . . , n. Moreover,for n ≥ n,

xn ∈ xn +W ⊂ tW +W ⊂ tW + tW ⊂ tU ,

which shows that xn ∈ tU for all n ∈ N.

Topologically bounded sets play an important role in the normability oflocally convex topological vector spaces (see Theorem A.40 below):

Definition A.25. A topological vector space X is locally convex if every pointx ∈ X has a neighborhood that is convex.

Proposition A.26. A locally convex topological vector space admits a localbase at the origin consisting of balanced convex neighborhoods of zero.

Let X be a vector space over R and let E ⊂ X. The function pE : X → R

defined bypE (x) := inf t > 0 : x ∈ tE , x ∈ X,

is called the gauge or Minkowski functional of E.

Definition A.27. Let X be a vector space over R. A function p : X → R iscalled a seminorm if

p (x+ y) ≤ p (x) + p (y)

for all x, y ∈ X and p (tx) = |t| p (x) for all x ∈ X and t ∈ R.

Remark A.28. Let X be a vector space over R and let E ⊂ X. The gauge pE

of set E is a seminorm if and only if E is balanced, absorbing, and convex.

Theorem A.29. If F is a balanced, convex local base of 0 for a locally convextopological vector space X, then the family pU : U ∈ F is a family of con-tinuous seminorms. Conversely, given a family P of seminorms on a vectorspace X, the collection of all finite intersections of sets of the form

V (p, n) :=x ∈ X : p (x) <

1n

, p ∈ P, n ∈ N,

is a balanced, convex local base of 0 for a topology τ that turns X into a locallyconvex topological vector space such that each p is continuous with respect toτ .


We now give some necessary and sufficient conditions for the topology τgiven in the previous theorem to be Hausdorff and for a set to be topologicallybounded.

Corollary A.30. Let P be a family of seminorms on a vector space X andlet τ be the locally convex topology generated by P. Then

(i) τ is Hausdorff if and only if p (x) = 0 for all p ∈ P implies that x = 0;(ii) a set E ⊂ X is topologically bounded if and only if the set p (E) is bounded

in R for all p ∈ P.

We now introduce the notion of dual space.

Definition A.31. Let X and Y be two vector spaces. A map L : X → Y iscalled a linear operator if

(i) L (x+ y) = L (x) + L (y) for all x, y ∈ X;(ii) L (tx) = t L (x) for all x ∈ X and t ∈ R.

If X and Y are topological vector spaces, then the vector space of allcontinuous linear operators from X to Y is denoted by L (X;Y ) . In the specialcase Y = R, the space L (X; R) is called the dual space of X and it is denotedby X ′. The elements of X ′ are also called continuous linear functionals.

The bilinear (i.e., linear in each variable) mapping

〈·, ·〉X′,X : X ′ ×X → R, (A.1)(L, x) → L (x) ,

is called the duality pairing.The dual space L (X ′; R) of X ′ is called bidual space of X and it is denoted

by X ′′.

Definition A.32. Let X, Y be topological vector spaces. An operator L : X →Y is bounded if it sends bounded sets of X into bounded sets of Y .

Theorem A.33. Let X, Y be topological vector spaces and let L : X → Y belinear and continuous. Then L is bounded.

Proof. Let E ⊂ X be a bounded set and let W ⊂ Y be a neighborhood ofzero. Since L is continuous and L (0) = 0, there exists a neighborhood U ⊂ Xof zero such that L (U) ⊂W . By the boundedness of E there exists t > 0 suchthat E ⊂ tU . Hence

L (E) ⊂ L (tU) = tL (U) ⊂ tW .

This shows that L (E) is bounded in Y .


Exercise A.34. On the vector space C ([0, 1]) consider two topologies, thefirst, τ , given by the metric

d (f, g) =∫ 1

0

|f (x)− g (x)|1 + |f (x)− g (x)| dx, f , g ∈ C ([0, 1]) ,

and the second, τp, given by the family of seminorms pxx∈[0,1], where

px (f) := |f (x)| , f ∈ C ([0, 1]) .

Prove that

1. the identity I : (C ([0, 1]) , τp) → (C [0, 1] , τ) maps bounded sets intobounded sets;

2. the identity I : (C ([0, 1]) , τp) → (C [0, 1] , τ) is sequentially continuousbut not continuous.

3. Deduce that the topology τp is not compatible with any metric.

We now present several versions of the Hahn–Banach theorem. These areheavily used in the second part of the text, starting from Chapter 4.

Theorem A.35 (Hahn–Banach, analytic form). Let X be a vector space,let Y be a subspace of X, and let p : X → R be a convex function. Then forany linear functional L : Y → R such that

L (x) ≤ p (x) for all x ∈ Y

there exists a linear functional L1 : X → R such that

L1 (x) = L (x) for all x ∈ Y

andL (x) ≤ p (x) for all x ∈ X.

The finite-dimensional version of the next theorem may be found in Chap-ter 4.

Theorem A.36 (Hahn–Banach, first geometric form). Let X be a topo-logical vector space, and let E, F ⊂ X be nonempty disjoint convex sets.Assume that E has an interior point. Then there exist a continuous linearfunctional L : X → R, L = 0, and a number α ∈ R such that

L (x) ≥ α for all x ∈ E and L (x) ≤ α for all x ∈ F .

As a corollary of Hahn–Banach theorem one can prove the following result(see also Exercise 2.43).

Corollary A.37. Let X be a topological vector space. Then the dual X ′ of Xis not the null space if and only if X has a convex neighborhood of the originstrictly contained in X.


In view of the previous corollary it is natural to restrict attention to locallyconvex topological vector spaces.

Theorem A.38 (Hahn–Banach, second geometric form). Let X be alocally convex topological vector space, and let C, K ⊂ X be nonempty disjointconvex sets, with C closed and K compact. Then there exist a continuous linearfunctional L : X → R and two numbers α ∈ R and ε > 0 such that

L (x) ≤ α− ε for all x ∈ C and L (x) ≥ α+ ε for all x ∈ K.

A.1.4 Normed Spaces

Definition A.39. A norm on a vector space X is a map

‖·‖ : X → [0,∞)

such that

(i) ‖x+ y‖ ≤ ‖x‖+ ‖y‖ for all x, y ∈ X;(ii) ‖tx‖ = |t| ‖x‖ for all x ∈ X and t ∈ R;(iii) ‖x‖ = 0 implies x = 0.

A normed space (X, ‖·‖) is a vector space X endowed with a norm ‖·‖.For simplicity, we often say that X is a normed space.

If for every x, y ∈ X we define

d (x, y) := ‖x− y‖ ,

then (X, d) is a metric space. We say that a normed space X is a Banachspace if it is complete as a metric space.

A topological vector space is normable if its topology can be determinedby a norm.

Theorem A.40. A topological vector space X is normable if and only if it islocally convex and it has a topologically bounded neighborhood of 0.

Two norms ‖·‖1 and ‖·‖2 are equivalent if there exists a positive constantC > 0 such that

1C‖x‖1 ≤ ‖x‖2 ≤ C ‖x‖1 for all x ∈ X.

Equivalent norms induce the same topology on X.

Proposition A.41. Let X and Y be normed spaces with norms ‖·‖X and‖·‖Y , respectively.

(i) A linear operator L : X → Y is continuous if and only if

‖L‖L(X;Y ) := supx∈X\0

‖L (x)‖Y

‖x‖X

<∞;


(ii) the mapping L ∈ L (X;Y ) → ‖L‖L(X;Y ) is a norm;(iii) if Y is a Banach space then so is L (X;Y ); conversely, if X = 0 and

L (X;Y ) is a Banach space then so is Y .

