subdifferential of the conjugate function in general banach spaces

Top (2012) 20:328–346DOI 10.1007/s11750-011-0238-0

O R I G I NA L PA P E R

Subdifferential of the conjugate function in generalBanach spaces

Rafael Correa · Abderrahim Hantoute

Received: 17 March 2011 / Accepted: 6 October 2011 / Published online: 6 November 2011© Sociedad de Estadística e Investigación Operativa 2011

Abstract We give explicit formulas for the subdifferential set of the conjugate of notnecessarily convex functions defined on general Banach spaces. Even if such a subd-ifferential mapping takes its values in the bidual space, we show that, up to a weak∗∗closure operation, it is still described by using only elements of the initial space re-lying on the behavior of the given function at the nominal point. This is achieved bymeans of formulas using the ε-subdifferential and an appropriate enlargement of thesubdifferential of this function, revealing a useful relationship between the subdiffer-ential of the conjugate function and its part lying in the initial space.

Keywords Conjugate function · Subdifferential mapping · Banach spaces

Mathematics Subject Classification (2000) 26J25 · 49N15 · 90C25

1 Introduction

Our aim in this paper is to characterize the Fenchel subdifferential of the Legendre–Fenchel conjugate of a given function, not necessarily convex and defined on a gen-eral Banach space, by means only of primal information. This will be achieved in anumber of explicit formulas by using the ε-subdifferential together with an appro-priate enlargement of the Fenchel subdifferential of the initial function, which has

A Marco López amb mottu del seu seixanté aniversari (To Marco López, on the occasion of hissixtieth birthday).

R. Correa · A. Hantoute (�)Departamento de Ingeniería Matemática, Centro de Modelamiento Matemático (CMM), Universidadde Chile, Blanco Encalada 2120, Piso 7, Santiago, Chilee-mail: [email protected]

R. Correae-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

Subdifferential of the conjugate function in general Banach spaces 329

been introduced and investigated in Correa and Hantoute (2010a). To compensate thelack of continuity assumptions, these formulas will also include the normal cone tothe domain of the conjugate function, which describes the asymptotic behavior ofthe initial function. Hence, this analysis provides complete characterizations of thesubdifferential of the conjugate function without requiring explicit expressions of theconjugate itself or its domain. The desired formulas will allow a comprehensive un-derstanding of the behavior of the conjugate in a variety of interesting and practicalsituations which rely on the initial function and/or the underlying space.

The main feature of the present analysis is the ability to describe, up to a weak∗∗closure process, the subdifferential of the conjugate function using only elements ofthe initial space relying on the behavior of this function at the corresponding point. Ofcourse, all these questions make sense when the underlying space is not necessarilyreflexive and its dual is endowed with its norm topology. However, the case where thisBanach space and its dual form a dual topological pair has already been investigatedin Correa and Hantoute (2010a). Nevertheless, a first attempt to deal with the generalsetting of Banach spaces has given rise in Correa et al. (2011a) to expressions of thissubdifferential mapping by invoking an appropriate extension of the initial functionto the bidual space. Next, the results of Correa and Hantoute (2010a) were appliedby considering the topological dual pair formed by the dual and the bidual spacesendowed with the norm and the weak∗∗ topologies, respectively. These results havebeen used to give integration criteria which provide the coincidence of the proper lscconvex hull of not necessarily convex functions. But, regarding the characterizationof the subdifferential of the conjugate, the resulting formulas in Correa et al. (2011a)do not distinguish between the parts of this subdifferential set which do or do not liein the initial space; see Sect. 3 for more comparisons. This is why we follow in thepresent work a direct approach invoking only the behavior of the initial function. Itis also important in our analysis to characterize the subdifferential of the conjugateby only invoking the behavior of the initial function at the nominal point; that is,the point where the subdifferential of the conjugate is evaluated. Hence, these resultsmay be also useful in the convex setting and can be compared with the classical result(Rockafellar 1970b, Proposition 1), in which the subdifferential of the conjugate isdescribed by means of subgradients of the initial function lying in the predual spaceat nearest points.

The summary of the paper is as follows: after we fix below the main notation whichis used later on, we give in Sect. 2 the desired results which are stated in Theorems 3by invoking an enlargement of the Fenchel subdifferential, and Theorem 5 which usesthe ε-subdifferential. The proof of Theorem 3 is postponed to Sect. 4 at the end of thepaper. For comparative purposes, in order to show the main advantages of the presentformulas, we make in Sect. 3 a short review of some of the recent results given inCorrea and Hantoute (2010a), Correa et al. (2011a).

Throughout the paper, X is a real Banach space endowed with a norm ‖ · ‖. Thedual and bidual spaces are denoted by X∗ and X∗∗, respectively. The null vector inall these spaces is denoted by θ . With abuse of language, in view of the canonicalembedding of X in X∗∗, we identify X with a subspace of X∗∗. We shall frequentlyendow X∗ and X∗∗ with the norm and the weak∗∗ topologies, respectively. The dual-ity product in both pairs (X,X∗) and (X∗,X∗∗) is denoted by 〈·, ·〉.

330 R. Correa, A. Hantoute

Let f : X → R or (f : X∗ → R) be an extended real-valued function. We say thatf is proper if its (effective) domain

domf := {x ∈ X | f (x) < +∞}

is nonempty and f (x) > −∞ for all x ∈ X. The conjugate function of f is the weak*lsc convex function f ∗ : X∗ → R given by

f ∗(x∗) := supx∈X

{〈x, x∗〉 − f (x)}.

If ε ≥ 0 is given, the ε-subdifferential mapping of f is the multifunction ∂εf : X ⇒ X∗which assigns to x ∈ X the (possibly empty) set

∂εf (x) := {x∗ ∈ X∗ | f ∗(x∗) + f (x) ≤ 〈x∗, x〉 + ε

}

(with the convention that ∂εf (x) := ∅ if f (x) /∈ R); hence, when ε = 0, we recoverthe usual Fenchel subdifferential mapping, which we simply denote by ∂f (x). In thisway, the subdifferential of the conjugate function f ∗ is the mapping ∂f ∗ : X∗ ⇒ X∗∗given by

∂f ∗(x∗) = {x∗∗ ∈ X∗∗ | f ∗∗(x∗∗) + f ∗(x∗) ≤ 〈x∗, x∗∗〉 + ε

},

where f ∗∗ : X∗∗ → R is the conjugate of f ∗, that is, f ∗∗(x∗∗) = supx∗∈X∗{〈x∗, x∗∗〉− f ∗(x∗)}.

Finally, given subsets A, B in X (X∗ or X∗∗), we use the Minkowski sum of A

and B defined as

A + B := {a + b | a ∈ A,b ∈ B}.The normal cone to A at x is defined as

NA(x) := {x∗ ∈ X∗ | 〈x∗, y − x〉 ≤ 0 for all y ∈ A

}if x ∈ A; ∅ if x ∈ X \ A.

By coA, coneA, affA, and linA, we denote the convex, conic, affine, and linear hullsof A, respectively. By parA we denote the parallel subspace to affA; for instance,parA = affA − a for a ∈ A. We use clA, clwA and clw

∗∗A (or, indistinctly, A, A

w,

and Aw∗∗

) to respectively denote the norm, weak, and weak∗∗ closures of A. Hence,we write coA := cl(coA), cow∗∗

A := clw∗∗

(coA), etc.

2 Subdifferential of the conjugate function

We give in this section the desired formulas which express the subdifferential of theconjugate of any function defined on a given real Banach space X with norm ‖ · ‖.

To this aim, an important tool is the following enlargement of the usual Fenchelsubdifferential, introduced and investigated in Correa and Hantoute (2010a).

