Smooth and polyhedral norms via fundamental biorthogonal systems

Abstract

Let X be a Banach space with a fundamental biorthogonal system and let Y be the dense subspace spanned by the vectors of the system. We prove that Y admits a C ∞ -smooth norm that locally depends on finitely many coordinates (LFC, for short), as well as a polyhedral norm that locally depends on finitely many coordinates. As a consequence, we also prove that Y admits locally finite, σ-uniformly discrete C ∞ -smooth and LFC partitions of unity and a C 1 -smooth LUR norm. This theorem substantially generalises several results present in the literature and gives a complete picture concerning smoothness in such dense subspaces. Our result covers, for instance, every WLD Banach space (hence, all reflexive ones), ℓ ∞ (Γ) spaces for every set Γ, C(K) spaces where K is a Valdivia compactum, duals of Asplund spaces, or preduals of Von Neumann algebras. Additionally, under Martin Maximum MM, all Banach spaces of density ω 1 are covered by our result.

Smooth norms in dense subspaces of $\ell_p(Γ)$ and operator ranges

“…In particular, in the case of ℓ p , i.e., when Γ is countable, we obtain a dense subspace of dimension continuum. This is in sharp contrast with the results in [4,5,9,19], where the dense subspaces had dimension equal to the density character of the Banach space X . Comparing this result with [9], it seems conceivable to conjecture that for every separable Banach space X there is a dense subspace Y of dimension continuum and with a C ∞ -smooth norm.…”

“…The present paper is a continuation of the research of the authors [4,5,12], dedicated to the study of smoothness in (incomplete) normed spaces. The main question that we face in this ongoing project is the following: given a Banach space X and k ∈ N ∪ {∞, ω} is there a dense subspace Y of X such that Y admits a C k -smooth norm?…”

“…In particular, for a separable normed space X the existence of a C 1 -smooth norm does not imply that X * is separable, a result that is possibly surprising at first sight. Our goal in [4,5] was to push such a theory to the non-separable context and it culminated in the main result of [5] asserting that every Banach space with a fundamental biorthogonal system has a dense subspace with a C ∞ -smooth norm (let us also refer to the same paper for a more thorough introduction to the subject).…”

“…Moreover, if |||•||| is a norm on ℓ p (Γ) that coincides with the C ∞ -smooth and LFC one on Y p , then it is standard to verify (see, e.g., the proof of [5,Corollary 3.5]) that the norm |||•||| is C ∞ -smooth and LFC at every point of Y p (as a function on ℓ p (Γ)). In lineability terms, this assertion can be restated as stating that the set of points in ℓ p (Γ) where the norm |||•||| is C ∞ -smooth and LFC is maximal densely lineable in ℓ p (Γ).…”