Semi-smooth Points in Some Classical Function Spaces

The investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space C(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}(\Omega )$$\end{document}. Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions C(T,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}(\mathcal {T},E)$$\end{document}, where T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}$$\end{document} is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions C0(T,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_0(\mathcal {T},E)$$\end{document} (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.


Introduction
Smoothness in Banach space has been intensively studied for decades from many different points of view. We refer to the monographs [2,6] and the papers [10,11] for motivations, history, various aspects and problems connected with the subject.
In this paper, motivated by the results published by Banach [2], Singer [10], Sundaresan [11] and Miličić [8,9], we study the notion of semi-smooth points in Banach spaces of continuous functions.
It is necessary to mention a classical result: if Ω is a compact Hausdorff space, then

Preliminaries
The paper is organized as follows. In Sect. 3 of the paper, we discuss semismoothness in the space C 0 (T , E). In the Sect. 4 we study semi-smooth points and relate them to the set of supporting functionals.
Before stating the results, we recall some definitions and introduce our notation.
The above functional ρ is also denoted by y|x g and called an M -semi inner product -cf. Miličić [9] and also [7]. From the above properties of the mappings ρ ± we get: As a consequence of (nd6) we have Moreover, ρ is continuous with respect to the second variable. It is worth mentioning that M -semi inner product ρ was recently very helpful to extended results concerning Birkhoff-James orthogonally additive mappings; in particular, see [3,Theorem 3.3].
The unit sphere of X is denoted by S(X). Fix x ∈ X\{0}. We consider the set J(x) defined as follows It is easy to check that the set J(x) is convex, closed and J(x) ⊆ S(X * ). By the Hahn-Banach theorem we get J(x) = ∅ for all x ∈ X\{0}. Moreover, it is easy to see that J(αx) = J(x) for all α ∈ R\{0}, x ∈ X\{0}.
Let us recall the following result containing a representation of the norm derivatives ρ ± in terms of supporting functionals.
So, in particular,

Smoothness and Semi-smoothness
From now on we assume that the considered normed spaces are real and their dimensions are not less than 2. Let (X, · ) be a normed space over R. We say that X is smooth at point x o if J(x) contains a unique element in X * . Now, we consider a set N sm (X) := {x ∈ X : X is smooth at x} ∪ {0}. It will be helpful to reformulate the definition of smoothness as follows: If X is a separable real Banach space, then N sm (X) is dense. We say that a normed space X is smooth if X = N sm (X). Now we will give a characterization of smoothness at a point in terms of the norm derivatives (see [1,7]). Theorem 2.2. ( [1,7]). Let X be a real normed space and let x o ∈ X\{0}. Then the following statements are equivalent: It is clear that if X is smooth, then the M -semi-inner product in the sense of Miličić ρ ( , ·) is linear in the second argument. However, there also exists non-smooth spaces from which the mapping ρ ( , ·) is linear in the second variable too ( A real normed space X is called semi-smooth if ρ is additive (or, equivalently, linear; see (Msip2)) with respect to the second variable, i.e., x, y, z ∈ X.
Each smooth space is semi-smooth in the above sense but not conversely (l 1 is a suitable example; see [8,Example 8.1] and [7, p. 51]). The notion of semi-smooth spaces motivates this paper. We say that a normed space (X, · ) is semi-smooth at the point or, equivalently, if the mapping ρ (x o , ·): X → R is linear. As was mentioned, semi-smooth points played a crucial role in [12].
In the papers [4,5], ρ and ρ ± -orthogonalities were considered as well as other characterizations of semi-smooth spaces in terms of approximately ρ,ρ ±orthogonalities.