Definition 1 Given L ⊂ X∗ and f : X → R, a vector x∗ ∈ L is said to be a relativesubgradient of f at x ∈ X with respect to L if f ∗(x∗) ∈ R and there exists a net(xα) ⊂ X such that


lim〈xα − x, y∗〉 = 0 ∀y∗ ∈ par(L ∩ domf ∗), and

lim〈xα, x∗〉 − f (xα) = f ∗(x∗).The set of such relative subgradients, denoted by ∂r

Lf (x), is called the relative sub-differential of f at x with respect to L. If domf ∗ ⊂ L, we omit the reference to L

and simply write ∂rf (x).

Here, and throughout the paper, for x∗ ∈ X∗, we use the notation

Fτ (x∗) := {

L ⊂ X∗ τ -closed and convex | x∗ ∈ L,

f ∗|τ-ri(L∩domf ∗) is finite and τ -continuous

}, (1)

where τ -ri denotes the (topological) relative interior with respect to a given topol-ogy τ ; that is, for A ⊂ X∗, τ -ri(A) is the interior relative to affA when affA isτ -closed, and the empty set otherwise (see, e.g., Zalinescu 2002). Hence, if the inte-rior of A with respect to τ , denoted by τ -intA, is nonempty then τ -ri(A) = τ -intA.The function f ∗|A used above refers to the restriction of f ∗ to A with the conventionthat f ∗

|∅ ≡ +∞. In what follows, if M : X ⇒ X∗ (or M : X∗ ⇒ X) is a multifunction,

its inverse M−1 : X∗ ⇒ X is given by M−1(x∗) := {x ∈ X | x∗ ∈ M(x)}.Before characterizing the whole set ∂f ∗(x∗) in X∗∗, we recall formulas providing

the part of this subdifferential lying in X. The following result was given in Correaand Hantoute (2010a) for the more general setting of locally convex spaces.

Proposition 1 (Correa and Hantoute 2010a, Theorem 4) Given a function f : X →R, for any x∗ ∈ X∗, we have that

∂f ∗(x∗) ∩ X =⋂

L∈Fτ (x∗)co

{(∂r

Lf )−1(x∗) + X ∩ NL∩domf ∗(x∗)},

where τ is a topology on X∗ compatible with the duality pair (X,X∗). In particular,the following formula holds, provided that X∗ ∈ Fτ (x

∗):∂f ∗(x∗) ∩ X = co

{(∂rf )−1(x∗) + X ∩ Ndomf ∗(x∗)

}.

We also give the following extension of the above proposition established in Cor-rea et al. (2011b). We recall that IC : X∗ → R+ denotes the indicator function of asubset C ⊂ X∗; that is,

IC(x∗) := 0 if x∗ ∈ C and + ∞ otherwise.

Proposition 2 Let a function f : X → R and a τ -closed convex set C ⊂ X∗ be given,where τ is a locally convex topology on X∗ compatible with the duality pair (X,X∗).Then, for every x∗ ∈ X∗, we have that

∂(f ∗ + IC)(x∗) ∩ X =⋂

L∈Fτ (C,x∗)co

{(∂r

L∩Cf )−1(x∗) + X ∩ NL∩C∩domf ∗(x∗)},

where Fτ (C,x∗) := {L ⊂ X∗ τ -closed and convex | x∗ ∈ L, f ∗|τ- ri(L∩C∩domf ∗) is

finite and τ -continuous}. In particular, if C ∈ Fτ (C,x∗), we get

∂(f ∗ + IC)(x∗) ∩ X = co{(∂r

Cf )−1(x∗) + X ∩ NC∩domf ∗(x∗)}.


Now, we give the main result of the paper in which we characterize the whole set∂f ∗(x∗) in X∗∗. Its proof is postponed to Sect. 4 at the end of the paper.

Theorem 3 Let f : X → R be any function. Then, for every x∗ ∈ X∗, we have that

∂f ∗(x∗) =⋂

L∈Fτ (x∗)cow∗∗{

(∂rLf )−1(x∗) + X ∩ NL∩domf ∗(x∗)

},

where τ is any topology on X∗ compatible with the duality pair (X,X∗).

Remark 1 It is worth observing that the term X∩NL∩domf ∗(x∗) in the formula abovedoes not require an explicit knowledge of the domain of f ∗ nor the values of thefunction f ∗ itself. Indeed, on the one hand, such a term describes the asymptoticbehavior of the initial function f as it can be easily seen from the straightforwardrelationship (assuming that the involved functions are proper)

X ∩ NL∩domf ∗(x∗) = (∂(cof )∞

)−1(x∗),

where (cof )∞ denotes the usual recession function in the sense of convex analysis(see, e.g., Rockafellar 1970a) of the lsc convex hull of the function f , cof . In thisrespect, a general relation between the normal cone to domf ∗ and an appropriateconcept of asymptotic function of f can be found in Correa and Hantoute (2010b).On the other hand, it can be also checked that the formula above can be equivalentlywritten as

∂f ∗(x∗) =⋂

L∈Fx∗cow∗∗{

(∂rLf )−1(x∗) + X ∩ NL∩domf ∗(x∗)

},

where Fx∗ := {L ⊂ X∗ | L is a finite-dimensional subspace containing x∗}, confirm-ing that the current formulas do not depend on any explicit knowledge of the conju-gate function f ∗.

The following corollary illustrates Theorem 3.

Corollary 4 Let f : X → R be weakly lsc such that f ∗ is finite and continuous atsome point with respect to a topology on X∗ compatible with the pair (X,X∗). Then,for every x∗ ∈ X∗,

∂f ∗(x∗) = cow∗∗{(∂f )−1(x∗) + X ∩ Ndomf ∗(x∗)

}.

Proof Let τ be a topology on X∗ as stated in the theorem and fix x∗ ∈ X∗. Then,since X∗ ∈ Fτ (x

∗) and f is weakly lsc, we can easily check that ∂rf = ∂f , and, so,the inclusion “⊂” follows by applying Theorem 3. This finishes the proof since theopposite inclusion is straightforward. �

The following result gives an alternative for Theorem 3 where one uses the ε-subdifferential mapping instead of ∂r

Lf .


Theorem 5 Let f : X → R and the topology τ be as in Theorem 3. Then, for everyx∗ ∈ X∗, we have the formula

∂f ∗(x∗) =⋂

ε>0L∈Fτ (x∗)

cow∗∗{(∂εf )−1(x∗) + X ∩ NL∩domf ∗(x∗)

}.

Moreover, provided that f ∗ is finite and τ -continuous at some point, the formulaabove reduces to

∂f ∗(x∗) =⋂

ε>o

cow∗∗{(∂εf )−1(x∗) + X ∩ Ndomf ∗(x∗)

}.

Proof It suffices to prove the inclusion “⊂” of the main statement when x∗ ∈ X∗is such that ∂f ∗(x∗) �= ∅. For this aim, according to Theorem 3, we only needto show that for every given ε > 0 and L ∈ Fτ (x

∗), we have (∂rLf )−1(x∗) ⊂

cow∗∗{(∂εf )−1(x∗) + X ∩ NL∩domf ∗(x∗)}. Equivalently, it suffices to show that

σ(∂rLf )−1(x∗)(w

∗) ≤ σ(∂εf )−1(x∗)+X∩NL∩domf ∗ (x∗)(w∗) for all w∗ ∈ X∗, (2)

where σ refers to the support function with the convention that σ∅ = −∞. Indeed,if w∗ ∈ L ∩ domf ∗ − x∗, we pick x ∈ (∂r

Lf )−1(x∗) and let (xα) ⊂ X be a net suchthat limα〈xα − x, y∗〉 = 0 for all y∗ ∈ par(L ∩ domf ∗), and limα f (xα) + f ∗(x∗) −〈xα, x∗〉 = 0. Hence, we may suppose that xα ∈ (∂εf )−1(x∗), so that

〈x,w∗〉 = limα

〈xα,w∗〉 ≤ σ(∂εf )−1(x∗)(w∗) ≤ σ(∂εf )−1(x∗)+X∩NL∩domf ∗ (x∗)(w

∗).

