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Revista de Ciencias

Print version ISSN 0121-1935

rev. cienc. vol.22 no.1 Cali Jan./June 2018

http://dx.doi.org/10.25100/rc.v22i1.7100 

Artículo de investigación

Multivalued Usco Functions and Stegall Spaces

Diana Ximena Narváez1 

1Departamento de Matemáticas, Universidad del Valle, Cali - Colombia. dianaximena85@yahoo.com

Abstract

In this article we consider the study of the 𝐺 -differentiability and F-ifferentiability for convex functions, not only in the general context of topological vector spaces (𝑡. 𝑣. 𝑠.), but also in the context of Banach spaces. We study a special class of Banach spaces named Stegall spaces, denoted by 𝕾, which is located between the Asplund 𝐹-spaces and Asplund 𝐺-spaces (𝒢-Asplund). We present a self-contained proof of the Stegall theorem, without appealing to the huge number of references required in some proofs available in the classical literature (1). This requires a thorough study of a very special type of multivalued functions between Banach spaces known as usco multi-functions.

Key words: Bornology; usco mapping; subdifferential; Asplund spaces; Stegall spaces

1 Introducction

In this article, we take as reference the classic book Gâteaux Differentiability of Convex Functions and Topology: Weak Asplund Spaces 1, in which Marián J. Fabian talks about the Asplund spaces. First, we define the multivalued functions as follows: Let 𝑋 and 𝑌 be sets. A multivalued function from 𝑋 to 𝑌 is a relation which assigns to each 𝑥 ( 𝑋 a subset of 𝑌, denoted 𝑓(𝑥); 𝑓(𝑥) = ø for some 𝑥 ∊ 𝑋 is admitted. The graph of the multivalued functions 𝑓 is the set Graph(𝑓): = {(𝑥, 𝑦) ∊ 𝑋 ( 𝑌: 𝑦 ( 𝑓(𝑥)}. If for each 𝑥 ∊ 𝑋, 𝑓(𝑥) has only one element, we say the function is single-valued. A single-valued function from 𝑋 to 𝑌, is a relation 𝑥 ↦ 𝑦 = 𝑓(𝑥) which assigns to each 𝑥 ∊ 𝑋 a unique 𝑦 ∊ 𝑌. The graph of such functions is Graph(𝑓): = {(𝑥, 𝑦) ∊ 𝑋 ( 𝑌: 𝑦 = 𝑓(𝑥)}.

The objective of this article is the study of the properties of the usco functions. A usco function is a multivalued function 𝑓: 𝑋 → 𝑌 between Banach spaces which is upper semicontinuous and 𝑓(𝑥) is compact for each 𝑥 ∊ 𝑋. A usco function 𝑓: 𝑋 → 𝑌 is minimal if for each usco function 𝑔: 𝑋 → 𝑌 such that

𝑔(x)f(x) we have that f=𝑔.

One of the most important examples of the usco functions appears in the study of differentiability of the convex functions defined on an open convex subset Ω of a Banach space 𝑋. It is convenient to note that the subdifferential of a function 𝑓: Ω → ℝ ∪ {+∞} in 𝑥 ∊ Ω is the multivalued function ∂𝑓: 𝑋 → 𝑋´ defined as follows:

x'f(x) if and only if f(y) - f(x)  y - x, x' for every yΩ

It is also important to observe that in the completely regular topological spaces, singletons and closed sets can be separated by open sets. If 𝑋 is a Banach space, the topological dual with the weak topology is a completely regular space. Let us remember that a subset 𝐺 of a topological space 𝑋 is residual if there exists a countable family (𝓤n)n of open dense subsets such that Gn𝓤n .

The Asplud 𝐺-spaces are Banach spaces in which every continuous convex function, defined on an open convex set Ω, is differentiable in the Gâteaux sense in a residual subset of Ω. We denote by 𝖀G the set of Asplund 𝐺-spaces. The Asplund 𝐹-spaces are Banach spaces in which every continuous convex function, is differentiable in the Fréchet sense in a residual subset of Ω. We denote by 𝖀F the set of Asplund 𝐹-spaces. A Banach space 𝑋 is a Stegall space if for every Baire space 𝑍, every usco minimal function 𝑓: 𝑍 → 𝑋´ (with the weak topology) is single-valued in a residual subset. We denote by 𝔖 the set of Stegall spaces.

2 𝛽-Differentiability

Let us start with the following definition:

Definition 2.1 Let 𝑋 be a vector space on a field 𝕂. A vector bornology in 𝑋 is a subset 𝛽 of parts of 𝑋, denoted by ℘(𝑋), satisfying the following axioms:

𝑏𝑜𝑟𝑣 1. The union of all sets of 𝛽 is 𝑋:

Bβ B = X

𝑏𝑜𝑟𝑣 2. 𝛽 is stable under inclusions. That is, if 𝐵 ( 𝛽 and 𝐷 ⊂ 𝐵, then 𝐷 ( 𝛽.

𝑏𝑜𝑟𝑣 3. 𝛽 is stable under finite union. That is, if

{Sk: 1 kn}β , then

k=1nSkβ.

𝑏𝑜𝑟𝑣 4. 𝛽 is stable under the sum operation. That is, if Sk:1knβ then

k=1nSkβ.

𝑏𝑜𝑟𝑣 5. 𝛽 is stable under the scalar multiplication operation. That is, if 𝑆 ∊ 𝛽 and λ ( 𝕂, then λ𝑆 ∊ 𝛽.

𝑏𝑜𝑟𝑣 6 is stable under the formation of balanced envelope. That is, if 𝑆 ∊ 𝛽, then 𝑏𝑎𝑙 (𝑆) ∊ 𝛽. Where 𝑏𝑎𝑙 (𝑆) denote the balanced hull of a set 𝑆 defined by

balS=λ1λS.

In the context of a topological vector space (𝑋, τ), we have some natural bornologies.

Examples 2.2 Let (𝑋, τ) be an 𝑡.𝑣.𝑠., then the following collections of subsets of 𝑋 are vector bornologies.

1. The 𝐹-bornology βF is the collection of all the τ-bounded subsets of 𝑋.

2. The 𝐻-bornology βH is the collection of all compact subsets of 𝑋.

3. The 𝐺-bornology βG is the collection of all finite subsets of 𝑋.

From the definitions of these bornologies, it is easy to see that

βGβHβF.

For a proof of this result see 2.

In the sequel, 𝑋 and 𝑌 denote Banach spaces, Ω is an open subset of 𝑋 and ℒ(𝑋, 𝑌) denotes the space of linear transformation from 𝑋 to 𝑌 that are continuous.

Definition 2.3 Let 𝛽 be a bornology in 𝑋. We say that a function 𝑓: Ω → 𝑌 is 𝛽-differentiable in 𝑎 ∊ Ω if there exists a function 𝑢 ∊ ℒ(𝑋, 𝑌) such that for every 𝑆 ∊ 𝛽

uh=limt0fa+th-fat=0  uniformly in  hS.

The linear and continuous function 𝑢 is called 𝛽-derivative of 𝑓 at the point 𝑎 ∊ Ω and is denoted by u:=dβ(x) .

The following result provides a differentiability criterium with respect to a given bornology:

Theorem 2.4 Let 𝑋 and 𝑌 be 𝑡.𝑣.𝑠., Ω an open subset of 𝑋, and 𝛽 be a bornology in 𝑋. A necessary condition for 𝑓: Ω → 𝑌 to be 𝛽-differentiable in 𝑎 ∊ Ω is that for every 𝑆 ∊ 𝛽

limt0+fa+th+fa-th-2fat=0  uniformly in  hS.

This result follows doing a suitable modification in the proof of proposition 1.23 in 3.

In the case of a convex continuous functions 𝑓:Ω ⊂ 𝑋 → 𝑌, we have the following characterization of the 𝛽-differentiable in 𝑎 ∊ Ω:

Theorem 2.5 Let 𝑋 be an 𝑡.𝑣.𝑠., Ω an open convex subset of 𝑋, and 𝛽 be a bornology in 𝑋. A necessary and sufficient condition for a convex and continuous function 𝑓:Ω → ℝ be 𝛽-differentiable in 𝑎 ∊ Ω is that for every 𝑆 ∊ 𝛽

limt0+fa+th+fa-th-2fat=0  uniformly in  hS.

Similar considerations as theorem 2.4.

2.1 Single-Valued Maps Derivatives

Now, we will study the main notions of differentiation in a 𝑡.𝑣.𝑠. linked to the bornologies mentioned before: the Frechet derivative linked to the bornology of the bounded subsets of a 𝑡.𝑣.𝑠., the derivative of Hadamard linked to the bornology of the compact subsets and the derivative of Gâteaux linked to the bornology of the sets finite. Our interest in this work is to develop this theory in the case of Banach spaces.

Definition 2.1.1 Let 𝑋 and 𝑌 be Banach spaces, Ω ⊂ 𝑋 be an open subset and 𝑓: Ω → 𝑌 be a function.

We say that the function 𝑓: Ω → 𝑌 is 𝐺-differentiable (differentiable in the sense of Gâteaux) at the point 𝑎 ∊ Ω if there exists 𝑢 ∊ ℒ(𝑋, 𝑌) such that

uh=limt0fa+th-fat, for every  hX

and this limit is uniform on the finite subsets of 𝑋. In this case, we say that 𝑢 is the 𝐺-derivative of 𝑓 in 𝑎 and dGf(a)(h):=dGf(a),h:=u(h) .

We say that the function 𝑓: Ω → 𝑌 is 𝐻-differentiable (differentiable in the sense of Hadamard) at the point 𝑎 ∊ Ω if there exists 𝑢 ∊ ℒ(𝑋, 𝑌) such that

uh=limt0fa+th-fat, for every  hX

and this limit is uniform on the compact subsets of 𝑋. In this case, we say that 𝑢 is the 𝐻-derivative of 𝑓 in 𝑎 and dHf(a)(h):=dHf(a),h:=u(h) .

