V-Proximal Trustworthy Banach Spaces

In a recent work (2016), the first author proved the fuzzy sum rule for the V-proximal subdifferential under some natural assumptions on an equivalent norm of the Banach spaces. In the present paper, we are going to prove that the class of Banach spaces satisfying the fuzzy sum rule is very large and contains all Lp spaces 1 < p<∞ as well as the sequence spaces lð1 < p<∞Þ, the Sobolev spaces Wp,nð1 < p<∞Þ, and the Schatten trace ideals Cpð1 < p<∞Þ.


Preliminaries
Let X be a Banach space with dual X * , f : X ⟶ R ∪ f∞g a function, x ∈ dom f : fx ∈ X : f ðxÞ<∞g. Throughout the paper, B denotes the closed unit ball in X and h· , ·i is the dual pairing between X and its dual X * . We define (see [1,2]) the analytic (resp., the geometric) V-proximal subdifferential of f at x as follows: (resp. ∂ G π f ð xÞ ≔ fx * ∈ X * : ðx * ,−1Þ ∈ N π ðepif ; ð x, f ð xÞÞgÞ, where N π ðS ; uÞ ≔ ∂ A π ψ S ðuÞ is the V-proximal normal cone associated with S at u ∈ S. Here, J : X ⟶ X * is the normalized duality mapping and V : X * × X ⟶ R is a functional defined by We recall, respectively, the well-known concepts of proximal subdifferential and Fréchet subdifferential (see, for instance, [3]): if and only if for any ε > 0, there exists δ > 0 such that The Fréchet and proximal normal cones are defined as N F ðS ; xÞ = ∂ F ψ S ð xÞ and N P ðS ; xÞ = ∂ P ψ S ð xÞÞ. Notice that (see [3][4][5]) Fréchet and proximal subdifferential can be defined geometrically by the formulas Using the same terminology used in Ioffe [6], we will say that X is a V-proximal trustworthy space provided that for any ε > 0, any two functions f 1 , f 2 : X ⟶ R ∪ f∞g and any u ∈ X such that f 1 is lower semicontinuous and f 2 is Lipschitz around u, the following fuzzy sum rule holds: Here, U f i ðu, εÞ ≔ fx ∈ u + εB such that jf i ðxÞ − f i ðuÞj < εg and B * denotes the closed unit ball in X * . It has been proved in Theorem 2.3 in [1] that if X is a Banach space with an equivalent norm k⋅k such that k·k s (for some s ≥2) is C 2 -differentiable on X \{0} and let V be the functional associated to that norm k⋅k, then X is a V -proximal trustworthy space. Indeed, we have proved the following result. Theorem 1. Let X be a Banach space with an equivalent norm k⋅k such that k·k s (for some s ≥ 2) is C 2 -differentiable on X \ f0g, and let V be the functional associated to that norm k⋅k. For any ε > 0, any two functions f 1 , f 2 : X ⟶ R ∪ f∞g and any u ∈ X such that f 1 is lower semicontinuous and f 2 is Lipschitz around u, the following fuzzy sum rule holds: For a positive measure space (Ω, Σ, μ), we denote by L p , p ∈ ½1,∞Þ, the Banach space L p ðΩ, Σ, μÞ with its canonical norm kxk = ð Ð Ω jxðwÞj p dμðwÞÞ 1/p . We recall the following result from Theorem 1.1 in Section 5.1 in [7] (see also [8]).

Theorem 2.
(i) If p is an even integer, then where ½p is the integer part of p The following corollary follows directly from Theorem 2.
Unfortunately, for the case of p ∈ ð1, 2Þ, the function k·k p is not C 2 -differentiable on L p \ f0g and so the fuzzy sum rule cannot be covered by Theorem 1. This our objective in the next few lines, and so we obtain that L p is V-proximal trustworthy, for any p ∈ ð1,∞Þ. In the sequel of this section, we assume that X ≔ L p with p ∈ ð1, 2Þ. It is well-known that X is 2-uniformly convex (see, for instance, [7]), that is, there is a constant c > 0 such that The following lemma is taken from [9].

Lemma 4.
If E is a uniformly convex Banach space, then the inequality holds for all x and y in E, where C = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkxk 2 + kyk 2 Þ/2 q .
In page 51 in [9], the following parallelogram inequality is presented: It follows that and so the previous lemma with the inequality (8) yields the following important result.
Using Remark 7.7 in [9] and the fact that X = L p ðp ∈ ð1, 2Þ is p-uniformly smooth, we obtain and hence, for any x ∈ X and any δ > 0, we can write where σ depends on x, δ, and β. Consequently, as a direct corollary of Proposition 6, we have, in our setting of L p spaces (p ∈ ð1, 2), both inclusions that hold for any lower semicontinuous function f and any nonempty closed set S. Now, we recall from Ioffe [10] and Fabian [11] the following two important results.

Proposition 7.
Suppose that X is uniformly convex and that S is a closed subset of X. Let x¯∈S and x * ∈ N F ðS ;¯xÞ. Then, for any ε > 0, there exist x ε ∈ S and x * ε ∈ N P ðS ; x ε Þ such that kx ε − xk < ε and kx * ε − x * k < ε.
Theorem 8. Let X be an Asplund space. For any ε > 0, any two functions f 1 , f 2 : X ⟶ R ∪ f∞g and any u ∈ X such that f 1 is lower semicontinuous and f 2 is Lipschitz around u, the following fuzzy sum rule holds: Now we are ready to prove the main result of this section.