Therefore, (2) follows by the arbitrariness of x in (∂rLf )−1(x∗). Moreover, by the

positive homogeneity of the support function it follows that (2) also holds for ev-ery w∗ ∈ cone(L ∩ domf ∗ − x∗). Now, if w∗ ∈ cone(L ∩ domf ∗ − x∗), we pickw∗

0 ∈ τ -ri(cone(L ∩ domf ∗ − x∗)) (this set being nonempty by assumption), so thatby the accessibility lemma for each λ ∈ (0,1), we have w∗

λ := λw∗ + (1 − λ)w∗0 ∈

cone(L ∩ domf ∗ − x∗). Then, invoking the convexity of the support function, fromthe paragraph above we obtain that

σ(∂rLf )−1(x∗)(w

∗λ) ≤ λσ(∂εf )−1(x∗)+X∩NL∩domf ∗ (x∗)(w

∗)+ (1 − λ)σ(∂εf )−1(x∗)+X∩NL∩domf ∗ (x∗)(w

∗0). (3)

But, writing w∗0 = γ (u∗ − x∗) for some γ ≥ 0 and u∗ ∈ L ∩ domf ∗, and observing

that (∂εf )−1(x∗) ⊂ ∂εf∗(x∗), we get

σ(∂εf )−1(x∗)+X∩NL∩domf ∗ (x∗)(w∗0) ≤ γ σ(∂εf )−1(x∗)+NL∩domf ∗ (x∗)(u

∗ − x∗)

≤ γ σ∂εf ∗(x∗)(u∗ − x∗) ≤ γ

(f ∗(u∗) − f ∗(x∗)

)

< +∞.

So, taking the limit as λ ↘ 0 in (3), in view of the lsc of the support function, we getσ(∂r

Lf )−1(x∗)(w∗) ≤ lim infλ↘0 σ(∂r

Lf )−1(x∗)(w∗λ) ≤ σ(∂εf )−1(x∗)+X∩NL∩domf ∗ (x∗)(w

∗),showing that (2) holds.

Finally, it remains to check that (2) holds when w∗ �∈ cone(L ∩ domf ∗ − x∗). In-deed, in this case, by the classical bipolar theorem there exists w ∈ X∩NL∩domf ∗(x∗)


such that 〈w∗,w〉 > 0, and so σ(∂εf )−1(x∗)+X∩NL∩domf ∗ (x∗)(w∗) = +∞. Thus, (2) triv-

ially holds. �

Remark 2 The following formula, significantly different from the one given in The-orem 5, has been established in Correa et al. (2011a, Proposition 4):

∂f ∗(x∗) =⋂

ε>0Ł∈Fτ (x∗)

cow∗∗{(∂εf )−1(x∗) + NL∩domf ∗(x∗)

}

for every x∗ ∈ X∗, where τ is any topology on X∗ compatible with the duality pair(X,X∗). Indeed, while Theorem 5 uses only the part of NL∩domf ∗(x∗) lying in X,the term NL∩domf ∗(x∗) in the last formula is a subset of X∗∗ which possibly containspoints that are not in the predual space X. In this respect, the formulas in Theorems 3and 5 agree with the classical result in convex analysis (Rockafellar 1970b, Propo-sition 1), corresponding to f being convex, where the subgradients of f ∗ at x∗ arewritten as a weak∗∗ limit of subgradients of f ∗, at nearby points of x∗, that belongto X. However, note that in Theorems 3 and 5 only the nominal point x∗ is concernedin the closure process. We refer to Sect. 3 for more comparisons.

Remark 3 If Argminf ∗∗ denotes the minimizer set of the biconjugate function f ∗∗,and f ∗ is proper, then in view of the relationship Argminf ∗∗ = ∂f ∗(θ), Theorem 5easily leads us to the following characterization of Argminf ∗∗ by means of the ε-minimizer sets ε-Argminf of f :

Argminf ∗∗ =⋂

ε>0L∈Fτ (θ)

cow∗∗{ε- Argminf + X ∩ NL∩domf ∗(θ)

},

where τ is any topology on X∗ compatible with the duality pair (X,X∗). Similarly,invoking Corollary 4, if f is weakly lsc and f ∗ is finite and τ -continuous at somepoint, then we have the following relationship which gives the characterization ofArgminf ∗∗ by means of the minimizer set Argminf of f :

Argminf ∗∗ = cow∗∗{Argminf + X ∩ Ndomf ∗(θ)

}.

We close this section by giving the finite-dimensional counterpart of Theorem 3.Namely, the following corollary has already been stated in Correa and Hantoute(2010a, Corollary 7), where a small gap appeared in the proof.

Corollary 6 (Correa and Hantoute 2010a, Corollary 7) Let f : Rn → R be such that

int(domf ∗) �= ∅. Then, for every x∗ ∈ X∗, we have the formula

∂f ∗(x∗) = co{(∂rf )−1(x∗)

} + Ndomf ∗(x∗).

In addition, if f is lsc, then

∂f ∗(x∗) = co{(∂f )−1(x∗)

} + Ndomf ∗(x∗).

Proof We shall denote by Bγ (z) (Bγ if z = 0) the ball of radius γ > 0 centered atz and by ‖ · ‖ the Euclidean norm in R

n. It is enough to prove the inclusion “⊂”


when x∗ = θ , ∂f ∗(θ) �= ∅, and f ∗(θ) = 0; thus, f ∗ is proper, and infRn f = 0. Byassumption, we fix x0 ∈ int(domf ∗) and ρ > 0 such that

f ∗(x0 + v) ≤ f ∗(x0) + 1 for all v ∈ Bρ. (4)

Then, according to Correa and Hantoute (2010a, Corollary 6), it suffices to show that

co{(∂rf )−1(θ)

} ⊂ co{(∂rf )−1(x∗)

} + Ndomf ∗(x∗). (5)

For this, we pick a sequence (xk)k in co{(∂rf )−1(θ)} which converges to a given x.By Carathéodory’s theorem, we have that for each k ≥ 1, there are xk,1, . . . , xk,n+1 ∈(∂rf )−1(θ) and

(λk,1, . . . , λk,n+1) ∈ Δn+1

:= {(δ1, . . . , δn+1) ∈ R

n+1 | δ1, . . . , δn+1 ≥ 0, δ1 + · · · + δn+1 = 1}

(6)

such that xk = λk,1xk,1 +· · ·+λk,n+1xk,n+1, 〈x0, xk〉 ≥ 〈x0, x〉−1 and λk,i > 0 (with-out loss of generality). We also may assume that the sequence (λk,1, . . . , λk,n+1)kconverges to some (λ1, . . . , λn+1) ∈ Δn+1. By the definition of (∂rf )−1(θ), for eachi ∈ {1, . . . , n + 1}, there exists yk,i ∈ B 1

k(xk,i) such that f (yk,i) ≤ 1

k, and so, by the

Fenchel inequality,

〈x0, xk,i〉 ≤ 〈x0, yk,i〉 + k−1‖x0‖ ≤ f ∗(x0) + ‖x0‖ + 1 for all k. (7)

Let us denote I := {i | λi > 0}, J := {i | λi = 0}, so that I �= ∅. If i ∈ I , taking intoaccount (7) and multiplying the equation xk = λk,1xk,1 + · · · + λk,n+1xk,n+1 by x0

for each k, we get λk,i〈x0, xk,i〉 ≥ 〈x0, x〉−1−max{f ∗(x0)+‖x0‖+1,1}. So, giventhat λk,i → λi > 0, there exists α > 0 independent of k such that 〈x0, xk,i〉 ≥ α. Thus,by invoking once again the Fenchel inequality together with (4) and (7), for everyv ∈ Bρ , we get 〈v, xk,i〉 ≤ f ∗(x0 + v) + f (yk,i) − α + k−1‖x0‖ + ρ ≤ f ∗(x0) − α +‖x0‖ + ρ + 2 < +∞; that is, we may suppose that the sequence (xk,i)k converges tosome xi and, consequently, the corresponding sequence (yk,i)k also converges to xi .Therefore, since limk→+∞ f (yk,i) = 0, we infer that xi ∈ (∂∗f )−1(θ).