We say that the function 𝑓: Ω → 𝑌 is 𝐹-differentiable (differentiable in the sense of Fréchet) at the point 𝑎 ∊ Ω if there exists 𝑢 ∊ ℒ(𝑋, 𝑌) such that

uh=limt0fa+th-fat, for every  hX

and this limit is uniform on the bounded subsets of 𝑋. In this case, we say that 𝑢 is the 𝐹-derivative of 𝑓 in 𝑎 and dF f(a)(h):=dF f(a),h:=u(h) .

When differentiability holds for any 𝑎 ∊ Ω, we say that 𝑓 is 𝐺-differentiable in Ω (resp. 𝐻-differentiable) (resp. 𝐹-differentiable).

It is straightforward to see that the condition for uniform convergence is expressed as follows:

For all bounded set 𝐵 of 𝑋 and every ε > 0 there is a δ(𝐵, ε) > 0 such that

fa+th-fat-dFfahY<ε,  if  t<δ  forany  hB. (1)

It is clear that if 𝑓 is 𝐹-differentiable in 𝑎 ∊ Ω with derivative 𝑢, then it is 𝐺-differentiable in 𝑎 with derivative 𝑢.

The following elementary theorem is fundamental to the study of differentiability of convex functions:

Theorem 2.1.2 Let 𝑋 be a normed space. A convex continuous function defined on an open convex set with values in ℝ is necessarily locally Lipschitz.

For a proof of this theorem see the work by R. Phelps in 3.

3 USCO Functions

In this section we will go deeper into the study of the class of usco functions. We start giving some basic definitions.

Definition 3.1 (Multivalued function) Let 𝑋 and 𝑌 be topological spaces. A multivalued function of a set 𝑋 in a set 𝑌 is a correspondence 𝑥 ↦ (𝑥), which assigns to each 𝑥 ∊ 𝑋 a subset 𝑓(𝑥) of the set 𝑌. It is possible that the set (𝑥) is the empty set. The effective domain of this function is the set of the 𝑥 ∊ 𝑋 such that (𝑥) ≠ Ø.

Definition 3.2 (Upper/Lower semicontinuous multivalued function) Let 𝑋 and 𝑌 be topological spaces. A multivalued function b: 𝑋 → 𝑌 is upper semicontinuous in 𝑎 ( 𝑋 if for every open set 𝑊 in 𝑌 such that 𝑊 ( (𝑎) (open neighborhood of 𝑓(𝑎)) there exists an open set 𝑉 of 𝑋 such that 𝑎 ∊ 𝑉 (neighborhood of 𝑎) and 𝑓(𝑥) ( 𝑊 for all 𝑥 ∊ 𝑉. In the case that 𝑓 is upper semicontinuous for any 𝑎 ∊ 𝑋, we say that 𝑓 is upper semicontinuous in 𝑋.

A multivalued function 𝑓: 𝑋 → 𝑌 is lower semicontinuous in 𝑎 ∊ 𝑋 if for every open set 𝑊 in 𝑌 such that 𝑓(𝑎) ∩ 𝑊 ≠ Ø, there exists an open set 𝑉 of 𝑋 such that 𝑎 ∊ 𝑉 (neighborhood of 𝑎) and 𝑓(𝑥) ∩ 𝑊 ≠ Ø for all 𝑎 ∊ 𝑉. In the case that 𝑓 is lower semicontinuous for any 𝑎 ∊ 𝑋, we say that 𝑓 is lower semicontinuous in 𝑋.

The first remark is that 𝑓 is continuous in 𝑎 ∊ 𝑋 if it is both upper semicontinuous and lower semicontinuous at the point 𝑎.

Definition 3.3 (Graph of a multivalued function) Let 𝑋 and 𝑌 be topological spaces. The graph of a multivalued function 𝑓: 𝑋 → 𝑌 is the subset Graph (𝑓) of 𝑋 ( 𝑌 of the pairs (𝑥, 𝑦) ∊ 𝑋 ( 𝑌 such that 𝑦 ∊ (𝑥).

Definition 3.4 The limit values of a net (yt)tT in a topological space (𝑌, 𝛼) are the elements of the set

tTys:st¯

Where ≽ is the partial order relation in 𝛵.

We note that 𝑦 is a limit value of the indicated net if and only if there exists a subnet that converges to 𝑦.

Theorem 3.5 Let 𝑋 and 𝑌 be topological spaces. A multivalued function 𝑓: 𝑋 → 𝑌 is upper semicontinuous if and only if for any closed set 𝐶 ⊂ 𝑌, the set

xX:fxC

is closed on 𝑋.

Proof. Suppose that 𝑓 is upper semicontinuous, we want to show that

xX:fxC  is closed on  X

Let 𝐶 be a closed subset of 𝑌 and 𝐴 = 𝐶𝐶, then 𝐴 is an open set in 𝑌 and if (𝑥) ∩ 𝐶 = Ø, then 𝑓(𝑥) ( 𝐴. Since 𝑓 is upper semicontinuous, there is a neighborhood open 𝑉 of 𝑥 such that (𝑧) ( 𝐴 for all 𝑧 ∊ 𝑉. But this means that (𝑧) ∩ 𝐶 = Ø for all 𝑧 ( 𝑉 and, therefore,

xX:fxCc

is open set and so xX:fxC is closed on 𝑋. Suppose now that

xX:fxC

is closed with 𝐶 is closed, we want to establish that 𝑓 is upper semicontinuous. Let 𝑎 ∊ 𝑋 and 𝑊 is an open subset of 𝑌 such that (𝑎) ( 𝑊. Then, 𝑊𝐶 is a closed set and by hypothesis

xX:fxWc

is closed in 𝑋 and its complement 𝑉 is an open set. Now, 𝑎 ∊ 𝑉 since (𝑎) ∩ 𝑊𝐶 = Ø and 𝑓(𝑥) ∩ 𝑊𝐶 = Ø for all 𝑥 ∊ 𝑉, so that 𝑓(𝑥) ⊆ 𝑊 for all 𝑋 ∊ 𝑉.

Observe that now we are able to introduce a relation of order on the set of multivalued functions (𝑋, 𝑌) with 𝑋 and 𝑌 being topological spaces. For 𝑓, 𝑔 in (𝑋, 𝑌), se define the order ≼ as

fg,  if  fxgx  for any  xX.

Definition 3.6 (Usco Function) Let 𝑋 and 𝑌 be topological spaces. A multivalued function 𝑓: 𝑋 → 𝑌 is a usco function if 𝑓 is upper semicontinuous such that 𝑓(𝑥) ≠ Ø and is compact for any 𝑥 ∊ 𝑋. We denote the set of usco functions from 𝑋 to 𝑌 by 𝒰 (𝑋,𝑌).

We say that 𝑓 ∊ 𝒰 (𝑋,𝑌) is a minimal usco function if 𝑓 is a minimal element of the ordered set (𝒰 (𝑋,𝑌), ≼). This means that, if 𝑔 ∊ 𝒰 (𝑋, 𝑌) and 𝑔≼ 𝑓, then 𝑓 = 𝑔.

Now, we will establish some results about usco functions.

Theorem 3.7 For every 𝑢 ∊ 𝒰 (𝑋,𝑌), there exists a minimal usco function 𝑓 ∊ 𝒰 (𝑋,𝑌) such that 𝑓≼ 𝑢.

Proof. Let 𝑢 ∊ 𝒰 (𝑋,𝑌) and let ℋ be the collection of usco functions ℎ such that ℎ≼𝑢. Let us show that every chain ℒ contained in ℋ is bounded below. If ℱ is a finite subset in ℒ, then f𝓕 (fx) is a closed and nonempty subset contained in the compact set 𝑢(𝑥) and which we can order ℱ linearly. Then h𝓛h(x) is a closed and nonempty set contained in the compact set 𝑢(𝑥) and is therefore compact. We apply theorem 3.5 to the chain {ℎ: ℎ ∊ ℒ} to conclude that the function

xgx=hLhx

is upper semicontinuous and thus, a usco function that minorizes ℒ. By Zorn’s lemma ℋ has a minimal element.

Lemma 3.8 Let (𝑋, 𝛼) and (𝑌, τ) be topological spaces and 𝑓: 𝑋 → 𝑌 be a usco function. If (𝑥𝑡,𝑦𝑡)𝑡∊𝛵 is a net in 𝐺𝑟𝑎𝑝ℎ(𝑓) and 𝑥𝑡 → 𝑥, then (𝑦𝑡)𝑡∊𝛵 has at least one limit value in 𝑓(𝑥).

Proof. By contradiction, we assuming that the net (𝑦𝑡)𝑡∊𝛵 has not limit values in 𝐺𝑟𝑎𝐺ℎ(𝑓). That is, for all 𝑧 ∊ 𝑓(𝑥), there exists a 𝑡𝑧 such that zys:stz¯ . Let 𝑊𝑧 be an open neighborhood of 𝑧 such that Wzys:stz¯= . Since 𝑓(𝑥) is compact, there exists a finite subset Ff(x) such that f(x)zFWz=A . For the upper semicontinuity of 𝑓, there exists an open neighborhood 𝑉 ( 𝑋 of 𝑥 such that

fbW  for any  bV.

As 𝑥𝑡 → 𝑥, there exists a t¯T (where 𝛵 is a directed set) such that 𝑥𝑡 ∊ 𝑉 provided that tt¯ . We can suppose that tt¯suptz:zF . Consequently, fxtA if  tt¯ . By hypothesis, we have (xt,yt)Graph(f) , i.e., 𝑦𝑡 ∊ 𝐴 and 𝑦𝑡 ∊ 𝑊𝑧 for some 𝑧 ∊ 𝐹, which implies that Wzys:stz¯ , which is a contradiction.