As a corollary of the Hahn–Banach theorem one has the following result.

Corollary A.42. Let X be a normed space. Then for all x ∈ X,

‖x‖X = maxL∈X′, ‖L‖X′≤1

|L (x)|‖x‖X

.

Remark A.43. This corollary is especially useful for Lp spaces.

Theorem A.44 (Banach–Steinhaus). Let X, Y be Banach spaces and letLαα∈I be a family of linear continuous operators Lα : X → Y such that

supα∈I

‖Lα (x)‖Y <∞

for every x ∈ X. Thensupα∈I

‖Lα‖L(X;Y ) <∞.

Definition A.45. Let X and Y be normed spaces with norms ‖·‖X and ‖·‖Y .A continuous linear operator L ∈ L (X;Y ) is said to be compact if it mapsevery bounded subset of X onto a relatively compact subset of Y .

In particular, if L is compact then from every bounded sequence xn ⊂ Xwe may extract a subsequence xnk

such that L (xnk) converges in Y .

Definition A.46. We say that the normed space X is embedded in thenormed space Y and we write

X → Y

if X is a vector subspace of Y and the immersion

i : X → Y ,x → x,

is continuous.

Note that since the immersion is linear, in view of Proposition A.41 thecontinuity of i is equivalent to requiring the existence of a constant M > 0such that

‖x‖Y ≤M ‖x‖X for all x ∈ X.

We say that X is compactly embedded in Y if the immersion i is a compactoperator. Compact embeddings will play an important role in the study ofSobolev spaces in [FoLe10].


A.1.5 Weak Topologies

Given a locally convex topological vector space X, for each L ∈ X ′ the func-tion pL : X → [0,∞) defined by

pL (x) := |L (x)| , x ∈ X, (A.2)

is a seminorm. In view of Theorem A.29, the family of seminorms pLL∈X′

generates a locally convex topology σ (X,X ′) on the space X, called the weaktopology , such that each pL is continuous with respect to σ (X,X ′). In turn,this implies that every L ∈ X ′ is σ (X,X ′) continuous.

Theorem A.47. Let X be a locally convex topological vector space and letE ⊂ X. Then

(i) E is bounded with respect to the (strong) topology if and only if it weaklybounded;

(ii) if E is convex then E is closed if and only if it weakly closed.

Definition A.48. Given a locally convex topological vector space X, a se-quence xn ⊂ X converges weakly to x ∈ X if it converges to x with respectto the weak topology σ (X,X ′).

We write xn x. In view of Theorem A.29 and (A.2), we have the fol-lowing result.

Proposition A.49. Let X be a locally convex topological vector space. A se-quence xn ⊂ X converges weakly to x ∈ X if and only if

limn→∞

L (xn) = L (x)

for every L ∈ X ′.

Similarly, given a locally convex topological vector space X, for each x ∈ Xthe function px : X ′ → [0,∞) defined by

px (L) := |L (x)| , L ∈ X ′, (A.3)

is a seminorm. In view of Theorem A.29, the family of seminorms pxx∈X

generates a locally convex topology σ (X ′,X) on the space X ′, called the weakstar topology , such that each px is continuous with respect to σ (X ′,X).

Definition A.50. Let X be a locally convex topological vector space. A se-quence Ln ⊂ X ′ is weakly star convergent to L in X ′ if it converges to Lwith respect to the weak star topology σ (X ′,X).

We write Ln∗ L. In view of Theorem A.29 and (A.3), we have the

following result.


Proposition A.51. Let X be a locally convex topological vector space. A se-quence Ln ⊂ X ′ converges weakly star to L ∈ X ′ if and only if

limn→∞

Ln (x) = L (x)

for every x ∈ X.

Theorem A.52 (Banach–Alaoglu). If U is a neighborhood of 0 in a locallyconvex topological vector space X, then

K := L ∈ X ′ : |L (x)| ≤ 1 for every x ∈ U

is weak star compact.

Corollary A.53. If X is a normed space then the closed unit ball of X ′,

L ∈ X ′ : ‖L‖X′ ≤ 1 ,

is weak star compact.

If X is separable, it actually turns out that weak star compact sets aremetrizable, and thus one can work with the friendlier notion of sequentialcompactness.

Theorem A.54. Let X be a separable, locally convex topological vector spaceand let K ⊂ X ′ be weak star compact. Then (K,σ (X ′,X)) is metrizable.

Hence, also in view of the Banach–Alaoglu theorem, we have the following:

Corollary A.55. Let U be a neighborhood of 0 in a separable locally convextopological vector space X and let Ln ⊂ X ′ be such that

|Ln (x)| ≤ 1 for every x ∈ U and for all n ∈ N.

Then there exists a subsequence Lnk that is weakly star convergent. In par-

ticular, if X is a separable Banach space and Ln ⊂ X ′ is any boundedsequence in X ′, then there exists a subsequence that is weakly star convergent.

For Banach spaces the converse of Theorem A.54 holds:

Theorem A.56. Let X be a Banach space. Then the unit ball B (0; 1) in X ′

endowed with the weak star topology is metrizable if and only if X is separable.

Proposition A.57. Let X be a Banach space. If a sequence Ln ⊂ X ′ con-verges weakly star to L ∈ X ′, then it is bounded and

‖L‖X′ ≤ lim infn→∞

‖Ln‖X′ .

Proposition A.58. Let X be a Banach space. If X ′ is separable then so isX.


The converse is false in general (take, for example, the separable spaceL1

(R

N)

and its dual L∞ (R

N)).

We now study analogous results for the weak topology. An infinite-dimensional Banach space when endowed with the weak topology is nevermetrizable. However, we have the following:

Theorem A.59. Let X be a Banach space whose dual X ′ is separable. Thenthe unit ball B (0; 1) endowed with the weak topology is metrizable.

Definition A.60. Let X be a locally convex topological vector space. A setK ⊂ X is called sequentially weakly compact if every sequence xn ⊂ K hasa subsequence converging weakly to a point in K.

Theorem A.61. Let X be a Banach space. If K ⊂ X is weakly compact thenit is weakly sequentially compact.

Using Banach–Alaoglu’s theorem one can prove the following theorem:

Theorem A.62 (Eberlein–Smulian). Let E be a subset of a Banach spaceX. Then the weak closure of E is weakly compact if and only if for any se-quence xn ⊂ E there exists a subsequence weakly convergent to some elementof X.

As an immediate application of the Hahn–Banach theorem we have thefollowing:

Proposition A.63. Let X be a normed space and consider the linear operatormapping J : X → X ′′ defined by

J (x) (L) := L (x) , L ∈ X ′.

Then ‖J (x)‖X′′ = ‖x‖X for all x ∈ X. In particular, J is injective andcontinuous.

Definition A.64. A normed space X is reflexive if J (X) = X ′′.

In this case it is possible to identify X with its bidual X ′′.

Theorem A.65 (Kakutani). A Banach space is reflexive if and only if theclosed unit ball x ∈ X : ‖x‖ ≤ 1 is weakly compact.

In view of the previous theorem and the Eberlein–Smulian theorem wehave the following corollary:

Corollary A.66. Let X be a reflexive Banach space and let xn ⊂ X be abounded sequence. Then there exists a subsequence that is weakly convergent.

Proposition A.67. A normed space X is reflexive if and only if X ′ is reflex-ive.


The following proposition is used throughout the text, sometimes withoutmention.

Proposition A.68. Let X be a Banach space. If a sequence xn ⊂ X con-verges weakly tox ∈ X, then it is bounded and

‖x‖ ≤ lim infn→∞

‖xn‖ .

An important family of reflexive Banach spaces that includes Lp(R

N),

1 < p <∞, is given by uniformly convex Banach spaces.

