A Banach space whose set of norm-attaining functionals is algebraically trivial
Miguel Martin
TL;DR
This work constructs a smooth renorming of cโ, producing a Banach space ๐ for which the set of norm-attaining functionals NA(๐,โ) is algebraically trivial: its intersection with any two-dimensional subspace of ๐* lies in the union of two lines, so no nontrivial cones exist and no interior segment between two NA functionals remains norm-attaining. The construction uses a dense operator-range in the dual, a carefully defined map R into an โโ-sum W, and a sum norm p that preserves smoothness while annihilating larger NA structures; proximinality and finite-rank operator implications are analyzed, linking to longstanding open problems. The results extend to a family of spaces isomorphic to cโ and highlight limitations for extending NA-based arguments to finite-rank operator density, while offering insights into the geometry of norm attainment and the obstructions posed by algebraic triviality. This work thus advances the understanding of when NA functionals can be forced to avoid rich linear structures, with potential implications for the theory of norm-attaining operators and proximinality in Banach spaces.
Abstract
We construct a Banach space $X$ for which the set of norm-attaining functionals $NA(X,\mathbb{R})$ does not contain any non-trivial cone. Even more, given two linearly independent norm-attaining functionals on $X$, no other element of the segment between them attains its norm. Equivalently, the intersection of $NA(X,\mathbb{R})$ with a two-dimensional subspace of $X^*$ is contained in the union of two lines. In terms of proximinality, we show that for every closed subspace $M$ of $X$ of codimension two, at most four elements of the unit sphere of $X/M$ have a representative of norm-one. We further relate this example with an open problem on norm-attaining operators.