Theorem 3.9 (Characterization of usco functions) Let (𝑋, 𝛼) and (𝑌, 𝛽) be topological spaces. A multivalued function 𝑓: 𝑋 → 𝑌 is a usco function if and only if its graph 𝐺𝑟𝑎𝑝ℎ(𝑓) is a closed set. Moreover, there exists a usco function 𝑢 ∊ 𝒰 (𝑋, 𝑌) such that 𝑓 ≼ 𝑢.

Proof. Assume that 𝑓 is a usco function. If prove that x,yGraphf¯ , then we will show that (𝑥, 𝑦) ∊ 𝐺𝑟𝑎𝑝ℎ(𝑓), i.e., 𝑦 ∊ 𝑓(𝑥). There is a net (𝑥𝑡, 𝑦𝑡)𝑡∊𝛵 such that 𝑥𝑡 → 𝑥, 𝑦𝑡 → 𝑦 and ytf(xt) . By lemma 3.8, (𝑦𝑡)𝑡∊𝛵 has at least one adhesion value in 𝑓(𝑥). That is, there is a subnet ytssS such that ytsy'fx and by the uniqueness of the limits, y=y'fx . It is clear that 𝑓 ≼ 𝑢 = 𝑓.

In the other direction, if Graph (𝑓) is closed and there is a usco function 𝑢 such that 𝑓 ≼ 𝑢, then 𝑓 is a usco function. By theorem 3.5, it suffices to show that 𝑓(𝑥) is closed for all 𝑥 ∊ 𝑋. Let yfx¯ and we will show that 𝑦 ∊ 𝑓(𝑥). Then there is a net (𝑦𝑡)𝑡∊𝛵 in 𝑓(𝑥) such that 𝑦𝑡 → 𝑦. Let 𝑥𝑡 = 𝑥 for all 𝑡 ∊ 𝛵. Then (𝑥𝑡, 𝑦𝑡)𝑡∊𝛵 is a net in 𝐺𝑟𝑎𝑝ℎ(𝑓) that fulfills the conditions of lemma 3.8 and, therefore, a subnet (𝑦𝑡)𝑡∊𝛵 converge to 𝑦′ ∊ 𝑓(𝑥) and as 𝑦𝑡 → 𝑦, it follows that 𝑦′ = 𝑦 ∊ 𝑓(𝑥).

The following lemma will be helpful for the main result in this section.

Lemma 3.10 Let (𝑋, τ) be the topological space, (𝑌, σ) Hausdorff space and 𝑓: 𝑋 → 𝑌 a usco function. The following are equivalent:

(i)𝑓 is a minimal usco function.

(ii) If 𝐴 is an open subset of 𝑋, 𝑊 is an open subset of 𝑌 and 𝑓(𝑎) ∩ 𝑊 ≠ Ø for some 𝑎 ∊ 𝐴, then there exists an open nonempty subset 𝑉 ( 𝐴 such that 𝑓(𝑥) ( 𝑊 for all 𝑥 ∊ 𝑉.

(iii)If 𝐴 is an open subset of 𝑋 and 𝐶 is a closed subset of 𝑌 such that 𝑓(𝑎) ∩ 𝐶 ≠ Ø for all 𝑎 ∊ 𝐴, then 𝑓(𝑎) ( 𝐶 for all 𝑎 ∊ 𝐴.

Proof. We will show that (𝑖) ⇒ (𝑖 𝑖). Let 𝑎 ( 𝑋 and 𝑊 ( 𝑌 open subsets as in (ii). We need to establish that exists an 𝑎0 ∊ 𝐴 such that 𝑓(𝑎0) ( 𝑊 because the fact that 𝑓 is upper semicontinuous implies that there is an open neighborhood 𝑉 in 𝐴 of 𝑎0 such that

fxW  for all  xV.

We argue by contradiction, let us assume that this statement is false and let 𝐶:= 𝑊𝐶 closed in 𝑌. Then, 𝑓(𝑥) ∩ 𝐶≠ Ø for any 𝑥 ∊ 𝐴. We define the function ℎ: 𝑋 → 𝑌 by

hx=fxC   if xA       fx     if xAc.

It is clear that ℎ(𝑥) ≠ Ø, closed for all 𝑥 ∊ 𝑋 and ℎ ≼ 𝑓. Then, we conclude that ℎ ∊ 𝒰 (𝑋, 𝑌) by theorem 3.9. Since 𝑓 is a minimal usco function by hypothesis, then ℎ = 𝑓 and consequently, ℎ(𝑥)= 𝑓(𝑥) ( 𝐶 for all 𝑥 ∊ 𝐴. This is a contradiction, since by hypothesis 𝑓(𝑎) ∩ 𝑊 ≠ Ø for some 𝑎 ∊ 𝐴.

Now we show that (𝑖 𝑖) ⇒ (𝑖) . By hypothesis 𝑓: 𝑋 → 𝑌 is a usco function. From theorem 3.7, there exists a minimal usco function 𝑔 ≼ 𝑓 and 𝑔 = 𝑓, as we will show below. If 𝑓 and 𝑔 are not equal, there exists 𝑥0 ∊ 𝑋 such that 𝒈(x0)f(x0) . As 𝒈(x0)f(x0) there exists 28 ∊ 𝑓(𝑥0) such that 𝑧 ∉ 𝑔(𝑥0). Thus, there exists an open 𝛵 in 𝑌 such that 𝑧 ∊ 𝛵 and 𝑔x0T¯= since 𝑔x0 is compact. From this, it follows that 𝑔x0T¯c . Let W:=T¯cY be an open neighborhood of 𝑔x0 . By the upper semicontinuity of 𝑔, there exists an open neighborhood 𝐴 ( 𝑋 of 𝑥0 such that

gxW  for any  xA.

This construction allows us to see that 𝑥0 ∊ 𝐴 and 𝑓(𝑥0) ∩ 𝛵 ≠ Ø. Then, there is a subset open nonempty 𝑉 ( 𝐴 such that

fxT  for any  xV

which is a contradiction, since 𝑔xfxT, 𝑔xW and TW= if xV .

Now we see that (𝑖𝑖) ⇒ (𝑖𝑖𝑖). Let 𝐴 be a subset open in 𝑋 and 𝐶 be a subset closed in 𝑌 as in (iii). Suppose there exists 𝑎 ∊ 𝐴 such that 𝑓(𝑎) is not contained in 𝐶. Then, 𝑓(𝑎) ∩ 𝑊≠ Ø for some 𝑎 ∊ 𝐴, where 𝑊:= 𝐶𝐶 is an open subset in 𝑌. For (ii) there exists a nonempty open subset 𝑉 ( 𝐴 such that 𝑓(𝑥) ( 𝑊 for all 𝑥 ∊ 𝑉, i.e., 𝑓 (𝑉) ∩ 𝐶 = Ø , which is a contradiction, since by hypothesis 𝑓(𝑥) ∩ 𝐶≠ Ø for all 𝑥 ( 𝐴.

We will show that (𝑖𝑖𝑖) ⇒ (𝑖𝑖). Let 𝐴 ( 𝑋 and 𝑊 ( 𝑌, with 𝑓(𝑎0) ∩ 𝑊≠ Ø such that 𝑎0 ∊ 𝐴. Let us prove that there exists 𝑎 ∊ 𝐴 such that 𝑓(𝑎) ( 𝑊. Otherwise, we would have 𝑓(𝑎) ∩ 𝑊𝐶 ≠ Ø for all 𝑎 ∊ 𝐴, where 𝐶:= 𝑊𝐶 is a closed subset in 𝑌. Then, for (iii) we have 𝑓(𝑎) ( 𝑊𝐶 for all 𝑎 ∊ 𝐴 and hence 𝑓(𝐴) ∩ 𝑊= Ø, which is a contradiction. Therefore, 𝑊 is an open neighborhood in 𝑌 of 𝑓(𝑎0) and by the upper semicontinuity of 𝑓, there exists an open neighborhood VX  of a0 contained in 𝐴 such that 𝑓(𝑉) ( 𝑊. In other words, 𝑓(𝑥) ( 𝑊 for all 𝑥 ( 𝑉.

We also have the following theorem of minimal usco functions:

Theorem 3.11 (Characterization of minimal usco functions) Let (𝑋, τ) and (𝑌,σ) be topological spaces and 𝑓: 𝑋 → 𝑌 be a usco function. Then the following statements are equivalent:

  1. 𝑓 is a minimal usco function.

  2. For every topological space (𝑍, 𝛼) and any continuous single-valued function 𝑔 : 𝑌→ 𝑍, 𝑔 ∘ 𝑓 is a minimal usco function of 𝑋 in 𝑍.

Proof. We will show 1) ⇒ 2).

(i) 𝑔 ∘ 𝑓 is compact and nonempty for all 𝑥 ∊ 𝑋. Indeed, as 𝑓 is a usco function, then 𝑓(𝑥) is compact for all 𝑥 ∊ 𝑋. Then 𝑔(𝑓(𝑥)) is compact since 𝑔 is continuous.

(ii) 𝑔 ∘ 𝑓 is upper semicontinuous. In fact, let 𝑥0 ∊ 𝑋 and 𝑊 be an open neighborhood in 𝑍 such that 𝑔(𝑓(𝑥0)) ⊆ 𝑊. By continuity, we have 𝐴:= 𝑔-1 (𝑊) is open in 𝑌 such that 𝑓(𝑥0) ⊆ 𝐴. Since 𝑓 is upper semicontinuous, there is a neighborhood open 𝑉 in 𝑋 of 𝑥0 such that 𝑓(𝑥) ⊆ 𝐴 for all 𝑥 ∊ 𝑉 and, therefore, 𝑔(𝑓(𝑥)) ⊆ 𝑊 for any 𝑥 ∊ 𝑉. So, we have shown that (𝑔 ∘𝑓) ∊ 𝒰 (𝑋, 𝑍).

