Distance to spaces of continuous functions (original) (raw)

A note in approximative compactness and continuity of metric projections in Banach spaces

Journal of Convex Analysis

We give a counterexample to a recent statement in the metric approximation theory and provide a setting where the statement holds. Let (X, •) be a real Banach space. Our notation is standard. We follow, for example, [FHHMPZ01]. In this note, if no reference to a different topology on X is made, convergence in X means •-convergence. The following concepts in the geometry of Banach spaces are more or less standard. A non-empty subset C of X is said to be approximately compact if for every x ∈ X and every sequence (c n) in C such that x − c n → dist (x, C) (such a sequence is called an approximate sequence for x in C), then (c n) has a convergent subsequence. The set C is said to be proximinal if, for every x ∈ X, the set P C (x) := {c ∈ C; x − c = dist (x, C)} is non-empty (the multivalued mapping P C : X → 2 X is called the metric projection onto C). The set C is said to be semi-Chebyshev if P C (x) contains at most one point for every x ∈ X. The set C is said to be Chebyshev if it is simultaneously proximinal and semi-Chebyshev. In this case we put P C (x) = {π C (x)} for all x ∈ X. A Banach space X is said to be locally uniformly rotund (LUR, for short) if for every x ∈ S X and every sequence (x n) in S X such that x + x n → 2, then x n → x. A Banach space X is said to be midpoint locally uniformly rotund (MLUR, for short) if for any x 0 , x n and y n in S X , n ∈ N, such that x n + y n − 2x 0 → 0, then x n − y n → 0. Every LUR Banach space is MLUR. Recall, too, that a Banach space X has property (H) (sometimes also called Kadec-Klee property) if every sequence in S X that w-converges to a point x in S X converges (to x). As it is well known, every LUR space has property (H). A Banach space X is rotund (also called strictly convex) if every point x ∈ S X is extremal. Obviously, every MLUR space is rotund. Let C be a non-empty subset of a Banach space X and let x * ∈ S X * bounded above on C. We denote S(C, x * , δ) the δ-section defined by x * in C, i.e., S(C, x * , δ) := {x ∈ C; x, x * ≥ sup C x * − δ}.

Approximation of the limit distance function in Banach spaces

Journal of Mathematical Analysis and Applications, 2007

In this paper we study the behavior of the limit distance function d(x) = lim dist(x, C n) defined by a nested sequence (C n) of subsets of a real Banach space X. We first present some new criteria for the non-emptiness of the intersection of a nested sequence of sets and of their ε-neighborhoods from which we derive, among other results, Dilworth's characterization [S.J. Dilworth, Intersections of centred sets in normed spaces, Far East J. Math. Sci. (Part II) (1988) 129-136 (special volume)] of Banach spaces not containing c 0 and Marino's result [G. Marino, A remark on intersection of convex sets, J. Math. Anal. Appl. 284 (2003) 775-778]. Passing then to the approximation of the limit distance function, we show three types of results: (i) that the limit distance function defined by a nested sequence of non-empty bounded closed convex sets coincides with the distance function to the intersection of the weak *-closures in the bidual; this extends and improves the results in [J.M.F. Castillo, P.L. Papini, Distance types in Banach spaces, Set-Valued Anal. 7 (1999) 101-115]; (ii) that the convexity condition is necessary; and (iii) that in spaces with separable dual, the distance function to a weak *-compact convex set is approximable by a (non-necessarily nested) sequence of bounded closed convex sets of the space.

Seminar about the Bounded Approximation Property in Fréchet Spaces

2020

The purpose of this seminar, which was presented at the Universitat Politecnica de Valencia in late 2012, is to explain several results concerning the bounded approximation property for Frechet spaces. We give a full detailed proof of an important result due to Pelczynski that asserts that every separable Frechet space with the bounded approximation property is isomorphic to a complemented subspace of a Frechet space with a Schauder basis. We also explain Vogt's example of a nuclear Frechet space without the bounded approximation property. This example is simpler than the original counterexample due to Dubinski. These examples solved a long standing problem of Grothendieck. Vogt obtained later another simple example of a nuclear Frechet function space without the bounded approximation property. The relation of the bounded approximation property for Frechet spaces with a continuous norm and the countably normable spaces, including several results due to Dubinski and Vogt, is also...

3 on the Bounded Approximation Property in Banach Spaces

2016

We prove that the kernel of a quotient operator from an L 1-space onto a Banach space X with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky-case ℓ 1-and Figiel, Johnson and Pe lczyński-case X * separable. Given a Banach space X, we show that if the kernel of a quotient map from some L 1-space onto X has the BAP then every kernel of every quotient map from any L 1-space onto X has the BAP. The dual result for L∞-spaces also hold: if for some L∞-space E some quotient E/X has the BAP then for every L∞-space E every quotient E/X has the BAP.

On the Banach-Mazur distance between continuous function spaces with scattered boundaries

Czechoslovak Mathematical Journal, 2023

We study the dependence of the Banach-Mazur distance between two subspaces of vector-valued continuous functions on the scattered structure of their boundaries. In the spirit of a result of Gordon [16], we show that the constant 2 appearing in the Amir-Cambern theorem may be replaced by 3 for some class of subspaces. This we achieve by showing that the Banach-Mazur distance of two function spaces is at least 3, if the height of the set of weak peak points of one of the spaces is larger than the height of a closed boundary of the second space. Next we show that this estimate can be improved, if the considered heights are finite and significantly different. As a corollary, we obtain new results even for the case of C(K, E) spaces.