Definition A.69. A normed space X is uniformly convex if for every ε > 0there exists δ > 0 such that for every x, y ∈ X, with ‖x‖ ≤ 1, ‖y‖ ≤ 1, and‖x− y‖ > ε, ∥∥∥∥x+ y

2

∥∥∥∥ < 1− δ.

Theorem A.70. Let X be a uniformly convex Banach space. Then

(i) (Milman) X is reflexive;(ii) if xn ⊂ X converges weakly to x ∈ X and

lim supn→∞

‖xn‖ ≤ ‖x‖ ,

then xn converges strongly to x.

A.1.6 Dual Pairs

In Chapter 6 we will use the notion of dual pairs.

Definition A.71. Given two vector spaces X, Y , a duality pairing betweenX and Y is a bilinear map

〈·, ·〉X,Y : X × Y → R

with the following properties:

(i) for all x ∈ X \ 0 there exists y ∈ Y \ 0 such that 〈x, y〉X,Y = 0;(ii) for all y ∈ Y \ 0 there exists x ∈ X \ 0 such that 〈x, y〉X,Y = 0.

The triple(X,Y, 〈·, ·〉X,Y

)is called a dual pair. For simplicity we will

refer to (X,Y ) as a dual pair.

Given a dual pair it is always possible to endow X and Y with locallyconvex topologies as follows: for every y ∈ Y consider the seminorm

py (x) :=∣∣∣〈x, y〉X,Y

∣∣∣ , x ∈ X.


By Theorem A.29, the family of seminorms pyy∈Y generates a locally con-vex topology on X, denoted by σ (X,Y ). Note that a sequence xn ⊂ Xconverges to some element x ∈ X with respect to the topology σ (X,Y ) if andonly if

limn→∞

〈xn, y〉X,Y = 〈x, y〉X,Y

for every y ∈ Y . Similarly, for every x ∈ X consider the seminorm

px (y) :=∣∣∣〈x, y〉X,Y

∣∣∣ , y ∈ Y .

Then again by Theorem A.29 the family of seminorms pxx∈X generates alocally convex topology on Y , denoted by σ (Y,X), and a sequence yn ⊂ Yconverges to some element y ∈ Y with respect to the topology σ (Y,X) if andonly if

limn→∞

〈x, yn〉X,Y = 〈x, y〉X,Y

for all x ∈ X. The importance of duality pairs is given by the followingtheorem.

Theorem A.72. Let(X,Y, 〈·, ·〉X,Y

)be a dual pair. Then the topological dual

of (X,σ (X,Y )) is Y , that is, for every functional L : X → R linear andcontinuous with respect to σ (X,Y ) there exists a unique y ∈ Y such that

L (x) = 〈x, y〉X,Y

for all x ∈ X. Similarly, the topological dual of (Y, σ (Y,X)) is X.

The proof of the previous theorem is hinged on the following auxiliaryresult.

Lemma A.73. Let L,L1, . . . , Ln : X → R be linear functionals on a vectorspace X. Then there exist scalars t1, . . . , tn ∈ R such that

L =n∑

i=1

tiLi

if and only ifn⋂

i=1

kerLi ⊂ kerL.

Proof. If L =∑n

i=1 tiLi for some t1, . . . , tn ∈ R and if x ∈⋂n

i=1 kerLi, then

L (x) =n∑

i=1

tiLi (x) = 0,

so that x ∈ kerL. Thus⋂n

i=1 kerLi ⊂ kerL.


Conversely, assume that⋂n

i=1 kerLi ⊂ kerL and define the linear operatorT : X → R

n by

T (x) := (L1 (x) , . . . , Ln (x)) , x ∈ X.

On the range of T define the linear functional Φ : T (X) → R by

Φ (L1 (x) , . . . , Ln (x)) := L (x) .

Note that Φ is well-defined since⋂n

i=1 kerLi ⊂ kerL. Extend Φ to all of Rn.

Then there exist scalars t1, . . . , tn ∈ R such that

Φ (y1, . . . , yn) = t1y1 + . . .+ tnyn

for all y ∈ Rn. In particular, if y ∈ T (X) then we may find x ∈ X such that

y = T (x), so that

L (x) = Φ (L1 (x) , . . . , Ln (x)) = t1L1 (x) + . . .+ tnLn (x) .

This shows that L =n∑

i=1

tiLi.

We now turn to the proof of Theorem A.72.

Proof (Theorem A.72). Since L is continuous at zero, given ε = 1, by TheoremA.29 there exist k ∈ N and y1, . . . , yn ∈ Y such that for all x ∈ X withpyi

(x) :=∣∣∣〈x, yi〉X,Y

∣∣∣ ≤ 1k for all i = 1, . . ., n we have |L (x)| ≤ 1. Define the

linear functionals L1, . . . , Ln : X → R by

Li (x) := 〈x, yi〉X,Y , x ∈ X,

i = 1, . . . , n. If x ∈⋂n

i=1 kerLi and t > 0, then pyi(tx) = 0 ≤ 1

k for alli = 1, . . . , n, and so

t |L (x)| ≤ 1

for all t > 0, which implies that L (x) = 0. Thus⋂n

i=1 kerLi ⊂ kerL, and so bythe previous lemma there exist scalars t1, . . . , tn ∈ R such that L =

∑ni=1 tiLi.

Define

y :=n∑

i=1

tiyi.

Then

L (x) =n∑

i=1

tiLi (y) =n∑

i=1

ti 〈x, yi〉X,Y =

⟨x,

n∑i=1

tiyi

⟩

X,Y

= 〈x, y〉X,Y

for all x ∈ X.To prove uniqueness assume that there exists y1 = y such that

L (x) = 〈x, y1〉X,Y

for all x ∈ X. Then 〈x, y − y1〉X,Y = 0 for all x ∈ X, which contradicts thedefinition of duality.


The previous theorem helps to explain why the topologies σ (X,Y ) andσ (Y,X) are still referred to as the weak and weak star topologies.

Example A.74. (i) The most important example of a dual pair is given ofcourse by taking a normed space X and by taking as Y its dual, with(A.1) as duality pairing.

(ii) Given an open set Ω ⊂ RN with the Lebesgue measure as underlying

measure, in Section 6.4.5 we will also use the duality pair(L1 (Ω; Rm) , C0 (Ω; Rm)

)under the duality

〈u, φ〉L1(Ω;Rm),C0(Ω;Rm) :=∫

Ω

u · φdx

for u ∈ L1 (Ω; Rm) and C0 (Ω; Rm), as well as

(M (Ω; Rm) , C0 (Ω; Rm))

under the duality

〈λ, φ〉M(Ω;Rm),C0(Ω;Rm) :=∫

Ω

φdλ

for λ ∈M (Ω; Rm) and C0 (Ω; Rm).

A.1.7 Hilbert Spaces

Definition A.75. An inner product on a vector space X is a map

(·, ·) : X ×X → R

such that

(i) (x, y) = (y, x) for all x, y ∈ X;(ii) (sx+ ty, z) = s (x, z) + t (y, z) for all x, y, z ∈ X and s, t ∈ R;(iii) (x, x) ≥ 0 for every x ∈ X, (x, x) = 0 if and only if x = 0.

If for every x ∈ X we define

‖x‖ :=√

(x, x),

then X becomes a normed space. We say that a normed space X is a Hilbertspace if it is a Banach space.

Theorem A.76. Let (X, ‖·‖) be a normed space. Then there exists an innerproduct (·, ·) : X ×X → R such that ‖x‖ =

√(x, x) for all x ∈ X if and only

if ‖·‖ satisfies the parallelogram law

‖x+ y‖2 + ‖x− y‖2 = 2 ‖x‖2 + 2 ‖y‖2

for all x, y ∈ X.