(iii) It remains to show that 𝑔 ∘ 𝑓 is a minimal usco function. Using equivalences established in lemma 3.10, we must verify that 𝑔 ∘ 𝑓 satisfies (ii). Let 𝐴 be an open subset in 𝑋 and 𝑊 subset open in 𝑍 such that ((𝑔 ∘ 𝑓(𝑎)) ∩ 𝑊≠ Ø for some 𝑎 ∊ 𝐴. For continuity of 𝑔, we have 𝑔-1(𝑊) is an open subset in 𝑌 such that 𝑓(𝑎) ( 𝑔-1 (𝑊) (this is because ((𝑔 ∘ 𝑓)(𝐴)) ∩ 𝑊≠ Ø if and only if 𝑓(𝐴) ∩ 𝑔-1 (𝑊) ≠ Ø. Since 𝑓 is upper semicontinuous, there is an open neighborhood 𝑉 ⊆ 𝐴 of 𝑎 such that

𝑓(𝑥) ⊆ 𝑔-1 (𝑊) for any 𝑥 ∊ 𝑉

and, therefore, 𝑔(𝑓(𝑥)) ⊆ 𝑊 for any 𝑥 ∊ 𝑉 .

We will show 2) ⇒ 1). Let 𝑍=𝑌 and 𝑖𝑌: 𝑌→𝑌 be the identity function 𝑖𝑌(𝑦)=𝑦.

We observe that 𝑖𝑌 ∘ 𝑓 = 𝑓, which by hypothesis is a minimal usco function.

Definition 3.12 (Residual space) Let (𝑋, 𝛼) be a topological space. We said that 𝐺 ( 𝑋 is a residual set, if there is a family of open sets (𝒰𝑛)𝑛(ℕ which are dense in 𝑋 such that ⋂𝑛∊ℕ 𝒰𝑛 ⊆ 𝐺.

Definition 3.13 (Baire space) A Baire space is a topological space (𝑋, 𝛼) with the property that any residual is dense in 𝑋.

Regarding Baire space, we have the following well known examples (see 4):

1. Any locally compact topological space is a Baire space.

2. Any open subset of a Baire space is a Baire space.

3. Any complete metric space is a Baire space.

Theorem 3.14 Let (𝑋, 𝛼) be a Baire space and (𝑌, 𝑑) be a complete metric space. Then any minimal usco function 𝑓: 𝑋 → 𝑌 is single-valued in some residual subset 𝑅 of 𝑋.

Proof. Let 𝑓: 𝑋 → 𝑌 be a minimal usco function, we will see that, for every ε > 0, there exists an open set 𝑉 such that diameter of 𝑓(𝑉) is less than ε (𝑑𝑖𝑎𝑚(𝑓(𝑉)) < ε. In fact, let 𝑦 ∊ 𝑓(𝑋) and By,ε2 . There is at least one 𝑎 ∊ 𝑋 such that f(a)  By,ε2  , otherwise f(x)  By,ε2c for all 𝑥 ∊ 𝑋. By the lemma 3.10.(ii), there exists an open nonempty subset 𝑉 of 𝑋 such that f(x)  By,ε2c for all 𝑥 ∊ 𝑉 and consequently

diamfVε2

Let 𝐴ε = ⋃{ 𝑉 ⊆ 𝑉 : 𝑋 𝑜p𝑒𝑛 and 𝑑𝑖𝑎𝑚(𝑓(𝑉)) < ε}. It is clear that this set is open. We will show that 𝐴ε is dense. Let 𝑥 ∊ 𝑋 and 𝑈 be an open neighborhood of 𝑥. If y  f(x) and By,ε2 then f(x)  By,ε2  and again by lemma 3.10.(ii), there is an open nonempty subset 𝑉 of 𝑈 such that f(v)  By,ε2 for all 𝑣 ∊ 𝑉, i.e.,

diamf(v) ε2

and, therefore, 𝑉 ( 𝐴ε. We have shown that 𝑈 ∩ 𝐴ε ≠ Ø. If ε = 𝑛-1, we will write 𝐴𝑛 instead of A1n . Then {𝐴𝑛: 𝑛 ∊ ℕ} is a countable collection of open subsets which are dense in 𝑋. Since this is a Baire space, then 𝑅: = ⋂ 𝑛∊ℕ 𝐴𝑛 is a subset residual dense. If 𝑥 ∊ 𝑅, then

diamf(x) <1n for any n ,

and, therefore, 𝑓(𝑥) consists of only one element. That is, 𝑓|𝑅 is a single-valued function.

3.1 Usco Functions and Continuity of Subdifferential of a Convex Function

Definition 3.1.1 (Dual space) Let X be a Banachspace. The topological dual of X is defined as the Banach space X'=𝓛(X,) . Hereafter we set the notation

x'x:=x',x,    x'X',    xX

If 𝑋 is a Banach space, the weak-* topology of the dual 𝑋′ denoted by σ (𝑋′, 𝑋) is generated by the family {𝑝𝑥: 𝑥 ∊ 𝑋} of seminorms, where pxx'=x',x .

We note that (𝑋′, σ (𝑋′, 𝑋)) is a completely regular topological space. A fundamental weak neighborhood of 𝑎 ∊ 𝑋 is any set of the form

BF',εa=x'F'xX:x',x-a<ε,

Where 𝐹′ is a finite subset of 𝑋′ and ε > 0.

Definition 3.1.2 (The subdifferential) Let Ω be a convex open subset of a Banach space 𝑋 and let 𝑓: Ω → ℝ ∪ {+∞} be a convex and continuous function, and let 𝑥 ∊ 𝑑𝑜𝑚𝑓. The subdifferential of 𝑓 in the point 𝑎 ∊ Ω, is the set of 𝑢 ∊ 𝑋′ such that

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We denote by ∂ 𝑓(𝑎) the subdifferential of 𝑓 at point 𝑎. Any function xuxX' such that uxf(x) for all 𝑥 ∊ Ω is called subgradient of 𝑓 at 𝑥. A function 𝑓 is called subdifferentiable in 𝑎 ∊ Ω, if there is at least one subgradient in 𝑎. A function 𝑓 is called subdifferentiable, if it is subdifferentiable at each 𝑥 ∊ 𝑑𝑜𝑚<𝑓.

Definition 3.1.3 Let 𝑋 be a Banach space and Ω be a convex open subset of 𝑋. A function 𝑓: Ω → ℝ called is radially differentiable at 𝑎 ∊ Ω in the direction ℎ if there exists the limit

hf(a+th)-f(a)t, when t0+ (2)

The value of this limit is called the radial derivative of 𝑓 at point 𝑎 in the direction ℎ. We denote

d+fa,h:=limt0+fa+th-fat (3)

the radial derivative of 𝑓 at point 𝑎 in the direction ℎ. The function 𝑑+𝑓(𝑎, ℎ) is well defined, it is sublineal, convex and continuous. If 𝑑+𝑓(𝑎, ℎ) exists for all ℎ ∊ 𝑋, then we say that 𝑓 is radially differentiable at 𝑎 ∊ Ω.

We know that every convex real valued function defined in an open is radially differentiable if

d+fa,h=limt0+fa+th-fat    and    d-fa,h=limt0-fa+th-fat (4)

then

d-fa,h    d+fa,h,  for any  aΩ  and any  hX.

The radial derivative 𝑑±𝑓(𝑎, ℎ) it allows us to locate the subgradient at the point 𝑎 ∊ Ω in the following sense:

Lemma 3.1.4 Let 𝑋 be a Banach space, Ω be a convex open subset of 𝑋, 𝑓: Ω → ℝ be a convex and continuous function. Then 𝑢 ∊ ∂ 𝑓(𝑎) if and only if

d-fa,h    uh    d+fa,h,  for all  aΩ  and all  hX

Proof. If 𝑢 ∊ ∂ 𝑓(𝑎), then 𝑢: 𝑋 → ℝ is linear and continuous. Also if 𝑎+ 𝑡ℎ ∊ Ω for 𝑡 small enough and

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Dividing by 𝑡 > 0 and taking 𝑡 → 0+, we have 𝑢(ℎ) ≤ 𝑑+𝑓(𝑎, ℎ). On the other hand,

fa+th-fat=fa+-t-h-fa-t-1

and, therefore, when 𝑡 tends to zero by the left, we obtain that

d-fa,h=-d+fa,-h.

Consequently,

u-h=-uh    d+fa,-h=-d-fa,h

and, therefore, d-fa,h  uh

Remark 3.1.5 For a Banach space 𝑋, we consider in 𝑋′ the weak-* topology σ (𝑋′, 𝑋). If 𝑓: 𝑋→ ℝ is a convex and continuous function, we will show that ∂𝑓: 𝑋 → 𝑋′ is a usco function, considering in 𝑋 the norm topology and in 𝑋′ the weak-* topology.

Lemma 3.1.6 Let 𝑋 be a normed space, let Ω be an open and convex subset and 𝑓: Ω → ℝ be a convex and continuous function. Then,

1.The subdifferential function is a locally bounded function multivalued. In other words, for all 𝑎 ∊ Ω, exists 𝑟 > 0 and 𝑚 > 0 such that

Ba,rΩ  and  x'm  for any  x'fx  and any  xBa,r

2. If xnn is a sequence in Ω that converges to 𝑥 ∊ Ω and vnn is a sequence in 𝑋′ such that vnfxn for all 𝑛 ∊ ℕ, then vnn is bounded and the set of the σX',X -adherence values is nonempty and is contained in ∂𝑓(𝑥).