Now, we suppose that i ∈ J . If (λk,ixk,i)k is bounded, then the sequence (λk,ixk,i)khas an accumulation point xi which is also an accumulation point of the se-quence (λk,iyk,i)k since yk,i ∈ B 1

k(xk,i). Hence, since (cof )(yk,i) ≤ f (yk,i) ≤ 1

kand

λk,i → 0, we infer that (recall that cof ∈ 0(Rn) is the lsc convex hull of f )

σdomf ∗(xi) = σdom(cof )∗(xi) = (cof )∞(xi) ≤ lim infk

λk,i(cof )(λ−1

k,i λk,iyk,i

)

= lim infk

λk,i(cof )(yk,i ) ≤ limk

k−1λk,i = 0,

showing that xi ∈ Ndomf ∗(θ). Next, we denote J0 := {i ∈ J | (λk,ixk,i)k is notbounded}; hence, since yk,i ∈ B 1

k(xk,i), we get that J0 = {i ∈ J | (λk,iyk,i)k is not

bounded}. We also choose j ∈ J0 such that λk,j‖yk,j‖ = max{λk,i‖yk,i‖ | i ∈ J0}.Thus, for each i ∈ J0, there exists an accumulation point zi of the sequence (zk,i )k

defined by zk,i := λk,iyk,i

λk,j ‖yk,j ‖ such that ‖zi‖ ≤ 1 (with equality when i = j). Moreover,


writing

σdomf ∗(zi) = σdom(cof )∗(zi) = (cof )∞(zi)

≤ lim infk

λk,i

λk,j‖yk,j‖ (cof )

(λk,j‖yk,j‖

λk,i

zk,i

)

= lim infk

λk,i

λk,j‖yk,j‖ (cof )(yk,i ) ≤ lim infk

λk,i

kλk,j‖yk,j‖ = 0,

we deduce that zi ∈ Ndomf ∗(θ). On the other hand, observing that xk = λk,1yk,1 +· · · + λk,n+1yk,n+1, dividing both sides of this last equality by λk,j‖yk,j‖, andthen passing to the limit on k, we get

∑i∈J0

zi = θ . But, invoking (4), sincezi ∈ Ndomf ∗(θ) for each i ∈ J0, we can write 〈zi, x0〉 ≤ −ρ

2 ‖zi‖, and so by summingup on i ∈ J0 we get the contradiction 0 ≤ −ρ

2

∑i∈J0

‖zi‖ ≤ −ρ2 ‖zj‖ = −ρ

2 < 0.Consequently, J0 = ∅, and so by passing to the limit on k in the equation xk =λk,1xk,1 + · · · + λk,n+1xk,n+1 we get

x = limk

∑

i∈J\J0

λk,ixk,i +∑

i∈I

λk,ixk,i

=∑

i∈J\J0

xi +∑

i∈I

λixi ∈ Ndomf ∗(θ) + co{(∂rf )−1(θ)

},

establishing the desired relation (5). �

3 Further remarks and comparisons with previous results

We give in this short section some remarks and compare the preceding formulas ofSect. 2 with those previously established in Correa and Hantoute (2010a), Correa etal. (2011a).

Hereafter, f : X → R is a given function defined on the Banach space X, and τ isa topology on X∗ which is compatible with the duality pair (X,X∗).

Remark 4 (i) Proposition 1 can be immediately obtained from Theorem 3 in thefollowing way: if x∗ ∈ X∗ is fixed, we write

X ∩ ∂f ∗(x∗) = X⋂[ ⋂


(∂rLf )−1(x∗) + X ∩ NL∩domf ∗(x∗)

}]

=⋂

L∈Fτ (x∗)

[X

⋂cow∗∗{

(∂rLf )−1(x∗) + X ∩ NL∩domf ∗(x∗)

}].

Hence, since the weak∗∗ topology coincides with the weak topology on X, by invok-ing Mazur’s theorem we obtain that

X ∩ ∂f ∗(x∗) =⋂

L∈Fτ (x∗)co

{(∂r

Lf )−1(x∗) + X ∩ NL∩domf ∗(x∗)},

which is the statement of Proposition 1.


(ii) Similarly, as in (i) above, the main formula in Theorem 5 yields the followingcharacterization for every x∗ ∈ X∗:

X ∩ ∂f ∗(x∗) =⋂

ε>0L∈Fτ (x∗)

co{(∂εf )−1(x∗) + X ∩ NL∩domf ∗(x∗)

};

this last formula is also a simple consequence of the characterization given inHantoute et al. (2008) for the subdifferential of the supremum of convex functions.

In the following remark we exhibit the relationship between the subdifferential setof the conjugate function and its part lying in the initial space.

Remark 5 Remark 4(i) above, together with Theorems 3–5 and Proposition 1, pro-vides a quite natural relationship between the subdifferential set ∂f ∗(x∗) and its partlying in the initial space, X ∩ ∂f ∗(x∗). Indeed, it is well known that the set ∂f ∗(x∗)may in general be strictly larger than X ∩ ∂f ∗(x∗)w

∗∗; for example (Phelps 1989,

Example 1.4(b)), the conjugate of ‖ · ‖∞ in X = c0(N) is ‖ · ‖1, while X∗ = l1 andX∗∗ = l∞. Moreover, f ∗ ≡ ‖ · ‖1 is Gâteaux-differentiable at any x∗ = (xn) withxn �= 0 for all n, and ∂f ∗(x∗) = {(sgnxn)} �⊂ c0(N). Nevertheless, our analysis showsthat the set ∂f ∗(x∗) can be still recovered by a weak∗∗ closure procedure on sub-sets entering the expression of X ∩ ∂f ∗(x∗). To put this in one picture, for instance,Theorems 3 and Proposition 1 respectively give us

∂f ∗(x∗) =⋂


(∂rLf )−1(x∗) + X ∩ NL∩domf ∗(x∗)

},

X ∩ ∂f ∗(x∗) =⋂

L∈Fτ (x∗)co

{(∂r

Lf )−1(x∗) + X ∩ NL∩domf ∗(x∗)},

showing that ∂f ∗(x∗) and X ∩ ∂f ∗(x∗) are built upon the same elements of the initialspace X but with closures invoking different topologies. So, the choice of the topol-ogy when taking the closure of the sets co{(∂r

Lf )−1(x∗) + X ∩ NL∩domf ∗(x∗)} isdecisive in the structure of ∂f ∗(x∗): the norm topology gives us the set X ∩ ∂f ∗(x∗),while the weak∗∗ topology provides us with the whole subdifferential set ∂f ∗(x∗). In

other words, in order the equality ∂f ∗(x∗) = X ∩ ∂f ∗(x∗)w∗∗

to hold, one needs tomanage the intersection over the sets L ∈ Fτ (x

∗). For example, according to Corol-lary 4, in the simple case where f ∗ is finite and continuous at some point, with respectto a topology compatible with the duality pair (X,X∗), we have that

∂f ∗(x∗) = X ∩ ∂f ∗(x∗)w∗∗

for every x∗ ∈ X∗.

Let us recall that when x∗ ∈ int(domf ∗), a characterization of (proper lsc convex)functions f whose the conjugates satisfy the last relationship is given in Chakrabartyet al. (2007, Proposition 5.2) by means of the behavior at 0 of the multifunctionε ⇒ X ∩ ∂εf

∗(x∗).