A.2 Wellorderings, Ordinals, and Cardinals 567

Remark A.77. If ‖·‖ satisfies the parallelogram law, then the inner product inthe previous theorem is defined as

(x, y) :=14

[‖x+ y‖2 − ‖x− y‖2

]

for all x, y ∈ X.

A.2 Wellorderings, Ordinals, and Cardinals

(by Ernest Schimmerling)We give an overview of some basic results from set theory and we state spe-

cific theorems that are used in the text, namely Propositions A.82 and A.84.Consider an arbitrary set X and a binary relation ≺ on X. We call (X,≺)

a linear ordering if

(i) ≺ is transitive on X:

for all x, y, z ∈ X, if x ≺ y and y ≺ z then x ≺ z;

(ii) trichotomy holds:

for all x, y ∈ X we have that x ≺ y or y ≺ x or x = y;

(iii) ≺ is irreflexive:

for all x ∈ X the property x ≺ x does not hold.

Some authors call this a strict linear ordering to distinguish it from theassociated relation , whereby

for all x, y ∈ X we have that x y if and only if x ≺ y or x = y.

We say that (X,≺) is a wellordering if it is a linear ordering and ≺ is well-founded on X:

for every E ⊂ X with E = ∅ there exists x ∈ Esuch that x y for all y ∈ E.

The element x is called the ≺-least element of S. The property that there isno infinite descending sequence

. . . ≺ xn ≺ . . . ≺ x1 ≺ x0

is equivalent to wellfoundedness. The usual proofs by induction and recursivedefinitions may be extended to wellorderings in the following way.


Proposition A.78 (Proofs by induction). Let (X,≺) be a wellordering.Let P (x) be a statement about a variable x. Suppose that for all y ∈ X,

if P (x) holds for all x ≺ y then P (y) holds.

Then P (y) holds for all y ∈ X.

Proposition A.79 (Recursive definitions). Let (X,≺) be a wellordering.Let F : X×G → Y be a function such that Y is a set and G is the family of allpartial functions from X to Y 2. Then there is a unique function G : X → Ysuch that

G(y) = F (y, G x ∈ X : x ≺ y))for all y ∈ X.

The function F : X × G → Y in Proposition A.79 tells us how to defineG(y) based only on the knowledge of y and of the function x → G(x) forx ≺ y.

Define X to be a transitive set if every element of X is also a subset of X,that is,

for all y ∈ X and for all x ∈ y we have x ∈ X.

We say that α is an ordinal if α is a transitive set and (α,∈) is a wellordering.It is a fundamental assumption of mathematics that ∈ is wellfounded; this iscalled the foundation axiom in set theory. Therefore, α is an ordinal if andonly if α is a transitive set and (α,∈) is a linear ordering. Starting from ∅,we use the operation α+ 1 := α ∪ α at successor stages and take unions atlimit stages to generate all the ordinals beginning with the natural numbers0 := ∅, 1 := 0, 2 := 0, 1, 3 := 0, 1, 2, etc. The next ordinal after all thenatural numbers is the set of nonnegative natural numbers3,

ω := 0, 1, 2, . . . .

After ω come

ω + 1 := 0, 1, 2, . . . , ω,ω + 2 := 0, 1, 2, . . . , ω, ω + 1,ω + 3 := 0, 1, 2, . . . , ω, ω + 1, ω + 2,

etc., followed by

ω + ω := 0, 1, 2, . . . , ω, ω + 1, ω + 2, . . . ,ω + ω + 1 := 0, 1, 2, . . . , ω, ω + 1, ω + 2, . . . , ω + ω,

2 We say that g is a partial function on X if g is a function with domain containedin X.

3 In the text ω is denoted by N0.


etc. Notice that ∈ coincides with the usual ordering < on the natural num-bers. For this reason, when it comes to ordinals, we often write α < β insteadof α ∈ β. It is also worth mentioning that addition, multiplication, and ex-ponentiation on ω lift to operations on the class of ordinals. We do not givethe details, but we remark that care is needed in ordinal arithmetic, since, forexample, 1 +ω = ω = ω+ 1. The relationship between ordinals and arbitrarywellorderings is summarized by the following result.

Proposition A.80. Let (X,≺) be a wellordering. Then there is a unique or-dinal α and a unique order isomorphism (X,≺) ! (α,∈).

Another fundamental assumption of mathematics is that for every set X,there exists a binary relation ≺ on X such that (X,≺) is a wellordering.This is the axiom of choice (AC) in set theory. By Proposition A.80, AC isequivalent to the statement that for every set X, there exists an ordinal αand a bijection f : α → X. In plain language, this says that we can list theelements of X as

f(0), f(1), . . . , f(ω), f(ω + 1), . . . ,

where α is the length of the list.The power set of X, written P(X), is defined to be the set of all subsets

of X. In other words, P(X) := Y : Y ⊂ X.

Proposition A.81 (Cantor). For all X, there is no surjection from X toP(X).

We call a set X countable if either X is finite or X is in one-to-one cor-respondence with ω. It is a corollary to Proposition A.81 that P(ω) is un-countable and that R is uncountable since R and P(ω) are in one-to-onecorrespondence. From this and what we said about Proposition A.80 and AC,it follows that there are uncountable ordinals, that is, infinite ordinals thatare not in one-to-one correspondence with ω. The least uncountable ordinal iscalled ω1. Note that the ordinals ω+1, ω+ω, ω ·ω, ωω, etc., are all countableand hence strictly less than ω1.

The following versions of Propositions A.78 and A.79 for ω1 are used inthe text. They are easy consequences of the general theory we have outlined.

Proposition A.82 (Proofs by induction). Let P (α) be a statement abouta variable α. Suppose that for all β < ω1,

if P (α) holds for all α < β then P (β) holds.

Then P (β) holds for all β < ω1.

Remark A.83. If α and β are ordinals and β = α + 1, then β is called asuccessor ordinal. Nonzero ordinals that are not successor ordinals are calledlimit ordinals. Often, in applications, the verification of the hypothesis ofProposition A.82 is broken up into three cases: β = 0, β = α + 1, and β alimit ordinal.


Proposition A.84 (Recursive definitions). Let F : ω1 × G → Y be afunction, where Y is a set and G is the family of all partial functions from ω1

to Y . Then there is a unique function G : ω1 → Y such that

G(β) = F (β, Gβ)

for all β < ω1.

Remark A.85. Often, in applications of Proposition A.84, the definition ofF (β, ·) is broken up into three cases: β = 0, β = α+ 1, and β a limit ordinal.

Given a set X, the least ordinal in one-to-one correspondence with X iscalled the cardinality of X, which is written cardX. An ordinal α is a cardinalif and only if cardα = α. Clearly, ω is a cardinal because it is the least infiniteordinal. When we think of ω as a cardinal, we may call it ℵ0. It is also clearthat ω1 is a cardinal because it is the least uncountable ordinal. When wethink of ω1 as a cardinal, we may call it ℵ1. In general, if α is an ordinal,then ℵα := ωα is the αth infinite cardinal. The distinction between ωα andℵα is nonmathematical (they are equal), but it is useful nonetheless becausecardinal arithmetic is different from ordinal arithmetic. A few examples sufficeto give the flavor of cardinal arithmetic, which does not play an explicit rolein the text:

ℵ0 = cardω = card (ω + 1) = card (ω + ω) = card (ω · ω) = cardωω

andℵ1 = cardω1 = card (ω1 + ω1) = card (ω1 · ω1) = cardωω1

1 .