For a proof of this lemma see the work by R Phelps 3.

Theorem 3.1.7 Let 𝑋 be a normed space, let Ω be an open and convex subset. If 𝑓: Ω → ℝ is a convex and continuous function, then 𝑥 ↦∂𝑓(𝑥) of 𝑋 in 𝑋′ is a usco function, considering in Ω the norm topology and in 𝑋′ the weak-* topology σ (𝑋′, 𝑋).

Proof. We must show that:

i. ∂𝑓(𝑥) is σ (𝑋′, 𝑋)-compact and ∂𝑓(𝑥) ≠ Ø for all 𝑥 ∊ Ω.

ii. ∂𝑓 is upper semicontinuous.

Let us show (i). Since 𝑓 is continuous at 𝑎 ∊ Ω and convex, then 𝑒𝑝𝑖(𝑓) ( 𝑋 ( ℝ is a convex set such that 𝑖𝑛𝑡(𝑒𝑝𝑖(𝑓)) ≠ Ø. In particular if (𝑎, 𝑓(𝑎)) ∉ 𝑖𝑛𝑡(𝑒𝑝𝑖(𝑓)), then under the Hahn-Banach theorem, it exists 𝑢 ∊ 𝑋′ which separates (but not strictly) the point (𝑎, 𝑓(𝑎)) of the convex set. We can observe that 𝑢 ∊ 𝑋′ is the subgradient of 𝑓 at 𝑎 and, therefore, ∂ 𝑓(𝑥) ≠ Ø for all 𝑥 ∊ Ω. Now, since ∂ 𝑓(𝑥) ( 𝑋′ is bounded, by the Banach-Alaoglu theorem, this set is σ (𝑋′, 𝑋)-relatively compact. Now, to conclude the proof we need to prove that σ (𝑋′, 𝑋)-closed. In addition, ∂ 𝑓(𝑥) is σ (𝑋′, 𝑋)-compact. We recall that, 𝑢 ∊ ∂ 𝑓(𝑥) if and only if 𝑢(𝑦) ≤ 𝑑+𝑓(𝑥, 𝑦) for 𝑦 ( 𝑋 (by lemma 1.3) and

fx=yXuX':uyd+fx,y

Is σ (𝑋′, 𝑋)-closed since for each 𝑦 ∊ 𝑋, the function 𝑢 ↦ 𝑢 (𝑦) is σ (𝑋′, 𝑋)-continuous.

Let us show (ii). We will argue by contradiction, assuming that 𝑥 ↦ ∂𝑓(𝑥) is not upper semicontinuous in a 𝑎 ∊ Ω. This means that there exists a σ (𝑋′, 𝑋)-neighborhood open 𝑊 of ∂𝑓(𝑎), a sequence (Xn)n in Ω and a sequence (vn)n in 𝑋′ such that Vnf(xn) and vnW for all 𝑛 ∊ ℕ. By the lemma 1.4.2, the sequence (𝑣𝑛) 𝑛∊ℕ has at least one σ (𝑋′, 𝑋)-adherence value 𝑢 ∊ ∂𝑓(𝑎). As 𝑊 is a σ (𝑋′, 𝑋)-neighborhood of 𝑢, then 𝑣𝑛 ∊ 𝑊, which is a contradiction.

Theorem 3.1.8 Let 𝑋 be a Banach space, Ω an open and convex subset and 𝑓: Ω → ℝ is a convex and continuous function. If 𝑓 is 𝐹-differentiable in 𝑎 ∊ Ω, then the subdifferential function 𝑥↦∂𝑓(𝑥) of Ω in 𝑋′ is upper semicontinuous at 𝑥 = 𝑎 with respect to the norm topologies.

For a proof of this theorem see 3.

Lemma 3.1.9 Let w be a normed space, Ω an open and convex subset and 𝑓:Ω→ℝ a continuous convex function. A sufficient condition for that 𝑓 to be 𝐹-differentiable in 𝑎∊Ω is that there is a continuous selection in 𝑎 with respect to the norm topologies.

For a proof of this lemma see 3.

4 The Asplund Spaces

We recall that a subset 𝐺 of a topological space (𝑋, 𝛼) is a set 𝐺δ if it can be expressed as a countable intersection of open and dense subsets. In complete metric spaces and so, in Banach spaces, any residual subset is dense by the Baire’s theorem.

Definition 4.1 (Asplund 𝐹-space/𝒢-space/𝒢𝒟-space) An Asplund 𝐹-space is a Banach space with the following property: If 𝑓 is a continuous convex function defined in open and convex subset 𝑈 ⊂ 𝑋, then 𝐹-differentiable in a dense subset 𝐺δ of 𝑈.

A 𝒢-Asplund is a Banach space with the following property: If 𝑓 is a continuous convex function defined in open and convex subset 𝑈 ⊂ 𝑋, then 𝐺-differentiable in a dense subset 𝐺δ of 𝑈.

An Asplund 𝒢𝒟-space is a Banach space with the following property: If 𝑓 is a continuous convex function defined in open subset 𝑈 ⊂ 𝑋, then 𝐺-differentiable in a dense subset 𝐺δ of 𝑈.

We note that a 𝒢-Asplund space is also an Asplund 𝒢𝒟-space. We want to point out that in the usual literature, the Asplund 𝐹-spaces are called the Asplund spaces, the 𝒢-Asplund are called weak Asplund spaces and Asplund 𝒢𝒟-spaces are called weak differenciable spaces. We could talk about the 𝐻-Asplund spaces, but it does not make much sense because it has been shown that continuous convex function wich are 𝐺-differentiables are 𝐻-differentiables. In the original article by E. Asplund 5, a 𝒢-Asplund is a weak differentiability spaces (𝑊𝐷𝑆) and a Asplund 𝐹-space is called strong differentiability spaces (𝑆𝐷𝑆). The definition we use comes from R. Phelps 6.

We will denote by 𝔘𝐺 the class of the 𝒢-Asplund and 𝔘𝐹 the class of Asplund 𝐹-spaces.

4.1 Asplund 𝐹-Spaces and Dentables Subset of a Banach Space

In this section we present the proof of the Namioka-Phelps theorem, which characterizes Asplund 𝐹-spaces by means of the geometric condition of dentability. First, we discuss the notion of the dentable set. To do that, if 𝑀 is a subset of a Banach space 𝑋, we define the support function of this set 𝑀 as follows:

x'pMx':=supxMx,x'

of 𝑋′ in ℝ ∪{∞}. If we replace 𝑀 by T=adhσX',XcvM , then 𝛵 is convex, closed and bounded in the norm topology and 𝑝𝑀 = 𝑝𝛵. This function is sublineal and lower semicontinuous considering in 𝑋′ the weak-* topology denoted by σ (𝑋′, 𝑋) because it is the upper lower bound of lower semicontinuous functions.

Suppose now that 𝑋 is a Banach space and 𝑀′ is a subset of 𝑋′. Proceeding as before, we define the support function of the set 𝑀′ as follows:

x'pM'x':=supx'M'x', x

of 𝑋′ in ℝ ∪{∞}. If T'=adhσX,X'cvM' , then 𝛵′ is convex, σ (𝑋′, 𝑋)-closed (topology in 𝑋) and bounded in the norm topology and 𝑝𝑀′ = 𝑝𝛵′.

Definition 4.1.2 (𝑋-slice) Let 𝑋 be a Banach space, 𝛼 > 0 and 𝑥 ∊ 𝑋. For a nonempty bounded subse t 𝑀 of 𝑋, we define the (𝛼, 𝑥′)-slice of 𝑀 as the set

𝓡α,x';M:=xM:x,x'>pMx'-α.

Is clear that every slice is a relatively open set of 𝑀 with the topology σ (𝑋′, 𝑋), i.e., σ (𝑋, 𝑋′)|𝑀 (restriction of σ (𝑋, 𝑋′) to 𝑀).

Definition 4.1.3 (𝑋′-slice) Let 𝑋 be a Banach space, 𝛼 > 0 and 𝑥 ∊ 𝑋. For a nonempty bounded subset 𝑀′ of 𝑋′, we define the weak*- (𝛼, 𝑥)-slice of 𝑀′ as the set

𝓡'α,x;M':=x'M':x',x>pM'x-α.

We note that ℛ′(𝛼, 𝑥; 𝑀′) is open in the topology σ (𝑋′, 𝑋)|𝑀′. On the other hand, if 𝛼 < 𝛽, then

𝓡'α,x;M'𝓡' (β, x; M').

Moreover, if T'=adhσ(x,x') o cv(M'), then 𝓡'(α,x;M')𝓡'(α,x;T') since 𝑝𝑀′ = 𝑝𝛵′.

Definition 4.1.3 (Dentable set) Let 𝑋 be a Banach space. A subset 𝑀 ⊂ 𝑋 is called dentable if it admits slices of arbitrarily small diameter. We say that a subset 𝑀′ ⊂ 𝑋′ is weak *-dentable if it admits weak *-(𝛼, 𝑥)-slice of arbitrarily small diameter. A Banach space is dentable if any bounded subset is dentable.

Theorem 4.1.4 (Namioka - Phelps). A Banach space 𝑋 is an Asplund 𝐹-space if and only if its dual 𝑋′ is weak *- dentable.

Proof. Suppose 𝑋 is an Asplund 𝐹-space and show that all bounded subset of 𝑋′ admits slices of arbitrarily small diameter, otherwise, out so there would be a bounded subset 𝑀′ of 𝑋′ in which all (𝛼, 𝑥)-slice has a diameter 𝑟 > 0.

