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arXiv · 1710.07852

Nonlinear weakly sequentially continuous embeddings between Banach spaces

Bruno de Mendonça Braga
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Abstract

In these notes, we study nonlinear embeddings between Banach spaces which are also weakly sequentially continuous. In particular, our main result implies that if a Banach space $X$ coarsely (resp. uniformly) embeds into a Banach space $Y$ by a weakly sequentially continuous map, then every spreading model $(e_n)_n$ of a normalized weakly null sequence in $X$ satisfies \[\|e_1+\ldots+e_k\|_{\overline{\delta}_Y}\lesssim\|e_1+\ldots+e_k\|_S,\] where $\overline{\delta}_Y$ is the modulus of asymptotic uniform convexity of $Y$. Among other results, we obtain Banach spaces $X$ and $Y$ so that $X$ coarsely (resp. uniformly) embeds into $Y$, but so that $X$ cannot be mapped into $Y$ by a weakly sequentially continuous coarse (resp. uniform) embedding.

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Bruno de Mendonça Braga. 2017-10-21. Nonlinear weakly sequentially continuous embeddings between Banach spaces. https://arxiv.org/abs/1710.07852

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