In the examples above, we mean ordinal arithmetic inside card (·). Cardinalexponentiation is defined so that

2card X = cardf : f : X → 0, 1 = cardP(X).

To see that this makes sense consider the case in which X is a natural number:there are 2n subsets of n = 0, . . . , n−1. Cardinalities of sets that are familiarto analysts include:

ℵ0 = card N = card Z = card Q,

2ℵ0 = card R

= card C

= cardB ⊂ R : B is a Borel set= cardf : f : N −→ R= cardf : R → R : f continuous,

and


22ℵ0 = cardP(R) = cardf : f : R → R.From Proposition A.81, we know that 2ℵ0 ≥ ℵ1. However, by deep results ofKurt Godel and Paul Cohen, the value of α such that 2ℵ0 = ℵα cannot bedetermined from the underlying assumptions about mathematics used in thistextbook except in the unlikely event that these underlying assumptions areinconsistent. Collectively, these underlying assumptions are called Zermelo–Fraenkel set theory with the axiom of choice or ZFC. For more on this, see[Ku83] or [DW87].

B

Notes and Open Problems

Unfortunately what is littlerecognized is that the mostworthwhile scientific books arethose in which the author clearlyindicates what he does not know;for an author most hurts hisreaders by concealing difficulties.

Evariste Galois

Chapter 1

For more information on abstract measure theory and for the proofs omittedin this chapter we refer to [AliBo99], [AmFuP00], [DB02], [DuSc88], [Ed95],[EvGa92], [Fe69], [Fol99], [GiMoSo98], [Rao04] [Ru87]; [Z67]. The readershould be warned that in several of these books (e.g., [DuSc88], [EvGa92],[Fe69]) outer measures are called measures.

Section 1.1

Section 1.1.1: For more information on measures with the finite subset prop-erty we refer to [Rao04] and [Z67]. Theorem 1.12 is due to Hewitt and Yoshida[He-Yo52]. The present proof is due to Heider [Hei58]. Exercise 1.18 is takenfrom [Fol99]. In the literature there are different definitions of atoms (seeRemark 1.19). The proof of Proposition 1.20 has been taken from a paperof Farkas [Fa03]. Proposition 1.22 is due to Johnson [J70] (although with adifferent definition of atoms).Section 1.1.2: The standard proof of Proposition 1.52 is based on the Rieszrepresentation theorem in C0 (see [Fol99], [Ru87]). The approach followed hereis from [Coh93]. Exercises 1.58 and 1.61 are taken from [Fol99] and [AliBo99],respectively. The De Giorgi–Letta theorem in this version may be found in

574 B Notes and Open Problems

[AmFuP00], [BrDF98], and [Bu89]. Theorem 1.64 is due to Fonseca and Maly[FoMy97].Section 1.1.3: The proof of Proposition 1.87 is adapted from a paper ofButtazzo and Dal Maso [BuDM83]. Corollary 1.90 is due to Alberti [Al93],while Proposition 1.91 is taken from a paper of Halmos [Hal48].Section 1.1.4: The proof of Step 3 in the Radon–Nikodym theorem is takenfrom [Rao04], to which we refer for further extensions. Lemmas 1.102, 1.103,1.113 and Theorem 1.114 are due to De Giorgi (see [Mo96]). The proofs pre-sented here have been obtained in collaboration with Massimiliano Morini.

Proposition 1.110 and Theorem 1.111 are taken from [Rao04], to which werefer for more information on localizable measures. Theorem 1.118 is due toMaynard [May79].Section 1.1.5: Proposition 1.128 is due to Luther [Lu67], while Theorem1.130 is due to Mukherjea [Muk73].

♠ Find necessary and sufficient conditions for the validity of Fubini’s andTonelli’s theorems. See [Muk72], [Muk73] for some partial results.

Section 1.1.6: Theorem 1.133 is taken from [Rog98], while the remainingproofs are taken from a paper of Leese [Lee78].

Section 1.2

Section 1.2.1: The Morse covering theorem was originally proved by Morse in[Mor47] (see also the Errata in the paper of Bledsoe and Morse [BleMor52]).The current proof is a modified version of the one presented in a paper ofBliedtner and Loeb [Bli-Lo92]; in particular, it avoids transfinite induction.We thank Danut Arama for useful discussions on this part.Section 1.2.2: The proof of the Besicovitch derivation theorem is an adap-tation of the one due to Ambrosio and Dal Maso [AmDM92] (see also[AmFuP00]).

Section 1.3

Sections 1.3.1–1.3.3: Some of the material in these sections is taken from[AliBo99], [DB02], and [DuSc88].Section 1.3.4: Some of the material in this section is taken from [AmFuP00]and [Bi99]. We thank Gordan Zitkovic for useful conversations on this partand for suggesting Exercise 1.211.

Chapter 2

For more information on Lp spaces and for the proofs omitted in this chapterwe refer to [AmFuP00], [Bar95], [DB02], [DuSc88], [Ed95], [EvGa92], [Fol99],[Ru87], [Z67].

Section 2.1

B Notes and Open Problems 575

Section 2.1.1: Theorem 2.5 is due to Subramanian [S78] (see also the papersof Romero [R83] and of Villani [Vi85]). We thank Peter Lumsdaine for a cleanproof of Step 3 of Theorem 2.16.Section 2.1.2: The decomposition lemma was originally proved by Kadec’and Pelcin’ski [KP65]. The present proof may be found in a paper of Delbaenand Schachermayer [DS99] (see also the paper of Fonseca, Muller, and Pedre-gal [FoMuPe98]). We thank Gordan Zitkovic for bringing the reference [DS99]to our attention.Section 2.1.3: The proof of Theorem 2.34 is due to W. Rudin (see [Le56]).Theorem 2.35 is due to Leach [Le56]. The Riesz representation theorem in L1

is due to Schwartz [Sc51], while the proof of Corollary 2.41 may be found in[Rao04] and [Z67]. Exercise 2.43 is taken from a paper of Farkas [Fa03]. TheRiesz representation theorem in L∞ is due to Hewitt and Yoshida [He-Yo52](see also [H94] and [Rao04]).Section 2.1.4: The proof of the Dunford–Pettis theorem in this generality isdue to Ambrosio, Fusco, and Pallara [AmFuP00] except for Step 4.

Theorem 2.59 is due to Dal Maso (see [Am89]).Section 2.1.5: For alternative proofs of the biting lemma and for historicalbackground we refer to [BaMu89], [GiMoSo98], [Pe97]. Exercises 2.64 and 2.69are taken from a paper of Ball and Murat [BaMu89]. Proposition 2.71 maybe found in a paper of Muller [Mu90].

♠ Find an appropriate version of the biting lemma in the case in which themeasure µ is infinite.

Section 2.2

Section 2.2.2: The proof of Theorem 2.88 is taken from a paper of Serrin[Ser62].Section 2.2.3: The material in this subsection is taken from [Ste70], [Ste93].

Section 2.3

With the exception of the proof of the Riesz representation theorem, all thematerial in this section is taken from [DuSc88], [DieU77], [Ed95], and [SY05].

Chapter 3

For more information on the material of this chapter we refer to [BrDF98],[Br02], [Bu89], [CaDA02], [DM93], [GiMoSo98]. The reader should be warnedthat we use here a different definition of coercivity.

Chapter 4

For more information on the material of this chapter we refer to [AliBo99],[CaDA02], [CasVa77], [Clar90], [Dac89], [EkTe99], [Roc97], [RocWe98].

Section 4.1


The proof of Proposition 4.7 is taken from [Bo].

Section 4.2

The proofs in this section are taken from [Bo].