Remark 6 In the current Banach space setting, the approach of Correa and Hantoute(2010a) applies when X∗ is endowed with a topology which is compatible with the


duality pair (X,X∗), in particular, with the norm topology on X∗ when X is a reflex-ive Banach space. The resulting formulas of this method provide different character-izations of the part of the subdifferential set ∂f ∗(x∗) lying in X; see Proposition 1.There is another way to overcome the difficulty raised in the case where the normtopology is considered on X∗. Indeed, as we explained in the introduction, an al-ternative approach was undertaken in Correa et al. (2011a) by using an appropriateextension to X∗∗ of the function f , f : X∗∗ → R, given by

f (x∗∗) := f (x∗∗) if x∗∗ ∈ X and +∞ otherwise.

So, if we respectively endow X∗ and X∗∗ with the norm and the weak∗∗-topologies sothat (X∗∗,X∗) forms a dual pair of locally convex spaces, according to Correa et al.(2011a, Lemma 2), both the functions f and f have the same conjugate and the sameε-subdifferential mapping. Consequently, we can characterize the subdifferential off ∗ by using either Correa and Hantoute (2010a) (evoking the subdifferential enlarge-ment as in Proposition 1) or Hantoute et al. (2008) (by means of the ε-subdifferential).In this way, it was recently shown in Correa et al. (2011a, Proposition 3) that, for ev-ery x∗ ∈ X∗,

∂f ∗(x∗) =⋂


(∂rLf )−1(x∗) + NL∩domf ∗(x∗)

}

or, equivalently, according to Correa et al. (2011a, Proposition 4),

∂f ∗(x∗) =⋂

ε>0L∈Fτ (x∗)

cow∗∗{(∂εf )−1(x∗) + NL∩domf ∗(x∗)

}.

It is clear that the main notable difference between these two formulas and the asser-tions of Theorems 3 and 5 is that the last formulas evoke the terms NL∩domf ∗(x∗) and(∂r

Lf )−1(x∗) which may contain points in X∗∗ \ X. So, in some sense Theorems 3and 5 could provide an accurate estimation of the subdifferential of the conjugatefunction since they only require the access to the (∂r

Lf )−1(x∗), (∂εf )−1(x∗) andX ∩ NL∩domf ∗(x∗) which lie in X.

4 Proof of Theorem 3

In this section we complete the proof of Theorem 3. We recall that f : X → R is agiven function and τ is a locally convex topology on X∗ which is compatible withthe duality pair (X,X∗). The family Fτ (x

∗), x∗ ∈ X∗, is defined in (1). If g : X → R

is another function, we denote by f �g : X → R the inf-convolution of f and g, thatis, f �g := infx∈X{f (x) + g(· − x)}.

Lemma 7 For every x∗ ∈ X∗, we have that

∂f ∗(x∗) =⋂

L∈Fτ (x∗)∂(f �σL)∗(x∗).

Moreover, if L ∈ Fτ (x∗), then we have

∂(f �σL)∗(x∗) ∩ X = co{(∂r (f �σL))−1(x∗) + X ∩ NL∩domf ∗(x∗)

}.


Proof We fix x∗ ∈ X∗ and denote by S the subset on the right-hand side in the firstformula. We pick L ∈ Fτ (x

∗). Then, for z ∈ S, we have that ∅ �= ∂(f �σL)∗(x∗) =∂(f ∗ + IL)(x∗) and, so, x∗ ∈ L ∩ domf ∗. Given y∗ ∈ domf ∗, we denote M :=aff{x∗, y∗}, so that M ∈ Fτ (x

∗). Then, since z ∈ S ⊂ ∂(f �σM)∗(x∗), we ob-tain that 〈z, y∗ − x∗〉 ≤ (f �σM)∗(y∗) − (f �σM)∗(x∗) = f ∗(y∗) − f ∗(x∗), show-ing that z ∈ ∂f ∗(x∗). Conversely, we pick z ∈ ∂f ∗(x∗) and L ∈ Fτ (x

∗) so thatx∗ ∈ L ∩ domf ∗ and (f �σL)∗(x∗) = f ∗(x∗) + IL(x∗) = f ∗(x∗). Thus, sincef ∗ ≤ f ∗ + IL = (f �σL)∗, we deduce that z ∈ ∂(f �σL)∗(x∗), and, so, by the ar-bitrariness of L ∈ Fτ (x

∗) it follows that z ∈ S.The second formula remains to be checked. For this aim, we fix x∗ ∈ X∗ and

L ∈ Fτ (x∗) so that x∗ ∈ L ∩ domf ∗. Then, in view of the relationship (f �σL)∗ =

f ∗ + IL, the desired conclusion follows by applying Proposition 1 to the functionf �σL. �

In the remainder of the proof we fix (x∗, x∗∗) ∈ ∂f ∗, L ∈ Fτ (x∗) and define the

sets AL and A as

AL := (∂rLf )−1(x∗) + X ∩ NL∩domf ∗(x∗), A :=

⋂

L∈Fτ (x∗)cow∗∗

(AL). (8)

Lemma 8 We have that A ⊂ ∂f ∗(x∗).

Proof According to the first statement of Lemma 7, we write

AL ⊂ co{(

∂r (f �σL))−1

(x∗) + X ∩ Ndom(f �σL)∗(x∗)

}

= ∂(f �σL)∗(x∗) ∩ X ⊂ ∂(f �σL)∗(x∗), (9)

which gives A ⊂ ⋂L∈Fτ (x∗) ∂(f �σL)∗(x∗). So, the desired inclusion follows by in-

voking the second statement of Lemma 7. �

We continue with the proof of Theorem to prove the opposite of the inclusiongiven in the lemma above. Equivalently, we shall establish the inequality

〈x∗∗,w∗〉 ≤ σAL(w∗) for all w∗ ∈ X∗. (10)

In the following corollary we show that it is enough to prove this last inequality onthe set (L ∩ domf ∗ − x∗) ∩ domσAL

.

Lemma 9 Inequality (10) holds if and only if it holds for w∗ lying in (L ∩ domf ∗ −x∗) ∩ domσAL

.

Proof Let us first observe that for all z1 ∈ X ∩ NL∩domf ∗(x∗) and z2 ∈ (∂rLf )−1(x∗)

(⊂ (∂r (f �σL))−1(x∗) ⊂ ∂(f �σL)∗(x∗), by (9)), and every y∗ ∈ L ∩ domf ∗, wehave

〈z1 + z2, y∗ − x∗〉 ≤ 〈z2, y

∗ − x∗〉 ≤ (f �σL)∗(y∗) − (f �σL)∗(x∗)= f ∗(y∗) − f ∗(x∗) < +∞,


showing that cone(L ∩ domf ∗ − x∗) ⊂ domσAL. Now, we fix w∗ ∈ domσAL

∩coneτ (L ∩ domf ∗ − x∗) and pick w∗ ∈ τ -ri(cone(L ∩ domf ∗ − x∗)) (⊂ domσAL

)so that w∗

λ := (1 − λ)w∗ + λw∗ ∈ cone(L ∩ domf ∗ − x∗) (⊂ domσAL, as shown

in the last inequality). Hence, since (10) holds on cone(L ∩ domf ∗ − x∗) for everyλ ∈ (0,1), we can write

〈x∗∗,w∗λ〉 ≤ σAL

(w∗λ) ≤ (1 − λ)σAL

(w∗) + λσAL(w∗),

which leads us, as λ → 0+, to 〈x∗∗,w∗〉 ≤ limλ→0+(1 − λ)σAL(w∗) + λσAL

(w∗) =σAL

(w∗). Therefore, (10) holds on coneτ (L ∩ domf ∗ − x∗). Finally, if w∗ ∈ X∗ \coneτ (L ∩ domf ∗ − x∗), then by the separation theorem applied in the (locally con-vex) space (X∗, τ ) we find x ∈ X \ {θ} and α ∈ R such that

〈x,w∗〉 > α ≥ 〈x, v∗〉 for all v∗ ∈ coneτ (L ∩ domf ∗ − x∗).