Approximation property in Banach Spaces.pdf

The notion of a basis in a Banach space was introduced by J. Schauder in 1927. If a Banach space has a basis then it is also separable. The problem whether every separable Banach space has a Schauder basis appeared for the first time in 1931 in Banach’s book "Theory of Linear Operations". It was clear to Banach, Mazur and Schauder that this question was related to the "approximation problem". This is mentioned in Banach’s book as a remark to the chapter on compact operators. If a Banach space X has a Schauder basis it also has the approximation property, since the natural projections of X onto its finite dimensional subspaces form a bounded sequence of finite rank operators converging pointwise on X to the identity operator. The approximation problem is equivalent to whether a Banach space has the "approximation property". A Banach space X has the approximation property if the identity operator on X is the limit for the topology of uniform convergence on compact subsets of X of a sequence of finite rank operators. The approximation property in its various forms was thoroughly analyzed by A. Grothendieck in his thesis "Produits tensoriels topologiques et espaces nucléaires" published in 1955. But it was not until 1972, that P. Enflo solved both questions in the negative. He found a subspace of co, which he showed does not have the approximation property and consequently does not have a basis. Almost immediately, T. Figiel and A. M. Davie greatly simplified his proof. The later using a probabilistic lemma constructed a separable closed subspace of l1 without the approximation property. Moreover, they also showed that both co and every lp space with 1  p < 1 , (p 6= 2) has subspaces without the approximation property. In this work we present some of the equivalent properties to the approximation property due to A. Grothendieck, and we make a detailed exposition of the proof by A. M. Davie.

On the bounded approximation property in Banach spaces

Israel Journal of Mathematics, 2013

We prove that the kernel of a quotient operator from an L 1 -space onto a Banach space X with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky -case ℓ 1 -and Figiel, Johnson and Pe lczyński -case X * separable. Given a Banach space X, we show that if the kernel of a quotient map from some L 1 -space onto X has the BAP then every kernel of every quotient map from any L 1 -space onto X has the BAP. The dual result for L∞-spaces also hold: if for some L∞-space E some quotient E/X has the BAP then for every L∞-space E every quotient E/X has the BAP.

Approximation Property in Banach Spaces

The notion of a basis in a Banach space was introduced by J. Schauder in 1927. If a Banach space has a basis then it is also separable. The problem whether every separable Banach space has a Schauder basis appeared for the first time in 1931 in the Polish edition of Banach’s book "Theory of Linear Operations". It was clear to Banach, Mazur and Schauder that this question was related to the "approximation problem". This is mentioned in Banach’s book as a remark to the chapter on compact operators. If a Banach space X has a Schauder basis it also has the approximation property, since the natural projections of X onto its finite dimensional subspaces form a bounded sequence of finite rank operators converging pointwise on X to the identity operator. The approximation problem is equivalent to whether a Banach space has the "approximation property". A Banach space X has the approximation property if the identity operator on X is the limit for the topology of uniform convergence on compact subsets of X of a sequence of finite rank operators. The approximation property in its various forms was thoroughly analyzed by A. Grothendieck in his thesis "Produits tensoriels topologiques et espaces nucléaires" published in 1955. But it was not until 1972, that P. Enflo solved both questions in the negative. He found a subspace of co, which he showed does not have the approximation property and consequently does not have a basis. Almost immediately, T. Figiel and A. M. Davie greatly simplified his proof. The later using a probabilistic lemma constructed a separable closed subspace of l1 without the approximation property. Moreover, they also showed that both co and every lp space with 1  p < 1 , (p 6= 2) has subspaces without the approximation property. In this work we present some of the equivalent properties to the approximation property due to A. Grothendieck, and we make a detailed exposition of the proof by A. M. Davie.

Realcompactness and Banach-Stone theorems

2000

For realcompact spaces X and Y we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on X and Y, where special attention is paid to spaces of vector-valued bounded continuous functions. These results are applied to describe the linear isometries between spaces of vector-valued bounded continuous and uniformly continuous functions.

On the near differentiability property of Banach spaces

Journal of Mathematical Analysis and Applications, 2006

Let μ be a scalar measure of bounded variation on a compact metrizable abelian group G. Suppose that μ has the property that for any measure σ whose Fourier-Stieltjes transformσ vanishes at ∞, the measure μ * σ has Radon-Nikodým derivative with respect to λ, the Haar measure on G. Then L. Pigno and S. Saeki showed that μ itself has Radon-Nikodým derivative. Such property is not shared by vector measures in general. We say that a Banach space X has the near differentiability property if every X-valued measure of bounded variation shares the above property. We prove that Banach spaces with the Radon-Nikodým property have the near differentiability property, while Banach spaces with the near differentiability property enjoy the near Radon-Nikodým property. We also show that the Banach spaces L 1 [0, 1] and L 1 /H 1 0 have the near differentiability property. Lastly, we show that Banach spaces with the near differentiability property have type II-Λ-Radon-Nikodým property, whenever Λ is a Riesz subset of type 0 of G.