Let T'=adhσX',XcvM' . This set is convex, σ (𝑋′, 𝑋)-closed and bounded. By the Banach-Alaoglu theorem, 𝛵′ is σ (𝑋′, 𝑋)-compact. Since 𝑀′ ( 𝛵′, all (𝛼, 𝑥)-slice of 𝛵′ has a diameter ≥ 𝑟, we will show that the continuous sublinear function

x'px=pM'x:=supx'M'x',x

Of 𝑋 in ℝ is not 𝐹-differentiable in any point, which contradicts that 𝑋 is an Asplund 𝐹-space. With this purpose in mind, let ε=r4 and δ > 0 arbitrary. We will choose 𝛼 and 𝛽 appropriately. For now, we assume that 𝑟 − 2 𝛽 > 0 , 𝛼 > 0 and 𝛽 > 0. If 𝑥 ∊ 𝑋, then 𝑑𝑖𝑎𝑚ℛ′(𝛼, 𝑥; 𝛵′) ≥ 𝑟, so that there are 𝑎′, 𝑏′ ∊ ℛ′(𝛼, 𝑥; 𝛵′) such that ‖ 𝑎′− 𝑏′‖𝑋′ ≥ 𝑟 − 𝛽,

a',x  >  px-α    and    b',x  >  px-α

There exists a yX such that yx= 1 and a'-b', y>r-2β Now, if 𝑡 > 0, then

px+ty+px-ty-2px    a',x+ty+b',x-ty-a'+b',x-2α =  ta'-b',y-2α    tr-2β-2α.

We can now choose 𝛼 and 𝛽 appropriately. Let β=r4 and α=rδ4 . So

px+ty+px-ty-2pxt    r-2β-2αt=r2-rδ8t (5)

Since inequality (5) holds for all 𝑡 > 0, in particular if t=δ2 , so we have shown that if ε=r4 and δ is arbitrary, then there is always a 𝑡 and a 𝑦 such that

0<t<δ,  yX=1    and    px+ty+px-ty-2pxt    r4.

In virtue of theorem 2.4 with the bornology 𝛽𝐹, we conclude that 𝑝 is not 𝐹-differentiable in 𝑥, which is a contradiction.

Now, we will show that 𝑋 is an Asplund 𝐹-space. Let Ω be an open subset of 𝑋 and 𝑓: Ω → ℝ a convex function. Theorem 2.5 makes possible to locate the points of differentiability of 𝑓. The bornology that we are going to consider is 𝛽𝐹, of all bounded subsets 𝑋 since here the 𝐹-derivative is defined. Let 𝐷(𝑓, ε) be the set of 𝑥 ∊ Ω for which there is a δ (𝑥, ε) > 0 such that

fx+th+fx-th-2fxt < ε  if  0<t<δ  for any  hX  with  h   1

Then:

(i) 𝐷(𝑓, ε) is open for all ε > 0.

(ii) ε>0Df,ε is the set of 𝐹-differentiability of function 𝑓.

The proof of (ii) is obvious. We will proof (i), i.e., 𝐷(𝑓, ε) is open. Since 𝑓 is locally Lipschitzian by theorem 1.1, there exists a 𝑟 > 0 and a constant 𝑘 > 0 such that 𝐵 (𝑥, 𝑟) ( Ω and

a,bBx,r    implies    fb-fa    kb-aX.

By the definition of 𝐷(𝑓, ε), there exists δ > 0 such that δ<r2 and

fx+th+fx-th-2fxt    η  <  ε    if    hX1    and    0<t<δ.

Let zBx,r2. Then z+th,  z-thBx,r since

z±th-xX    r2+thX  <  r2+t  <  r2+δ  <  r.

Now let us observe that if 𝑡 = δ, then

fz+δh+fz-δh-2fzδ    fx+δh+fx-δh-2fxδ+fz+δh-fx+δhδ

+fz-δh-fx-δhδ+2fz-fxδ

  η+4kδz-xX.

Let α=minδ/4kε-η,r2 . Consequently,

fz+th+fz-th-2fzt  <  ε,  if  z-xX<α,  hX1  and  0<t<δ.

Once the points of 𝐹-differentiability of 𝑓 are located, it remains to show that 𝐴ε: = 𝐷(𝑓, ε) is dense for every ε > 0. On the one hand, we have that Ω is a Baire space. If ε = 𝑛−1, we will write An=Df,1n instead of A1n . Then {𝐴𝑛: 𝑛 ∊ ℕ} is a countable collection of open subsets in Ω. Since Ω is a Baire space, if we define

R:=nAn

(set of differentiability of 𝑓) is a set 𝐺δ.

We will show that 𝐷(𝑓, ε) is dense for every ε > 0. That is, if 𝑥0 ( Ω (fixed but arbitrary) any open neighborhood 𝑊 of 𝑥0 intersects 𝐷(𝑓, ε). As the subdifferential function is locally bounded, we can suppose that ∂ 𝑓(𝑊) = 𝛵′ is a bounded subset of 𝑋′. As 𝑋′ is weak *-dentable, there exists a weak- *(𝛼, 𝑧)-slice of 𝛵′ of arbitrarily small diameter, let say < ε. If 𝑎′∊ ℛ(𝛼, 𝑧; 𝛵′), then 𝑎′∊ ∂ 𝑓(𝑎) for some 𝑎 in 𝑊, since

𝓡α,z;T'=x'T':x',zsupx'T'x',z-α.

Since 𝑊 is open, there exists 𝑟 > 0 (sufficiently small) such that 𝑏:= 𝑎 + 𝑟𝑧 ∊ 𝑊. If 𝑏′ ∊ ∂ 𝑓(𝑏), therefore,

b',a-b    fa-fb    and    a',b-a    fb-fa,

consequently,

b',a-b+a',b-a    0. (6)

As 𝑎 − 𝑏 = −𝑟𝑧 and 𝑏 − 𝑎 = 𝑟𝑧 by replacing in (6)

0    b',-rz+a',rz=ra',z-b',z

and how 𝑟 > 0

b',za',z>supa'T'a',z-α.

We have, thus, shown that fbR'α,z;T' . Now, the set ℛ(𝛼, 𝑧; 𝛵′) is σ (𝑋′, 𝑋) -open in 𝑋′ and as the subdifferential function is  X,σX',X -continuous, there exists a δ > 0 such that

x-bX<δ    implies    fxRα,z;T'. (7)

Suppose that hX1. Then fb+th fb-th𝓡'α,z;T' if 0<t<δ since

b+th-bX=thX t<δ and b-th-bX=thX  t<δ.

If ufb,vfb+th,wfb-th and 0<t<δ , then

ub+th-ub=ub+th    fb+th-fbvb-ub+th=v-th          fb+th-fbwb-wth=wb+th    fb-fb-th.

Now, as 0 < 𝑡 < δ and by the linearity of 𝑢, 𝑣, 𝑤 we have

uh  fb+th-fbt  vh-uh fb-th-fbt-wh.

By (7) we know that 𝑣, 𝑤 ∊ ℛ′(𝛼, 𝑧; 𝛵′) and that the diameter of this set is < ε, then we obtain

0 fb+th+fb-th-2fbtvh-whvh-whX' <ε.

This last inequality is true for every ℎ ∊ 𝑋 such that ‖ℎ‖𝑋 ≤ 1 and every 0 < 𝑡 < δ. Therefore, 𝑏 ∊ 𝐷(𝑓, ε), i.e., 𝑊∩ 𝐷(𝑓, ε) ≠ Ø. This means that 𝐷(𝑓, ε) is dense and in consequence

R:=nAn

(set of differentiability of 𝑓) is a 𝐺δ-dense set. We have, thus, proved that 𝑋 is an Asplund 𝐹-space.

5 The Class of Stegall 𝕾 on Topological Spaces

This section is the central part of the article, since it introduces an intermediate class between the Asplund 𝐹-spaces and 𝒢-Asplund, called the class of Stegall 𝔖^ . This class establishes sufficient conditions for a Banach space to be a 𝒢-Asplund.

Let us recall that a topological space (𝑋, τ) is completely regular, if it is Hausdorff and for each closed set 𝐶 and every point 𝑝 that does not belong to 𝐶, there is a continuous function 𝑓: 𝑋 → [0,1] such that 𝑓|𝐶 = 0 and 𝑓(𝑝) = 1. The Urysohn’s lemma, every metric space is completely regular.

Theorem 5.1 If 𝑋 is a separable Banach space and 𝐾 ⊂ 𝑋′ is bounded, then the topology σ (𝑋′, 𝑋)|𝐾 is metrizable (the topology of the dual is metrizable on the bounded sets).

Theorem 5.2 Let 𝑋 be a Banach space. Then (𝑋′, σ (𝑋′, 𝑋)) is a space completely regular.

For the proof of previuos theorems see 1.

Definition 5.3 (𝑆-space) An 𝑆-space is a completely regular topological space 𝑋 that satisfies the following condition:

If 𝑍 is a Baire space and 𝑓: 𝑍 → 𝑋 is a minimal usco function, then 𝑓 is single-valued in a residual subset of 𝑍. We will denote by 𝔖, the set of all the 𝑆-spaces.

Even though Marian J. Fabian in 1 proves the following theorems, some important details were left out. We present here complete proofs.

Theorem 5.4 Every metric space (𝑋, 𝑑) is an 𝑆-space.