Hence, x ∈ (X ∩ NL∩domf ∗(x∗)) \ {θ} and 〈x,w∗〉 > 0, so that σAL(w∗) ≥

σ(∂rLf )−1(x∗)(w

∗) + supn≥1 n〈x,w∗〉 = +∞; that is, (10) trivially holds. �

From now on, we shall use the notation Y := aff(L ∩ domf ∗ − x∗), so that, bythe choice of L(∈ Fτ (x

∗)), Y is a τ -closed (a fortiori, norm-closed) subspace of X∗,and there exist z∗ ∈ L∩ domf ∗ and a convex symmetric τ -neighborhood V ⊂ X∗ ofθ such that

(z∗ + V ) ∩ aff(L ∩ domf ∗) ⊂ L ∩ domf ∗,(11)

f ∗(z∗ + z∗) ≤ f ∗(z∗) + 1

2for all z∗ ∈ V ∩ Y.

Then, according to Lemma 9, we only need to show in the following lemmas that forevery fixed w∗ ∈ L ∩ domf ∗,

〈x∗∗,w∗ − x∗〉 ≤ σAL(w∗ − x∗). (12)

Lemma 10 Let (x∗, x∗∗) ∈ ∂f ∗ and L ∈ Fτ (x∗) be fixed as in (8) above. Then, there

exist nets (xα, x∗α) ⊂ X × L, α ∈ (S1,≤), such that xα ∈ co{(∂r

Lf )−1(x∗α) + X ∩

NL∩domf ∗(x∗α)},

‖x∗α − x∗‖∗ → 0,

(‖xα‖) bounded, xα ⇀w∗∗x∗∗; (13)

lim infα

f ∗(x∗α) ≥ f ∗(x∗); (14)

f ∗(x∗α) ≥ f ∗(x∗) − 1

2for all α. (15)

Proof We recall that cl and co also refer to the closed and closed convex hulls of func-tions. By the current assumptions ∂f ∗(x∗) �= ∅ and x∗ ∈ L (∈ Fτ (x

∗)), from the rela-tionships ∂f ∗(x∗) ⊂ ∂(f ∗ + IL)(x∗) = ∂((cof )�σL)∗(x∗) = ∂(cl((cof )�σL))∗(x∗)we infer that the functions cof and cl((cof )�σL) are necessarily proper. Then, writ-ing x∗∗ ∈ ∂f ∗(x∗) ⊂ ∂(cl((cof )�σL))∗(x∗), by Rockafellar (1970b, Proposition 1)there exists a net (xα, x∗

α)α ⊂ X × X∗, α ∈ S1, satisfying (13) together with

xα ∈ X ∩ ∂(cl

((cof )�σL

))∗(x∗

α) = X ∩ ∂((cof )�σL

)∗(x∗

α) = X ∩ ∂(f ∗ + IL)(x∗α);


hence, x∗α ∈ L and (14), (15) hold in view of the weak* lsc of f ∗. Finally, by invoking

Proposition 2 we deduce that xα ∈ X ∩ ∂(f ∗ + IL)(x∗α) = co{(∂r

Lf )−1(x∗α) + X ∩

NL∩domf ∗(x∗α)}. �

Lemma 11 We fix ρ > 0 and let L ∈ Fτ (x∗) and (xα, x∗

α) ⊂ X × X∗ be as inLemma 10. Then, there exist nets uα ∈ X ∩ NL∩domf ∗(x∗

α), vα ∈ co{(∂rLf )−1(x∗

α)}(α ∈ S1) and u,v ∈ Y ∗ such that

uα + vα − xα ∈ ρBX; (16)

vα ∈ ∂(f ∗ + IL)(x∗α); (17)

〈vα − v, z∗〉 → 0 for all z∗ ∈ Y ; (18)

〈uα − u, z∗〉 → 0 for all z∗ ∈ Y. (19)

Proof Since xα ∈ co{(∂rLf )−1(x∗

α) + X ∩ NL∩domf ∗(x∗α)} as shown in Lemma 10,

we find uα ∈ X ∩ NL∩domf ∗(x∗α) and vα ∈ co{(∂r

Lf )−1(x∗α)} ⊂ X ∩ ∂(f ∗ + IL)(x∗

α)

(taking into account Proposition 2) such that (16) and (17) hold. Consequently, onthe one hand, without loss of generality on α, for each z∗ ∈ V ∩ Y (recall that z∗ andV,Y were defined in (11)), we write

〈vα, z∗〉 ≤ 〈vα, x∗α − z∗〉 + f ∗(z∗ + z∗) + IL(z∗ + z∗) − f ∗(x∗

α)

≤ 〈vα, x∗α − z∗〉 + f ∗(z∗) − f ∗(x∗) + 1

(by (11) and (15)

)

≤ 〈vα + uα, x∗α − z∗〉 + f ∗(z∗) − f ∗(x∗) + 1

(as uα ∈ NL∩domf ∗(x∗

α)).

In other words, in view of (13) and (16), we have that supz∗∈V ∩Y supα |〈vα, z∗〉| <

+∞. Therefore, invoking the Banach–Alaoglu theorem and passing to a subnet ifnecessary, we get the existence of v ∈ Y ∗ such that 〈vα − v, z∗〉 → 0, for all z∗ ∈ Y ,that is, (18) holds. Moreover, since uα ∈ NL∩domf ∗(x∗

α), for each z∗ ∈ V ∩ Y , wehave that

〈uα, z∗〉 ≤ 〈uα, x∗α − z∗〉 (

by (11))

≤ 〈vα + uα, x∗α − z∗〉 + f ∗(x∗) + IL(x∗)

− f ∗(x∗α) − IL(x∗

α) + 〈vα, z∗ − x∗〉 (by (17)

)

≤ 〈vα + uα, x∗α − z∗〉 + 〈vα, z∗ − x∗〉 + 1

2

(by (15)

).

This, together with supα{|〈vα, z∗ − x∗〉|, |〈vα + uα, x∗α − z∗〉|} < +∞ (recall (13),

(16) and (18)), gives us supz∗∈V ∩Y supα |〈uα, z∗〉| < +∞. Hence, arguing as above,we show the existence of u ∈ Y ∗ which satisfies (19). �

Lemma 12 The vector u which appears in Lemma 11 (19) can be extended to X, andthis extension, denoted in the same way, satisfies u ∈ NL∩domf ∗(x∗).

Proof First, the extension of u to X can be easily done by using the Hahn–Banachextension theorem in the space (X∗, τ ). So, we only need to show that such anextension, which is denoted in the same way, belongs to NL∩domf ∗(x∗). Indeed,


since uα ∈ NL∩domf ∗(x∗α) and vα ∈ ∂(f ∗ + IL)(x∗

α) (recall Lemma 11), for everyz∗ ∈ L ∩ domf ∗, we have that

〈uα, z∗ − x∗〉 ≤ 〈uα, x∗α − x∗〉 ≤ 〈uα + vα, x∗

α − x∗〉 + f ∗(x∗) − f ∗(x∗α).