Section 4.4

Remark 4.31 may be found in a paper of Artola and Tartar [ArtTa95]. Theo-rem 4.32 in this generality is due to C. Pucci (see [DB02]); see also the papersof McShane [Mc34] and Kirszbraum [Kir34] for the Lipschitz case.

Section 4.5

Theorem 4.36 is well known. The present proof, which provides an explicit con-stant, may be found in a paper of Ball, Kirchheim, and Kristensen [BaKiKr00].Theorem 4.56 is taken from a paper of Bauschke, Borwein, and Combettes[BBC01]. Proposition 4.64 was first proved by Marcellini [Mar85].

Section 4.6

The material of this section is taken from [Roc97], [RocWe98].

Section 4.7

Theorem 4.79 is due to De Giorgi [DG68](see also the paper of Gori andMarcellini [GorMar02]). We thank Virginia De Cicco for bringing the reference[DG68] to our attention

♠ Theorem 4.79 provides an explicit formula for the affine functions approx-imating a given convex function f . Does an analogous formula hold forconvex functions f : R

m → (−∞,∞]?

Section 4.8

Exercise 4.94, as well as Theorem 4.98 and Proposition 4.100 in Section 4.8, isdue to Carbone and De Arcangelis [CaDA99] (see also [CaDA02]). Proposition4.102 is due to Ambrosio [Am87]. The proof of Theorem 4.103 presented hereis due to Kirchheim and Kristensen [KiKr01].

Section 4.9

Theorem 4.107 may be found in a paper of Toranzos [T67].

Chapter 5Section 5.1

Theorems 5.1 and 5.6 have been adapted from a paper of Marcus and Mizel[MarMi79]. We thank an anonymous referee for significant simplifications ofthe original proof and for Lemma 5.2.


Section 5.2

Section 5.2.2: The blowup method was introduced in a paper of Fonseca andMuller [FoMu92] (see also the paper of Fonseca and Maly [FoMy97]). The firstpart of the statement of Proposition 5.16 may be found in two papers due toIoffe [Io77], [Io77a], while inequality (5.20) is due to Giuseppe Savare. Thenecessity part of Theorem 5.17 seems to be new.Section 5.2.3: Theorem 5.27 and the necessity part of Theorems 5.19, 5.25seem to be new.

♠ Prove a suitable version of Theorem 5.27 when E is unbounded. Note thatin this case the dual of Cb

(E; Rm

)is the space of regular finitely additive

measures rba(E; Rm

), where it is known that Radon–Nikodym theorem

fails.

Section 5.3

The proof of Theorem 5.29 is taken from a paper of Buttazzo and Dal Maso[BuDM83] (see also [Bu89]).

Section 5.4

Section 5.4.1: The proof of the second part of Theorem 5.32 has beenadapted from the work of Carbone and Dearcangelis [CaDA96], [CaDA99](see also [CaDA02]). Exercise 5.35 may be found in [CaDA96] and [CaDA02].Section 5.4.2: The proof of Theorem 5.36 draws upon a paper of Goffmanand Serrin [GoSer64] (see also the paper of Serrin [Ser61]).

♠ Is the function h in Theorem 5.36 the recession function of the functiong?

Section 5.5

Theorem 5.38 is due to Friesecke [Fri94], while Theorem 5.40 seems to be new.


For more information on multifunctions we refer to [AliBo99], [CasVa77].Section 6.1.1: The proof of Theorem 6.9 is taken from a paper of Leese[Lee78], while Theorem 6.13 may be found in a paper of Valadier [Va71].Section 6.1.2: The material of this subsection is due to Michael [Mi56],except for Theorem 6.18, which is taken from a paper of Fathi [F97].

Section 6.2

For more information on integrands we refer to [Bu89], [EkTe99], and [Roc76].Section 6.2.1: Proposition 6.24 is due to Buttazzo and Dal Maso [BuDM83].Section 6.2.2: The material of this subsection is taken from [Bu89], [EkTe99],and [Roc76].


Section 6.2.3: Theorem 6.36 is due to De Giorgi (see [GorMar02]). Theorem6.40 was proved in a paper of Dal Maso and Sbordone [DMSb95]. Proposition6.42 is due to Ambrosio [Am87] (see also the paper of Fonseca and Leoni[FoLe00]). Proposition 6.43 may be found in [Roc76].

♠ Under the hypotheses of Proposition 6.42 one can prove the existence oftwo sequences of continuous functions

ai : E → R, bi : E → Rm,

such thatf(x, z) = sup

i∈N

ai(x) + bi(x) · z

for all x ∈ E and z ∈ Rm. On the other hand, if we assume that f is also

real-valued, then De Giorgi’s theorem allows one to give an alternativeapproximation in which the functions ai and bi are now given by theexplicit formula (6.26). Under the hypotheses of Proposition 6.42 and whenf is real-valued, are the functions (6.26) continuous?

Section 6.3

Subsection 6.3.1: Theorem 6.45 is due to Alberti [Al93].

♠ Prove the analogue of Theorem 6.45 for sets E with infinite measure. Whathappens if the Lebesgue measure is replaced by an arbitrary measure?

♠ Prove the analogue of Exercises 5.5 and 5.8 for integrands f = f (x, z).

Section 6.4

♠ Throughout this section the underlying measure is the Lebesgue measure.Most of the proofs can be carried out for non atomic Radon measures. Butwhat happens when the underlying measure has atoms? See the papers ofBouchitte and Buttazzo [BouBu90] and [BouBu92] for some results in thisdirection.

Section 6.4.1: Theorem 6.49 is due to Alberti [Al93].Section 6.4.3: Theorem 6.54 is due to Ioffe [Io77], [Io77a]. The sufficiencypart of the proof is adapted from a paper of Fonseca and Muller [FoMu92](see also the paper the paper of Fonseca and Maly [FoMy97]). Step 1 of thenecessity part of the proof is taken from [Bu89]. We thank Giuseppe Savarefor simplifying the proof of Step 2.Section 6.4.5: The material of this subsection is taken from a paper of Bou-chitte and Valadier [BouVa88] (see also the paper of Ambrosio and Buttazzo[AmBu88] and [Bu89]).

♠ Condition (6.95) in Theorem 6.57 is not necessary. Find a necessary andsufficient growth condition from below. That is, find the analogue of thegrowth conditions in Theorems 5.19 and 5.25 in Section 5.2.3.


♠ Without the growth condition (6.95) is (6.96) still valid? If not, what isthe right formula?

♠ Characterize the class of integrands f for which condition (6.96) holds.♠ Prove the analogue of Theorem 5.27 for integrands f = f (x, z).

Section 6.5

Theorem 6.65 and Exercise 6.66 are taken from a paper of Buttazzo and DalMaso [BuDM83].

Section 6.6

Section 6.6.1: The second part of Theorem 6.68 has been adapted from thework of Carbone and Dearcangelis [CaDA96], [CaDA99] (see also [CaDA02]).

♠ What happens in Theorem 6.68 when the growth condition (6.145) is re-placed by the growth condition (6.49)?

Section 6.6.2: Theorem 6.70 is due to Bouchitte and Valadier [BouVa88](see also the paper of Ambrosio and Buttazzo [AmBu88] and [Bu89]).

♠ If the coercivity condition (6.150) is replaced by the hypothesis that f ≥ 0what is the relation between the integrands h and f in Remark 6.71(i)?Does h still coincide with the function f defined in (6.151)?

♠ More generally, what happens in Theorem 6.70 when the growth condition(6.150) is replaced by the growth condition (6.49) for p = 1?

♠ What is the analogue of Theorem 5.38 for integrands f = f (x, z)?


Section 7.3.2: Theorem 7.5 and Lemma 7.7 (except for Step 4) are due toIoffe [Io77].

