A NOTE ON L²-SUMMAND VECTORS IN DUAL SPACES

Abstract

Abstract. It is shown that every L2 -summand vector of a dual real Banach space is a norm-attaining functional. As consequences, the L 2 -summand vectors of a dual real Banach space can be determined by the L 2 -summand vectors of its predual; for every n ∈ ‫,ގ‬ every real Banach space can be equivalently renormed so that the set of norm-attaining functionals is n-lineable; and it is easy to find equivalent norms on non-reflexive dual real Banach spaces that are not dual norms.2000 Mathematics Subject Classification. Primary 46B20, 46C05, 46B04. Introduction and background.A vector e of a real Banach space X is said to be an L 2 -summand vector if there exists a closed vector subspace M of X such that X = ‫ޒ‬e ⊕ 2 M; in other words, λe + m 2 = λe 2 + m 2 for every λ ∈ ‫ޒ‬ and every m ∈ M. If e = 0, then the functional e * ∈ X * such that e * (e) = 1 and M = ker(e * ) is called the L 2 -summand functional associated to e. It satisfies e * =

Vector Subspaces of the Set of Non-Norm-Attaining Functionals

“…Before discussing the solution to the previous question, let us note that, as indicated in the next results, not every equivalent norm on a dual Banach space is a dual norm (see, for instance, [9, p. 27]). On this topic, in [2] the following result is shown. THEOREM 3.3 [2].…”

“…If x * ∈ X * is an L 2 -summand vector, then x * ∈ NA(X ). The previous theorem reveals another way to prove that there are always equivalent norms on nonreflexive dual Banach spaces that are not dual norms (see, again, [2]). COROLLARY 3.4 [2].…”

On the Existence of Non-Norm-Attaining Operators

“…As we have mentioned in the Introduction, these properties were already studied in norm-attaining framework. For instance, the interested reader may consult [2,5,9,27,33,34,47]. For more on the property of (α, β)-spacebility applied to families of operators, see [22,23].…”