Thus, combining (13), (14), (16), and (19), by taking limits on α we get 〈u, z∗ −x∗〉 ≤0, showing that u ∈ NL∩domf ∗(x∗). �

Lemma 13 Let (x∗α) and (vα) be the nets defined in Lemmas 10 and 11, re-

spectively. Then, for each α ∈ S1, there exist (λ1, λ2, λ3, λ4) ∈ Δ4 (see (6)),(λ1,α, λ2,α, λ3,α, λ4,α)α ⊂ Δ4, and (v1,α), . . . , (v4,α) ⊂ (∂r

Lf )−1(x∗α) such that

vi,α ∈ X ∩ ∂(f ∗ + IL)(x∗α) for i = 1, . . . ,4; (20)

limα

λi,α = λi ≥ 0 for i = 1, . . . ,4; (21)

〈vα, x∗α − x∗〉 =

∑

i=1,...,4

λi,α〈vi,α, x∗α − x∗〉; (22)

〈vα,w∗ − x∗α〉 =

∑

i=1,...,4

λi,α〈vi,α,w∗ − x∗α〉; (23)

〈vα, z∗ − x∗〉 =∑

i=1,...,4

λi,α〈vi,α, z∗ − x∗〉; (24)

supα

{∣∣〈vα,w∗ − x∗α〉∣∣, ∣∣〈vi,α,w∗ − x∗

α〉∣∣, i = 1, . . . ,4}

< +∞. (25)

Furthermore, one may suppose without loss of generality that λ1,α, λ2,α, λ3,α, λ4,α >

0 for all α.

Proof We fix α ∈ S1. By Lemma 11 we have that vα ∈ co{(∂rLf )−1(x∗

α)}, andso there exist lα ∈ N

∗, (λ1,α, . . . , λlα,α) ∈ Δlα and v1,α, . . ., vlα,α ∈ (∂rLf )−1(x∗

α)

(⊂X ∩ ∂(f ∗ + IL)(x∗α) by (17)) such that vα = ∑

1≤i≤lαλi,αvi,α . Moreover, in-

voking Carathéodory’s theorem applied in R3 and reordering if necessary, we find

(λ1,α, λ2,α, λ3,α, λ4,α) ∈ Δ4 such that (22)–(24) hold. In particular, we may assumethat each (λi,α)α (⊂ [0,1]) converges to λi ≥ 0, so that

∑1≤i≤1 λi = 1, that is, (21)

follows. We also may suppose that λ1,α, λ2,α, λ3,α, λ4,α > 0; for otherwise, we con-sider only the nonnul elements. Now, from (20) together with (15) we infer that (recallthat w∗ ∈ L ∩ domf ∗ and x∗

α ∈ L)

〈vi,α,w∗ − x∗α〉 ≤ f ∗(w∗) − f ∗(x∗

α) ≤ f ∗(w∗) − f ∗(x∗) + 1. (26)

But, since uα ∈ NL∩domf ∗(x∗α) and vα ∈ ∂(f ∗ + IL)(x∗

α) (recall Lemma 11), by (15)we have that 〈vα + uα,w∗ − x∗

α〉 ≤ 〈vα,w∗ − x∗α〉 ≤ f ∗(w∗) − f ∗(x∗) + 1, and so

supα |〈vα,w∗ − x∗α〉| < +∞, accordingly to (13) and (16). Therefore, by combining

(26) together with (23) we deduce that supα |〈vi,α,w∗ − x∗α〉| < +∞, i = 1, . . . ,4,

that is, (25) follows. �


Lemma 14 For all α ∈ S1 and i ∈ {1, . . . ,4}, there exists a net (vβi,α)β ⊂ X, β ∈

(S2,≤), such that

limβ

⟨v

βi,α − vi,α, z∗⟩ = 0 for all z∗ ∈ Y, (27)

limβ

f(v

βi,α

) + f ∗(x∗α) − ⟨

vki,α, x∗

α

⟩ = 0. (28)

Consequently, by passing to subnets if necessary,

limα

〈vi,α, x∗α − x∗〉 = lim

α〈vα, x∗

α − x∗〉 = 0; (29)

limα

limβ

(f

(v

βi,α

) − ⟨v

βi,α, x∗⟩) = −f ∗(x∗). (30)

Proof We fix α and i = {1, . . . ,4}. Since vi,α ∈ (∂rLf )−1(x∗

α), according to Lem-

ma 13, we find a net (vβi,α)β ⊂ X, β ∈ (S2,≤), such that (27)–(28) hold. Thus,

since x∗α , x∗ ∈ L ∩ domf ∗, we can write −f ∗(x∗

α) = limβ(f (vβi,α) − 〈vβ

i,α, x∗α〉) =

−〈vi,α, x∗α − x∗〉 + limβ(f (v

βi,α) − 〈vβ

i,α, x∗〉), so that, invoking the Fenchel inequal-ity, we have

−f ∗(x∗) ≤ limβ

(f

(v

βi,α

) − ⟨v

βi,α, x∗⟩) = −f ∗(x∗

α) + 〈vi,α, x∗α − x∗〉. (31)

Hence, in view of (14), we deduce that lim infα〈vi,α, x∗α − x∗〉 ≥ 0, and so by appeal-

ing to (22) together with (13) and the fact that uα ∈ NL∩domf ∗(x∗α) we get

0 ≤ λi lim infα

〈vi,α, x∗α − x∗〉

≤ lim infα

∑

i=1,···,4λi,α〈vi,α, x∗

α − x∗〉

= lim infα

〈vα, x∗α − x∗〉 ≤ lim sup

α〈vα + uα, x∗

α − x∗〉 = 0.

But λi > 0 (recall Lemma 13), and so we deduce that lim infα〈vi,α, x∗α − x∗〉 =

limα〈vα, x∗α − x∗〉 = 0; that is, (29) follows up to a subnet. Finally, (30) follows by

combining (29) and (31). �

Lemma 15 For each i ∈ {1, . . . ,4} there exists vi ∈ Y ∗ such that

limα

〈vi,α − vi, z∗〉 = 0 for all z∗ ∈ Y, (32)

where the net (vi,α)α appears in Lemma 13.

Proof For each α ∈ S1, by (17) we have that vi,α ∈ ∂(f ∗ + IL)(x∗α), and so, in view

of (11) together with (15), for every given z∗ ∈ V ∩ Y , we have that

〈vi,α, z∗ + z∗ − x∗α〉 ≤ f ∗(z∗ + z∗) + IL(z∗ + z∗) − f ∗(x∗

α) − IL(x∗α)

≤ f ∗(z∗) − f ∗(x∗) + 1. (33)

Moreover, by invoking the Fenchel inequality together with (27) and (30) we get


〈vi,α, z∗ − x∗〉 = limβ

⟨v

βi,α, z∗ − x∗⟩ ≤ f ∗(z∗) + lim sup

β

(f

(v

βi,α

) − ⟨v

βi,α, x∗⟩)

= f ∗(z∗) − f ∗(x∗) + 1. (34)

Now, since u ∈ NL∩domf ∗(x∗) (recall Lemma 12), we observe that

〈vα, z∗ − x∗〉 = 〈vα + uα, z∗ − x∗〉 + 〈uα − u,x∗ − z∗〉 + 〈u,x∗ − z∗〉≥ 〈vα + uα, z∗ − x∗〉 + 〈uα − u,x∗ − z∗〉.

But these last terms are bounded independently of α (according to (13), (16),and (18)), and so from (24) and (34) together with (29) we infer thatsupα{|〈vi,α, z∗ − x∗〉|, |〈vi,α, z∗ − x∗

α〉|} < +∞. Consequently, in view of (33), weget supz∗∈V ∩Y 〈vi,α, z∗〉 < +∞, so that, taking subnets if necessary and arguing as inthe proof of Lemma 11, we find vi ∈ Y ∗ such that 〈vi,α − vi, z

∗〉 → 0 for all z∗ ∈ Y .Thus, (32) holds. �

Lemma 16 For each i ∈ {1, . . . ,4}, the vector vi ∈ Y ∗ which appears in Lemma 15can be extended to X, and this extension, denoted in the same way, satisfies vi ∈(∂r

Lf )−1(x∗).