♠ Find a simpler proof of Lemma 7.7.♠ What is the analogue of Theorem 6.57 for integrands f = f (x, u, z)?

Section 7.4

Proposition 6.44 is taken from [Bu89], while Exercise 7.14 is from a paper ofMarcellini and Sbordone [MarSb80].

♠ What is the analogue of Theorem 6.70 for integrands f = f (x, u, z)?♠ What is the analogue of Theorem 5.38 for integrands f = f (x, u, z)?

Chapter 8

For more information on Young measures we refer to the monographs [Mu99],[Pe97], [Ta79], [Y69], as well as to the papers of Ball [Ba89], Berliocchi andLasry [BerLa73], and Tartar [Ta95].


Appendix

Basic references for the appendix are [AliBo99], [Bre83], [DuSc88], [Ru91],[Yo95].

Notation and List of Symbols

• N = the set of positive integers, N0 = N ∪ 0; R = real line; R :=[−∞,∞] = the extended real line; if x ∈ R, then x+ := max x, 0,x− := max −x, 0, |x| := x+ + x−, x = the integer part of x.

• RN = the N -dimensional Euclidean space, N ≥ 1; for x = (x1, . . . , xN ) ∈

RN ,

|x| :=√

(x1)2 + . . .+ (xN )2;

Ω = open set of RN (not necessarily bounded); BN (x, r) (or simply

B (x0, r) whenever the underlying space is clear) = open ball in RN of

center x0 and radius r;

QN (x0, r) := x0 +(−r

2,r

2

)N

(or simply Q (x0, r) whenever the underlying space is clear); SN−1 = unitsphere in R

N ; for a multi-index α = (α1, . . . , αN ) ∈ (N0)N ,

∂α

∂xα:=

∂|α|

∂xα11 . . . ∂xαN

N

, |α| := α1 + . . .+ αN ;

Cm (Ω) = the space of all functions that are continuous together withtheir partial derivatives up to order m ∈ N0;

C∞ (Ω) :=∞⋂

m=0Cm (Ω) ;

Cmc (Ω) and C∞

c (Ω) = the subspaces of Cm (Ω) and C∞ (Ω), respectively,consisting of all functions with compact support.

• X, Y usually denote sets or spaces, cardX = the cardinality of a set X,P (X) = the family of all subsets of a set X; F , G family of sets or offunctions;

• A, U usually denote open sets, K a compact set; B a Borel set, C a closedor convex set or an arbitrary constant; τ = topology;

582 Notation and List of Symbols

• M, N = algebras or σ-algebras; M⊗N = product σ-algebra of M and N

(not to be confused with M×N); B (X) = Borel σ-algebra;• µ, ν, υ = (positive) finitely additive measures or positive measures; ν ⊥ µ

means that µ, ν are mutually singular measures; ν µ means that themeasure ν is absolutely continuous with respect to the measure µ; dν

dµ isthe Radon–Nikodym derivative of ν with respect to µ; suppµ support ofa Borel measure; µ∗ outer measure;

• = restriction;• dist = distance; diam = diameter;• ‖·‖ norm or total variation; ‖·‖Lp norm in Lp spaces;• E, F , G usually denote sets; ∂E = boundary of E; co (E) = convex hull of

E; aff (E) = affine hull of E; riaff (E) = relative interior of E with respectto aff (E); rbaff (E) = the relative boundary of E with respect to aff (E),E∞ = the recession cone of E, kerE = kernel of a set;

• χE = characteristic function of the set E; IE = indicator function of theset E;

• u, v, and w usually denote functions or variables, z usually denotes anelement of R

m;• f , g, ϕ, ψ, φ usually denote functions; supp f = support of the function f ;

dome f = effective domain of the function f ; Lip f = Lipschitz constant ofthe function f ; ∂f (z) = subdifferential of f at z; osc (f ;E) = oscillationof f on E; f∞ = the recession function of f ; C f = the convex envelopeof f ; f∗ = the polar or conjugate function of f ; f∗∗ = the bipolar orbiconjugate function of f1; f ∗ g = convolution of the functions f and g;

• LNo = Lebesgue outer measure, LN = Lebesgue measure;

αN := LN (B (0, 1)) ;

|E| := LN (E) for E ⊂ RN Lebesgue measurable;

• M (u), MR (u) denote various types of maximal functions,• det = determinant of a matrix or a linear mapping;• 〈·, ·〉X,Y = duality pairing;• λ, ς usually denote signed measures or finitely additive signed measures;• M (X; R) = space of all (signed) finite Radon measures;• ba (X,M) = space of all bounded finitely additive signed measures;

rba (X,M) = space of all regular bounded finitely additive signed mea-sures; ba (X,M, µ) = space of all bounded finitely additive signed measuresabsolutely continuous with respect to µ;

• B (X,M) = space of all bounded measurable functions;• C (X) = space of all continuous functions; Cb (X) = space of all continuous

bounded functions; C0 (X) = space of all continuous functions that vanish

1 Unless otherwise specified, Cf (x, ·), f∗ (x, ·), f∗∗ (x, ·), and lsc f (x, ·) stand for theconvex envelope, polar, bipolar, and lower semicontinuous envelope, respectively,of f (x, ·).

Notation and List of Symbols 583

at infinity; Cc (X) = space of all continuous functions whose support iscompact;

• Lp (X,M, µ), Lp (X,µ), Lp (X) are various notations for Lp spaces; p′ =Holder conjugate exponent of p;

• spaces p := Lp (N,P (N) , counting measure);• Lp (X;Y ) = Lp space on Banach spaces; Lp

w (X;Y ′) = Lp space of weaklystar measurable functions;

• denotes weak convergence; ∗ denotes weak star convergence; b

denotesbiting convergence;

• I usually denotes an interval, a function, or a functional, epi I = epigraphof the function I; lsc I = lower semicontinuous envelope of the function I;slsc I = sequentially lower semicontinuous envelope of the function I; E ,Ep = types of relaxed energy of the function I;

• Γ = multifunction, GrΓ = graph of the multifunction.

Acknowledgments

This book grew out of a series of lectures, Ph.D. research topics courses andcourses taught at several summer schools, and for this the authors thankthe hospitality and support of Carnegie Mellon University and the Univer-sity of Pavia, Italy. Exceptional working conditions during the preparation ofthis book were offered by the Mathematisches Forschungsinstitut Oberwol-fach through the Research in Pairs program in the Spring of 2002 and by theInstitute for Advanced Study, Princeton, in Spring 2003.

Several iterations of the manuscript benefited from the input of many col-leagues and students in particular: Tom Bohman, Giuseppe Buttazzo, GianniDal Maso, Virginia De Cicco, Jan Kristensen, Giuseppe Savare, Enrico Vitali,Gordan Zitkovic and the students Danut Arama, Peter Lumsdaine, and AlexRand. The authors are profoundly indebted to Nicola Fusco and MassimilianoMorini for their careful reading and extremely insightful comments and sug-gestions that greatly improved the book, to Ernest Schimmerling for writingSection A.2, and to two anonymous referees for pointing out some errors andtypos in an earlier draft.

The authors thank the Center for Nonlinear Analysis (NSF Grants No.DMS-9803791, DMS-0405343) for its support during the preparation of thisbook. The research of I. Fonseca was partially supported by the NationalScience Foundation under Grants No. DMS-0103798, DMS-040171 and thatof G. Leoni under Grant No. DMS-0405423.

