Norm-attaining functionals need not contain 2-dimensional subspaces

Abstract

G. Godefroy asked whether, on any Banach space, the set of normattaining functionals contains a 2-dimensional linear subspace. We prove that a recent construction due to C.J. Read provides an example of a space which does not have this property.This is done through a study of the relation between the following two sentences where X is a Banach space and Y is a closed subspace of finite codimension in X:We prove that these are equivalent if X is the Read's space.Moreover, we prove that any non-reflexive Banach space X with any given closed subspace Y of finite codimension at least 2 admits an equivalent norm such that implication (B) =⇒ (A) fails.

Closed linear spaces consisting of strongly norm attaining Lipschitz functionals

“…According to Rmoutil's result [25], there exists an infinite-dimensional Banach space X (namely c 0 in the equivalent norm constructed by Read [24]) such that the set NA(X ) ⊂ X *…”

“…In this section, we will study the existence of n-dimensional linear subspaces in SNA(M), where M is a pointed metric space and n ∈ N. Our main result from the section states that if M contains at least 2 n points (in particular, if M is infinite), then SNA(M) contains an isometric copy of n 1 (see Theorem 1). This provides a shocking contrast when compared to the classical theory of norm-attaining functionals, where Rmoutil showed that an infinite-dimensional Banach space X introduced by Read satisfies that NA(X , R) has no 2-dimensional subspaces (see [25]). In order to prove our main result in this direction, we need a bit of preparatory work.…”

“…Observe that Proposition 4 applies to all normed spaces. This should be once more compared with the classical theory of norm-attaining functionals, where there exist Banach spaces X such that NA(X , R) does not have 2-dimensional subspaces (see [25]).…”

Pre-Schauder Bases in Topological Vector Spaces

“…Example 2. In [13], an equivalent renorming on the Banach space c 0 of sequences converging to 0 is found in such a way that NA(c 0 ) does not contain any vector subspace of dimension strictly greater than 1.…”

Dense lineability and spaceability in certain subsets of ℓ∞$\ell _{\infty }$

“…A subset $$M$$ of a vector space $$X$$ is said to be lineable (resp., $$\kappa$$ ‐lineable , for a cardinal $$\kappa$$) if $$M\cup \lbrace 0\rbrace$$ contains a vector space of infinite dimension (resp., of dimension $$\kappa$$). Lineability problems have been investigated in several areas of Mathematical Analysis; we refer to, for example, [1, 2, 6, 11–13, 24, 25] for a rather non‐exhaustive list of results. Let us just quote here the seminal result of Gurariy [14] that the set of continuous, nowhere differentiable functions is lineable in $$C([0,1])$$.…”