Proof Let (x∗α)α , (vα)α , (vi,α)α and (v

βi,α)(α,β), i ∈ {1, . . . ,4}, (α,β) ∈ S1 × S2, be

the nets defined in the previous lemmas. Then, by successively invoking (15), (29),and (32) we may assume that for all i ∈ {1, . . . ,4} and all α ∈ S1,

f ∗(x∗α) ≥ f ∗(x∗) − 1

2,

∣∣〈vi,α, x∗α − x∗〉∣∣ ≤ 1

3,

∣∣〈vi,α, z∗ − x∗〉∣∣ ≤ ∣∣〈vi, z∗ − x∗〉∣∣ + 1

3.

Moreover, accordingly to (27), (28), and (29) all together, for all z∗ ∈ Y , we have that

limβ

⟨v

βi,α − vi,α, z∗⟩ = lim

β

(f

(v

βi,α

) + f ∗(x∗α) − ⟨

vβi,α, x∗

α

⟩) = limα

〈vi,α, x∗α − x∗〉

= limα

〈vα, x∗α − x∗〉 = 0, (35)

and so, for every given α ∈ S1, there exists βα ∈ S2 such that the following statementholds for all β ≥ βα :

∣∣〈vi,α, x∗α − x∗〉∣∣ ≤ 1

2, f

(v

βi,α

) − ⟨v

βi,α, x∗

α

⟩ ≤ −f ∗(x∗α) + 1; (36)

hence, (f ∗)∗(vβi,α) − 〈vβ

i,α, x∗α〉 ≤ f (v

βi,α) − 〈vβ

i,α, x∗α〉 ≤ −f ∗(x∗

α) + 1, and so

vβi,α ∈ ∂1f

∗(x∗α). Moreover, by taking into account Lemma 15 we may assume that

|〈vi,α, z∗ − x∗〉| + |〈vi,α, z∗ − x∗α〉| ≤ m for some m ∈ R, so that, for all β ≥ βα ,

∣∣⟨vβi,α, x∗

α − x∗⟩∣∣ ≤ 1,∣∣⟨vβ

i,α, z∗ − x∗⟩∣∣ ≤ m + 1,(37)∣∣⟨vβ

i,α, z∗ − x∗α

⟩∣∣ ≤ m + 1

and, at the same time,

f(v

βi,α

) − ⟨v

βi,α, x∗

α

⟩ ≤ −f ∗(x∗α) + 1. (38)


At this moment, for fixed i ∈ {1, . . . ,4}, we define the net (wβi,α)(α,β)∈S as

wβi,α := v

βi,α where the corresponding index set

S := {(α,β) ∈ S1 × S2 | β ≥ βα for some βα satisfying (36)–(38)

}

is directed via the order relation (α1, β1) ≤ (α2, β2) ⇐⇒ α1 ≤ α2 and β1 ≤ β2. Then,recalling (11) and (15) together with the fact that w

βi,α ∈ ∂1f

∗(x∗α) (see (36)), for each

(α,β) ∈ S and all z∗ ∈ V ∩ Y , we have that⟨w

βi,α, z∗ + z∗ − x∗

α

⟩ ≤ f ∗(z∗ + z∗) − f ∗(x∗α) + 1 ≤ f ∗(z∗) − f ∗(x∗) + 2.

Hence, by (37) we get⟨w

βi,α, z∗⟩ ≤ ⟨

wβi,α, x∗

α − z∗⟩ + f ∗(z∗) − f ∗(x∗) + 2 ≤ m + f ∗(z∗) − f ∗(x∗) + 3,

and so, by taking a subnet if necessary we find wi ∈ Y ∗ such that the net (wβi,α)(α,β)∈S

weak* converges to wi in Y ∗. Let us show that wi = vi . Indeed, if this was not thecase, there would exist open neighborhoods in Y ∗, W1 of vi and W2 of wi such thatW1 ∩ W2 = ∅. Then, on one hand, we find (α0, β0) ∈ S such that w

βi,α ∈ W1 for every

(α,β) ∈ S satisfying (α,β) ≥ (α0, β0). On the other hand, by Lemma 15 there ex-ists α1 such that vi,α1 ∈ W2; we may assume without loss of generality that α1 ≥ α0.

We also find β1 ≥ βα1 with β1 ≥ β0 such that vβ1i,α1

∈ W2. Therefore, (α1, β1) ∈ S,

(α1, β1) ≥ (α0, β0), and we have that wβ1i,α1

= vβ1i,α1

∈ W2 ∩ W1, leading us to a con-tradiction. Hence, we must have wi = vi on Y .

Now, we are going to show that

lim(α,β)∈S

f(w

βi,α

) − ⟨w

βi,α, x∗⟩ + f ∗(x∗) = 0. (39)

Proceeding by contradiction, if this last inequality does not hold, taking into accountthe Fenchel inequality, we get that there would exist η > 0 such that

lim(α,β)∈S

f(w

βi,α

) − ⟨w

βi,α, x∗⟩ + f ∗(x∗) ≥ η.

Hence, we find (α2, β2) ∈ S such that f (wβi,α) − 〈wβ

i,α, x∗〉 + f ∗(x∗) ≥ η for ev-ery (α,β) ∈ S satisfying (α,β) ≥ (α1, β2). Moreover, since for all β ≥ β2, we havethat (α2, β) ∈ S and (α2, β) ≥ (α2, β2), by taking the limit on β in the inequalityf (v

βi,α2

) + f ∗(x∗α2

) − 〈vβi,α2

, x∗α2

〉 ≥ η from (35) we obtain the contradiction

0 < η ≤ limβ

f(w

βi,α2

) − ⟨w

βi,α2

, x∗⟩ + f ∗(x∗) = 0,

showing that (39) holds. Finally, to conclude that vi ∈ (∂rLf )−1(x∗), we only need

to observe that vi can be extended using the Hahn–Banach theorem in the space(X∗, τ ). �

Now, we are able to conclude the proof of Theorem 3:

Lemma 17 Inequality (12) holds.


Proof By invoking (29) together with Lemma 15 we take the limit over α in (23) toobtain that

〈v,w∗ − x∗〉 = limα

〈vα,w∗ − x∗〉 + 〈vα, x∗ − x∗α〉 = lim

α〈vα,w∗ − x∗〉

= limα

∑

i=1,...,4

λi,α〈vi,α,w∗ − x∗α〉 =

∑

i=1,...,4

λi〈vi,w∗ − x∗〉.

But, by Lemma 16 we have that vi ∈ (∂rLf )−1(x∗), and so

〈v,w∗ − x∗〉 ≤ σco{(∂rLf )−1(x∗)}(w∗ − x∗).

Therefore, taking into account Lemma 12, by (16) together with (18) and (19) we get

〈x∗∗,w∗ − x∗〉 ≤ limα

〈uα + vα,w∗ − x∗〉 + ρ‖w∗ − x∗‖= 〈u + v,w∗ − x∗〉 + ρ‖w∗ − x∗‖≤ σco{(∂r

Lf )−1(x∗)+X∩NL∩domf ∗ (x∗)}(w∗ − x∗) + ρ‖w∗ − x∗‖.Thus, the desired inequality holds a ρ → 0. �

Acknowledgements Research supported by Fondecyt Projects Nos. 1080173 and 1110019 and ECOS-Conicyt project No. C10E08.

We are grateful to two anonymous referees for many very helpful suggestions and constructive com-ments that have substantially improved the paper. We also would like to thank Professor C. Zalinescu formaking valuable suggestions and carefully reading a previous version of this paper, namely for kindlypointing out to us the gap in the proof given in Correa and Hantoute (2010a, Corollary 7) of Corollary 6.Finally, our grateful thanks also go to the Guest Editors of this Special Issue, Profs. M.J. Cánovas andJ. Parra, for their nice work.

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subdifferential of the conjugate function in general banach spaces

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