Proof. Let (𝑋, 𝑑) be a metric space, 𝑍 be a Baire space and 𝑓: 𝑍 → 𝑋 is a minimal usco function. For each 𝑛 ∊ ℕ, we define the open set

𝓤n=VZ:V  open  and  diamfV<1n. (8)

We will show that G=nUn Is residual. Note that each one 𝒰𝑛 is open and we will prove that each 𝒰𝑛 is dense for all 𝑛 ∊ ℕ. Let 𝑥 ∊ 𝒰𝑛 and 𝑉𝑥 ⊂ 𝑍 be an open neighborhood of 𝑥 with

diamfVx  <  1n

If yfx and By,13n, then fxBy,13n and by lemma 3.10.(ii), there is a nonempty open subset Ω of 𝑉𝑥 such that fx'  By,13n for each 𝑥′∊ Ω. So

diamfΩ  13n

Then, Ω ⊂ 𝒰𝑛 and, therefore, Ø ≠ 𝑉𝑥 ∩ 𝒰𝑛, which proof the density of the 𝒰𝑛. So {𝓤n:n} is a countable collection of open and dense subsets in 𝑍 and, therefore, the intersection 𝐺 of the 𝒰𝑛 is a residual subset. Finally, by vertue of theorem 3.14, the usco function 𝑓 is single-valued in 𝐺 and, consequently, 𝑋 ( 𝔖.

Theorem 5.5 Let 𝑋 be a completely regular space. If (Xn)n is a sequence of closed subsets of 𝑋𝑛 , with 𝑋 ( 𝔖 for every 𝑛 ∊ ℕ and 𝑋=⋃𝑛∊ℕ 𝑋𝑛, then 𝑋 ∊ 𝔖.

Proof. Let 𝑍 be a Baire space and 𝑓: 𝑍 → 𝑋 a minimal usco function. For each 𝑛 ∊ ℕ, we define the set

Zn:=zZ:fzXn.

Since 𝑓 is a upper semicontinuous function and by hypothesis each 𝑋𝑛 is closed in 𝑋, then each 𝑍𝑛 is a closed set in 𝑍 by theorem 3.5. If An=int(Zn), then A=nAn is open and dense in 𝑍, since 𝑍 is a space of Baire and 𝑍 = ⋃𝑛∊ℕ 𝑍𝑛 (see 7, p.63). It is clear that

𝐴𝑛 ≠ for some 𝑛 ∊ ℕ

Let 𝑛 ∊ ℕ be such that 𝐴𝑛 ≠ Ø, and let fn=fAn . We will show that 𝑓𝑛 is a minimal usco function and 𝑓𝑛(𝐴𝑛) ( 𝑋𝑛.

1) In general, if 𝑓: 𝑍 → 𝑋 is a usco function and 𝑔: 𝑋 → 𝑌 (with 𝑌 a regular space) is a continuous single-valued function, by theorem 3.11.2, we have that 𝑔 ∘ 𝑓 is a usco function. From this it follows that if 𝑋 is a subspace of 𝑌, then 𝑓|𝑋 is a usco function since fx=ix°f and 𝑖𝑋 is continuous.

2) We will show that 𝑓𝑛 is minimal. For this we will use lemma 3.10 several times. Let VAn open and CX closed such that

3) fnzC    for any zV

Then

fzC    for any zV

and as 𝑓 is minimal, by lemma 3.10.(iii) then 𝑓(𝑧) ( 𝐶 for every 𝑧 ∊ 𝑉. So

fnzC for any zV

and, hence, 𝑓𝑛 is minimal (using lemma 3.10 again).

1) 𝑓𝑛(𝐴𝑛) ( 𝑋𝑛. By the definition 𝑍𝑛, we have fnzC for every 𝑧 ∊ 𝐴𝑛 and by lemma 3.10.(iii), we obtain that fnAnXn since 𝑋𝑛 is closed by hypothesis.

Let

𝑅: = {𝑧 ∊ 𝑍: 𝑓(𝑧) is a singleton 𝑓 or all 𝑧}

And 𝑅𝑛: = 𝑅 ∩ 𝐴𝑛.

We note that 𝑅𝑛 is the set of 𝑧 ∊ 𝐴𝑛 such that fn(Z) consists of a single element. By hipothesis 𝑋𝑛 ∊ 𝔖, so 𝐴𝑛 is a Baire space because it is an open subset of a Baire space and 𝑓𝑛: 𝐴𝑛→ 𝑋𝑛 is a minimal usco function. Then we conclude that 𝑅𝑛 is a residual set in 𝐴𝑛 for each 𝑛∊ℕ. Now we can show that 𝑅 is residual in 𝑍. We have shown that there exists a countable collection {An}n of open sets of 𝑍 such that 𝑅 ∩ 𝐴𝑛 is residual in 𝐴𝑛 and A=nAn is dense in 𝑍. This implies that 𝑅 is residual as we shall proof continuation. We define

W1=A1  y  Wn=An-adhk=1n-1Ak  if  n2.

Then Wnn is an open and disjoint collection of open sets and 𝑊𝑛 ( 𝐴𝑛 for each 𝑛∊ℕ. Let us show that 𝑊 = ⋃𝑛∊ℕ 𝑊𝑛 is dense in 𝑍. Let 𝑧 ∊ 𝑍 and 𝑉 be a open neighborhood of 𝑧 contained in 𝑍. As 𝑉 ∩ 𝐴 ≠ Ø, then 𝑉 ∩ 𝐴𝑛 ≠ Ø for some 𝑛∊ℕ. Let 𝑝 be the first ordinal such that 𝑉 ∩ 𝐴𝑝 ≠ Ø. Then 𝑉 ∩ 𝑊𝑝 ≠ Ø and, therefore, 𝑉 ∩ 𝑊 = 𝑉 ∩ ⋃𝑛∊ℕ 𝑊𝑛 ≠ Ø which shows that 𝑊 is dense at 𝑍.

Since 𝑅 ∩ 𝐴𝑛 is residual in 𝐴𝑛, there exist a countable collection {Tkn:k} of open and dense subsets of 𝐴𝑛 such that RAn kTkn. Now since WnAn , then

RWnWnRAnkWnTkn. (9)

The set WnTkn is open and dense in 𝑊𝑛. In fact, let 𝑧 ( 𝑊𝑛 and 𝑉 be a open neighborhood of 𝑧 contained in 𝑊𝑛. Then 𝑉 is an open neighborhood of 𝑧 contained in 𝐴𝑛. Since 𝛵𝑘𝑛 is dense in 𝐴𝑛, then VTkn and, therefore, V(WnTkn) . Since 𝑊𝑛 is a Baire space, then

WnkTkn=kWnTkn.

This is the final part of the proof. Let Tk=n(TkWn) . Then we will see that 𝛵𝑘 is open and dense in 𝑍. To prove this statement, let 𝑉 be an open and nonempty subset of 𝑍. Since 𝑊 is dense in 𝑍, then WnV and, therefore, VWn for some 𝑛(ℕ (since 𝑊 is defined as the union of 𝑊𝑛). As WnTk is open and dense in 𝑊𝑛, it follows that

VWnTknWn=VTknWn.

We have, thus, shown that VTk for any open and nonempty of 𝑍, i.e., 𝛵𝑘 is open and dense in 𝑍. As 𝑍 is a Baire space, then T:=kTk is dense in 𝑍. Now we remark that

T=kTk=knTknWn.

As

WnTk=mWnTkmWm=TknWn

Since WnWm= if nm , then

kTk=knWnTkn=kTknnWn=nkWnTkn.

Using (9), we obtain that

kTknRWnR.

In other words, we have shown that 𝑅 is residual, since each 𝛵𝑘 is open and dense, accordingly 𝑋 = ⋃𝑛∊ℕ 𝑋𝑛 ∊ 𝔖.

5.1 The Class of Stegall

S^ of Banach Spaces

The class S^ of Stegall spaces consists of all of the Banach spaces whose dual with the weak-* topology are topological 𝔖-spaces. In other words, X  S^ , if 𝑋 is a Banach space and for all Baire space 𝑍, we have that any minimal usco function 𝛵 of 𝑍 in 𝑋´ (with weak-* topologie) is single-valued in a residual subset.

Next, we give a detailed proof of the Stegall’s theorem whose proof can be found in 1 citing numerous articles. We show that the usco multivalued functions, particulary the subdifferentials of convex functions, play an important role in this proof.

Theorem 5.1.1 The Stegall class lies between the Asplund 𝐹-spaces and the 𝒢-Asplund. In other words,

UF    S^    UG.

Proof. Let us first show that UF  S^ . Let 𝑋 be a Banach space and 𝑋′ be its dual. Recall that a bounded subset 𝑀 of 𝑋′ is weak *-dentable, if it admits weak- *(𝛼, 𝑥)-slice of arbitrarily small diameter. The weak-*(𝛼, 𝑥)-slice of 𝑀 is the set

𝕽α,x;M=x'M:x,x'>supx,M-α

A weak-*(𝛼, 𝑥)-slice of 𝑀 is a set σ (𝑋′, 𝑋)|𝑀 open due to the continuity of the function 𝑥′ ↦ ⟨ 𝑥, 𝑥′⟩ (i.e., the continuity of the support function).

Suppose 𝑋 is an Asplund 𝐹-space and let M=BX'=x'X':x'X'1 . Let 𝑍 be a Baire space and 𝛵: 𝑍 → (𝑀, τ), where τ = σ (𝑋′, 𝑋)|𝑀 (restriction of σ (𝑋′, 𝑋) to 𝑀) and let 𝛵 be a minimal usco function. We will show that 𝛵 is single-valued in a residual subset. This would show that 𝑀 is an 𝑆-space and since 𝑀 is σ (𝑋′, 𝑋)-closed and

X'=nnM,

(since 𝑋′ is fitted with the topology τ=σX',XBX' and 𝐵𝑋, ∊ τ(0) we conclude that 𝑋′ is an 𝑆-space by theorem 5.5, that is, XS^ .

Our aim now is to show that 𝛵 is single-valued in a residual subset. For this, let us consider the set defined in (8).

We claim that 𝒰𝑛 is an open set in (𝑀, τ), where τ = σ (𝑋′, 𝑋)|𝑀. In fact, If 𝑤 ∊ 𝑉, then 𝑉 is an open neighborhood of 𝑤.

