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The set of elementary tensors is weakly closed in projective tensor products

Colin Petitjean

TL;DR

The paper proves that the set of elementary tensors is weakly closed in the projective tensor product $X\widehat{\otimes}_\pi Y$ when one factor has the approximation property, and derives a convergence corollary for weakly null sequences. The authors then apply this result to vector-valued Lipschitz free spaces, establishing the weak closure of delta-like sets and providing conditions under which natural preduals for $\mathcal{F}(M,X^*)$ arise. The methods combine a rank-argument in the operator representation of tensors with intersections of weakly closed sets, and extend scalar predual results to the vector-valued setting under AP/RNP-type hypotheses. Altogether, the work clarifies the structure of vector-valued Lipschitz free spaces and their duals, with implications for Lipschitz operator theory and tensor product geometry.

Abstract

In this short note, we prove that the set of elementary tensors is weakly closed in the projective tensor product of two Banach spaces. As a result, we are able to answer a question from the literature proving that if $(x_n) \subset X$ and $(y_n) \subset Y$ are two weakly null sequences such that $(x_n \otimes y_n)$ converges weakly in $X \widehat{\otimes}_πY$, then $(x_n \otimes y_n)$ is also weakly null.

The set of elementary tensors is weakly closed in projective tensor products

TL;DR

The paper proves that the set of elementary tensors is weakly closed in the projective tensor product when one factor has the approximation property, and derives a convergence corollary for weakly null sequences. The authors then apply this result to vector-valued Lipschitz free spaces, establishing the weak closure of delta-like sets and providing conditions under which natural preduals for arise. The methods combine a rank-argument in the operator representation of tensors with intersections of weakly closed sets, and extend scalar predual results to the vector-valued setting under AP/RNP-type hypotheses. Altogether, the work clarifies the structure of vector-valued Lipschitz free spaces and their duals, with implications for Lipschitz operator theory and tensor product geometry.

Abstract

In this short note, we prove that the set of elementary tensors is weakly closed in the projective tensor product of two Banach spaces. As a result, we are able to answer a question from the literature proving that if and are two weakly null sequences such that converges weakly in , then is also weakly null.
Paper Structure (4 sections, 8 theorems, 29 equations)

This paper contains 4 sections, 8 theorems, 29 equations.

Table of Contents

  1. Weak convergence in projective tensor product
  2. Applications to vector-valued Lipschitz free spaces
  3. Weak closure of delta(M,X).
  4. Natural preduals

Key Result

Lemma 1.1

Let $X,Y$ be Banach spaces such that $X$ or $Y$ has the AP. Let $T \in X \widehat{\otimes}_\pi Y$. Then $T \in \mathcal{T}$ if and only if for every linearly independent families $\{x_1^*,x_2^* \} \subset X^*$ and $\{y_1^{*},y_2^{*} \} \subset Y^{*}$ we have:

Theorems & Definitions (17)

  • Lemma 1.1
  • proof
  • Proposition 1.2
  • proof
  • Corollary 1.3
  • proof
  • Lemma 1.4
  • Corollary 1.5
  • proof
  • Definition 2.1: Vector-valued Lispschitz free spaces
  • ...and 7 more