The epigraphs were taken from the Mathematical Quotations Server ofFurman University:

http://math.furman.edu/˜mwoodard/mquot.html

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Index

affine hull, 249Alexandroff compactification, 128algebra, 3atom, 10axiom of choice, 569

blowup method, 334

cardinal, 570Cauchy sequence, 553coercivity, 241conjugate exponent, 140convergence

almost uniform, 148biting, 185in measure, 148Mackey topology, 148weak, 560weak star, 560

convex combination, 247convex cone, 252

generated by, 253convex envelope, 300convex hull, 247correspondence, 380cover, 91

fine, 91fine Morse, 91Morse, 91

derivativeRadon–Nikodym, 108Radon-Nikodym, 56

diagonalization argument, 245

direct method, 245directional derivative

one-sided, 275distance, 553dual

pair, 563duality pairing, 556, 563

effective domain, 259element

least, 567embedding, 559

compact, 559epigraph, 232equi-integrability, 150essential supremum, 140

first axiom of countability, 551foundation axiom, 568function

affine continuous, 293approximately continuous, 111bipolar, 301Bochner integrable, 220Borel, 37Caratheodory, 412characteristic, 38concave, 258continuous, 550convex, 258cutoff, 552equivalent, 139essential infimum, 65essential supremum, 65

596 Index

Gateaux differentiable, 284

Holder continuous, 262

indicator, 260

integrable, 39

LN × B measurable, 401

Lebesgue integrable, 117

Lipschitz, 91

Lipschitz continuous, 262

locally integrable, 39

lower semicontinuous, 231

maximal, 208

measurable, 37, 38

polar, 301

proper, 258

Q-periodic, 199

recession, 290

selection, 380

separately convex, 261

sequentially lower semicontinuous,231

simple, 38, 218

strictly convex, 258

strongly measurable, 218

subdifferentiable, 275

upper semicontinuous, 231

weakly measurable, 218

weakly star measurable, 218

functional

additive, 354, 464

coercive, 241

local, 354, 464

locally bounded, 129

positive, 129

proper, 465

gauge, 555

immersion, 559

inner product, 566

integral

Bochner, 221

of a nonnegative function, 39

of a real function, 39, 40, 117

of a simple function, 38

integrand

equivalent in the Lp sense, 404

normal, 404

lemmabiting lemma, 184decomposition lemma, 154Fatou lemma, 42Riemann–Lebesgue lemma, 199Vitali, 209

limit inferior, 232locally finite, 552

measure, 5absolutely continuous, 55, 117Borel, 30Borel regular, 30complete, 18countably additive, 5counting, 5diffuse, 55Dirac delta, 10doubling property, 101finite, 5finite subset property, 10finitely additive, 5

signed, 120, 121, 169Lebesgue, 36localizable, 67lower variation, 116mutually singular, 55, 117nonatomic, 10probability, 5product, 77purely atomic, 13purely finitely additive, 8Radon, 30restriction, 6semifinite, 10σ-finite, 5, 117signed, 113support, 31total variation, 116upper variation, 116vectorial, 119Young, 518

generated, 521homogeneous, 533

metric, 552minimizing sequence, 246Minkowski functional, 555modulus of continuity, 263

concave, 263

Index 597

mollification, 190mollifier, 190

standard, 190multi-index, 190multifunction, 380

essential supremum, 393graph, 380lower semicontinuous, 395measurable, 380weakly measurable, 380

neighborhood, 550norm, 558

equivalent, 558

operatorbounded, 556linear, 556

compact, 559ordering

irreflexive, 567linear, 567strict linear, 567transitive, 567trichotomy, 567wellfounded, 567wellordering, 567

ordinal, 568limit, 569successor, 569

oscillation, 266outer measure, 18

Borel, 25Borel regular, 25finite, 18Lebesgue, 36Lebesgue–Stieltjes, 20metric, 22product, 77Radon, 25regular, 22restriction, 19σ-finite, 18

p-equi-integrability, 150parallelogram law, 566partition of unity, 552

locally finite, 552subordinated to a cover, 552

pointLebesgue, 110of approximate continuity, 111of density one, 111of density t, 111

polar decomposition, 119polar of a convex cone, 252

recession cone, 288relaxed energy, 365, 369, 473, 478

second axiom of countability, 551section, 78seminorm, 555sequence

Cauchy, 554convergent, 550

setabsorbing, 554affine, 247balanced, 554Borel, 4cardinality, 570closed, 549closure, 549compact, 551convex, 247convex component, 319countable, 569dense, 550essential union, 66inner regular, 25interior, 550kernel, 319Lebesgue measurable, 36Morse, 90µ∗-measurable, 20negative, 114open, 549outer regular, 25positive, 114precompact, 551regular, 25relative boundary, 249relative interior, 249relatively compact, 551sequentially weakly compact, 562σ-compact, 551

598 Index

star-shaped with respect to a set, 90,318

starshaped

radial function, 318

Suslin, 83

symmetric difference, 115

topologically bounded, 554

transitive, 568

σ-algebra, 3

Borel, 4

product, 77

σ-locally finite, 552

σ-algebra

generated by, 145

space

bidual, 556

ba(X, M, µ), 169

B (X, M), 119

B (X, M), 126

ba(X, M), 120

C∞ (Ω), 190

Cm (Ω), 190

Cc (X), 126

dual, 556

Hilbert, 566

L log L (E), 212

Lp, 139, 144

p, 140

Lploc, 145

Lpw (X; Y ′), 223

measurable, 4

separable , 145

measure, 5

metric, 553

complete, 553

normed, 558

Banach, 558

reflexive, 562

uniformly convex, 563

of Radon measures , 119

rba (X, M), 121

topological, 549

Hausdorff, 550

locally compact, 551

metrizable, 553

normable, 558

normal, 550

separable, 550

topological vector, 554complete, 554locally convex, 555

subgradient, 275Suslin

F set, 83scheme, 83set, 83

theoremAlexandroff, 122Aumann measurable selection, 391Baire category theorem, 553Banach–Steinhaus, 559Besicovitch covering theorem, 98Besicovitch derivation theorem, 103Calderon–Zygmund, 212Cantor, 569Caratheodory, 20Caratheodory, 306converse Holder inequality, 158De Giorgi, 72, 297De Giorgi–Letta, 32De la Vallee Poussin, 152Dunford–Pettis, 175Eberlein–Smulian, 562Egoroff, 149Fubini, 79fundamental

Young measures, 523Hahn decomposition theorem, 115Hahn–Banach, analytic form, 557Hahn–Banach, first geometric form,

557Hahn–Banach, second geometric

form, 558Hewitt–Yosida, 8Holder’s inequality, 141Jensen’s inequality, 298Jordan decomposition theorem, 117Kakutani, 562Lebesgue decomposition theorem, 72,

118Lebesgue differentiation theorem,

108, 216Lebesgue dominated convergence

theorem, 43Lebesgue monotone convergence

theorem, 41

Index 599

Lindelof, 551Lusin, 54Michael continuous selection theorem,

396Minkowski’s inequality, 144Morse covering theorem, 92Morse measure covering theorem, 98partition of unity, 396Pettis, 219Prohorov, 133Projection, 89Radon–Nikodym, 56, 68Riesz representation theorem

in B, 124in C, 126in C0, 127in Cb, 126in Cc, 129in L1, 164in L∞, 169in Lp, 159in Lp (X; Y ), 224

Scorza-Dragoni, 413smooth partition of unity, 194Tietze extension, 550Tonelli, 78Urysohn, 550Vitali convergence theorem, 150Vitali–Besicovitch covering theorem,

102Vitali–Caratheodory, 54Vitali–Hahn–Saks, 173Weierstrass, 236

topology, 549base, 550discrete, 549weak, 560weak star, 560

total variation norm, 120

weak-L1 inequality, 211

Yosida transform, 355, 358, 422Young’s inequality, 260

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