Semi-smoothness in Spaces of Continuous Functions
In this section we discuss the notion of semi-smoothness in continuous function spaces.
If T is a locally compact Hausdorff space and E a real normed linear space, we let C 0 (T , E) denote the space of all continuous functions f from T to E which vanish at infinity (namely, a function f vanishes at infinity if for In this section we want to focus our Proof. Take f = 0 and consider ε = f ∞ 2 . Take a compact set K ⊆ T such that f (t) E < ε if t ∈ T \K. We know that the mapping K t → f (t) E ∈ 0, f ∞ ⊆ R is continuous and K is compact. Therefore this mapping attains its maximum, i.e., there is t 0 ∈ K such that f ∞ = f (t 0 ) E . Thus the set M (f ) is not empty. Moreover, it is easy to check that the set M (f ) is closed. Therefore it is compact.
The next theorem will be useful for the study of semi-smooth points. The same symbol, ρ + , will be used to denote the norm derivatives on E and on C 0 (T , E).
Now we prove the first main result of this section.
The following statements are equivalent: E)). By Theorem 2.3, we may assume that f ∞ = 1, and then f ( On the other hand, from (3.2) it is very easy to prove that (B)⇒(A).
A topological space is called perfectly normal (or T 6 ) if it is normal and every closed subset is a G δ subset. The next result is from topology. It may be known to the reader, but it is presented here for the convenience. Of more immediate interest to us are the facts that metric spaces are perfectly normal spaces. Now we prove the second main result of this section.
Combining Theorems 3.3, 3.5 and 3.7, we immediately get the following characterizations.
Theorem 3.8. Suppose that T is a locally compact metric space and let E be a normed space. Assume that f ∈ C 0 (T , E), f = 0, and f (t) is a smooth point for every t ∈ M(f ). The following statements are equivalent: Theorem 3.9. Suppose that T is a locally compact metric space. Let E be a smooth normed space. Assume that f ∈ C 0 (T , E), f = 0. The following statements are equivalent: (A) f is a semi-smooth point, i.e., f ∈ N s sm (C 0 (T , E)), In this part of the section, we present another application of our main results in the classical sequence spaces, i.e., c 0 and c.
The sequence x = (x 1 , x 2 , . . .) ∈ c 0 can be identified with a function f x : N → R by f x (n) := x n . Clearly, f x ∈ C 0 (N, R) and such an identification is a linear and topological isomorphism, which is also an isometry. In the same manner we can see that the space c is identified with C(N∪{+∞}, R), and in particular C(N∪{+∞}, R) = C 0 (N∪{+∞}, R).
Proof. We can make the identification f x ∈ C 0 (N, R) by f x (n) := x n Theorem 3.9 implies f x is semi-smooth if and only if cardM(f x ) ≤ 2, which translates to conditions (a) and (b).
It is easy to verify that f α ∈ C 0 (Σ 2 , R). Furthermore, M (f α ) = {−1 ∞ , 1 ∞ } and thus by Theorem 3.9 we conclude that f α is a semi-smooth point in the space C 0 (Σ 2 , R). Clearly, one can take any f ∈ C 0 (Σ 2 , R) with the following representation: a j x j for some suitable sequence (a j ) j∈N with a j = 0 for all j ∈ N such that f (x) converges for any x ∈ Σ 2 . Then Thus such an f is a semi-smooth point.
The above considerations can be rewritten as the following result.
Theorem 3.13. Every unconditionally convergent series is a semi-smooth point in C 0 (Σ 2 , R).
a j x j . The rest is clear.
Let X be a real normed linear space. Let X ⊕ ∞ . . . ⊕ ∞ X denote the direct sum of X with itself n times, and in a natural way we introduce the norm (x 1 , . . . , x n ) ∞ := max{ x 1 , . . . , x n }. Note that X⊕ ∞ X is identified with C 0 ({1, 2}, X). As an immediate consequence of Theorems 3.7 and 3.9, we have the following.
We may identify X⊕ ∞ X⊕ ∞ X with a space C 0 ({1, 2, 3}, X), and applying again Theorem 3.5 and using a similar argument as in the proof of the above corollary, we obtain Corollary 3.15. Let X be any nontrivial normed space. Then X ⊕ ∞ X ⊕ ∞ X is not a semi-smooth space.
Bearing in mind that the space l 1 is semi-smooth, we have the following result.
Proposition 3.16. Let X be any nontrivial normed space. Then, there is no linear isometry μ from X ⊕ ∞ X ⊕ ∞ X into l 1 .

Semi-smooth Points in Normed Spaces and Supporting Functionals
The condition (2.3) motivates this section. When x o is not a smooth point, consider the set J(x o ). An interesting question in analysis is to study points for which J(x o ) is a "large set". The question is: What about spaces which are not smooth? Motivated by these, in this section we prove some characterization of semi-smoothness (instead of smoothness). Let us fix D ⊆ X and a ∈ D. We say that D is symmetric (with respect to the point a) if for all x ∈ D we have 2a − x ∈ D or, equivalently, D = 2a − D.
As a consequence of the above theorem, we get the following result. Fix a nonzero x ∈ X. If dim X = 2, then dim X * = 2. Since J(x) is a convex and J(x) ⊆ S(X * ), the equality dim X * = 2 shows that cardJ(x) = 1 or J(x) = conv{x * , y * }. Hence x ∈ N sm (X) or x ∈ N s sm (X). As an immediate consequence of this reasoning, we have the following.