Now, we claim that 𝒰𝑛 is dense in 𝑍. To prove this statement, let 𝒰 be an open and nonempty subset of 𝑍. Recall that 𝑋 is an Asplund space, so from theorem 1.1, 𝑋′ is weak-*-dentable. Now, since 𝛵(𝒰) ( 𝑀, then given ε > 0 there exists a weak-*(𝛼, 𝑥)-slice ℛ(𝛼, 𝑥; 𝛵(𝒰)) of arbitrarily small diameter, let say < ε (i.e., 𝑑𝑖𝑎𝑚[ ℛ(𝛼, 𝑥; 𝛵(𝒰))] < ε). If

W:=x'M:x,x'>supx,M-α

then 𝑊 is a subset τ-open of 𝑀 and TUW=Rα,x;TU since

Rα,x;TU=x'TU:x,x'>supx,TU-α

By lemma 3.10.(ii), there exists an open nonempty subset 𝑆 ( 𝒰 such that 𝛵(𝑆) ( 𝑊 and, therefore, 𝛵(𝑆) ( 𝛵(𝒰) and 𝛵(𝑆) ( 𝑊 ∩ 𝛵(𝒰) = ℛ(𝛼, 𝑥; 𝛵(𝒰)). So that,

diamTS<ε    and    SUnT,

which shows that 𝒰𝑛 is dense in 𝑍. From previous claims, we have that {𝒰𝑛: 𝑛∊ℕ} is a countable collection of open and dense subsets in 𝑍. We remark that

n𝓤nzZ:Tz  is a singleton for all  z:=𝓓.

Indeed, if zn𝓤n , then

diamTz<1n  for any n,

that is, 𝛵(𝑧) is a singleton for all 𝑧 from the dense residual subset 𝒟 of 𝑍. We have, thus, shown that 𝛵 is single-valued in a residual subset of 𝑍, and so XS^ .

Now, we will establish that S^UG . Let 𝑋 be a Banach space in Stegall’s class S^ . That is, (𝑋′, σ (𝑋′, 𝑋)) is an 𝑆-space ((𝑋′, σ (𝑋′, 𝑋)) ( 𝔖). That is, any minimal usco function of a Baire space 𝑍 in 𝑋′ is single-valued in a residual subset. We must show that 𝑋 is a 𝒢-Asplund space. Let ψ: Ω → ℝ be a continuous convex function, where Ω is a convex open subset of 𝑋 and show that this function is 𝐺-differentiable in a residual subset of Ω. The subdifferential of ψ is the multivalued function ∂ψ: Ω → 𝑋′ defined as:

xψx=ψx+h-ψx    uh  for any  h  such that  x+hΩ.

By theorem 3.1.5, the subdifferential ∂ ψ is a usco function considering in Ω the norm topology and in 𝑋′ the weak-* topology σ (𝑋′, 𝑋). By theorem 3.7, there exists a minimal usco function 𝛵: Ω → 𝑋′ such that 𝛵 ≼ ∂ ψ. Since Ω is a Baire space, (𝑋′, σ (𝑋′, 𝑋)) is completely regular and 𝛵: Ω → 𝑋′ is a minimal usco function, then 𝛵 is single-valued on a residual subset 𝒟 of Ω. It remains to verify that ψ is 𝐺-differentiable in 𝒟. Let 𝑥 ∊ 𝒟 (arbitrary but fixed) and ℎ ∊ 𝑋 . Then 𝛵(𝑥) = {𝑢}. As }.(𝑥) ( ∂ ψ (𝑥), then 𝑢 ∊ ∂ ψ (𝑥) and, therefore, for a 𝑡 > 0 small enough we have to

ψx+th-ψx    uth    with    x+thΩ. (10)

Let vtTx+th . Then vtψx+th and, therefore, by changing the variable 𝑧 = 𝑥 + 𝑡ℎ, we have that

ψz+-th-ψz    vt-thψx-ψx+th    vt-th,

which implies that

ψx+th-ψx    vtth (11)

Since 𝑡 > 0, from (10) and (11), we obtain that

uh    ψx+th-ψxt    vth

and, therefore,

0    ψx+th-ψxt-uh    vth-uh (12)

Now, let ε > 0 be such that

W=ξX':ξh-uh<ε

is a σ (𝑋′, 𝑋)-open neighborhood 𝑢 ∊ 𝑋′. Using the upper semicontinuity of 𝛵 in 𝑥 ∊ Ω, there exists an open neighborhood 𝑉 ( Ω of 𝑥 such that

TzW    for any    zV

There exists a δ > 0 such that x + th  V if 0<t<δ . So that,

Tx+thW    if    0<t<δ and, therefore,

vth-uh  <  ε    if    0<t<δ (13)

In virtue of (12) and (13), we conclude that

limt0+ψx+th-ψxt=uh (14)

Since 𝑢 ∊ 𝑋′, so

limt0ψx+th-ψxt=uh  and  ψ  is  G-differentiable in  x

We have, thus, shown that ψ is 𝐺-differentiable in a 𝐺δ-dense subset 𝒟 and hence 𝑋∊ 𝔘𝐺.

Proposition 5.1.2 Every separable Banach space belongs to the Stegall class.

Proof. Suppose 𝑋 is a separable Banach space. Let

BE'=x'X':xX'1    and    τ=σX',XBX'

By the Banach-Alaoglu theorem, the closed unitary ball 𝐵′ de 𝑋′ is σ (𝑋′, 𝑋)-compact, then by theorem 5.1 we have that (𝐵𝑋′, τ) is metrizable and (𝐵𝑋′, τ) ∊ 𝔖. As

X'=nnBX'    and    BX'  is    σX',X-closed

Then (𝑋′, σ (𝑋′, 𝑋)) ∊ 𝔖 by theorem 5.5 and therefore XS^ . We have, thus, shown that 𝑋 is a Stegall space.

Remarks 5.1.3

1) In general  UFS^ (strict inclusion). Indeed, let us consider 𝑋 = 𝓁1(ℝ). It has been showed that the ‖·‖1 is not 𝐹-differentiable at any point, but by theorem 5.1.2 we deduce that 𝓁1(ℝ) is a Stegall space because it is separable. Consequently, 𝓁1(ℝ) ∊ 𝔖, but 𝓁1(ℝ) ∉ 𝔘𝐹.

2) In general S^ UG (strict inclusion). To prove this result, we should go beyond the separable Banach spaces. Indeed, it was shown in theorem 5.1.2 that a separable Banach space is a Stegall space. Furthermore, Mazur's theorem assures us that the separable Banach spaces are also 𝒢-Asplund spaces (see 8). Kalenda in 8 showed that there are 𝒢-Asplund spaces whose dual with the weak weak-* topology do not belong to class S^ . That is, there exist 𝒢-Asplund spaces which are not in the S^ class.

3) There exist Banach spaces that are not 𝒢-Asplund spaces, for example, the nonseparable Banach space 𝑋 = 𝓁( ℝ) is not a 𝒢-Asplund space.

5.2 Example of Usco Function

The Supremum Mapping

As it is well known, the problem of the differentiability of convex functions is well addressed by Fabian in 1. Let 𝑋 be a compact and Hausdorff topological space, and 𝒞(𝑋) be the Banach space of the continuous functions of 𝑋 in ℝ with the norm f:=suptXft . We define the supreme function on 𝒞(𝑋) as

φ: 𝒞(𝑋) → ℝ

fφf:=suptXft

This function is sublineal and, therefore, convex. In addition, we define

ψ: 𝒞(𝑋) → 𝑋

fψf:=tX:φf=ft

This is a multivalued function that assigns to each function 𝑓 the set of points 𝑡 ∊ 𝑋 in which the function attains the supremum. We will call ψ the supremum mapping. We remark that the function 𝑓 ↦ ψ(𝑓) is a usco function. Indeed,

I. It is clear that ψ(𝑓) is compact because

tX:φf=β=ft

is closed in 𝑋. As 𝑋 is compact,

φf    for everything  f𝓒X

and, therefore, ψ(𝑓) ≠ Ø for all 𝑓 ( 𝒞(𝑋).

II. We will show that ψ is upper semicontinuous. We argue by contradiction assuming that for some point 𝑔 ∊ 𝒞(𝑋) there is an open neighborhood ψ(𝑔) ( 𝑊 such that on every open ball 𝐵(𝑔, ε) there exists a ℎ such that ψ(ℎ) is not contained in 𝑊. That is, there exists a sequence (𝑔𝑛)𝑛∊ℕ such that gn-g0 and ψgnWc . Let tnψgnWc and we remark that

φg    φg-gn-φgn=φg-gn+gn-gtn+gtn=2g-gn+gtn

As 𝑊𝐶 is compact, the sequence (𝑡)𝑛∊ℕ has limit value 𝑡 ∊ 𝑊𝐶. Since 𝑔 is a continuous function, 𝑔(𝑡) is an adhesion value of the sequence (𝑔(𝑡)) 𝑛∊ℕ in ℝ. Therefore, there is a subsequence tknn such that gtkngt and how gn-g0 , then φ𝑔𝑔t . In addition, 𝑔t φ𝑔 , and, therefore, φ𝑔 =𝑔t , i.e., t  ψ𝑔W which contradicts that t  WC . We have shown that ψ is upper semicontinuous.

In summary, ψ is a usco function. We emphasize that this example is closely related to the differentiability of the supremum norm in 𝒞(𝑋).

Acknowledgments

D. X. Narvaez was supported by the Graduate Mathematics Program, the Assistantship Program and the Mobility Program of the Office of International Relations (DRI) at Universidad del Valle (Cali, Colombia). This material is part of the doctoral thesis of D. X. N. under Dr. Guillermo Restrepo’s advise (+).

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Received: March 02, 2018; Accepted: June 22, 2018

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