# Questions tagged [banach-spaces]

A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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### Boundary points in $\overline{\operatorname{conv}\{z_i\}_{i\in I}}$

Let $X$ be an infinitely-dimensional Banach space and $\{z_i\}_{i\in I}$ be a set of linearly independent points in $X_{\leq 1}$, the closed unit ball of $X$. $I$ the index set is not necessarily ...
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### Continuity of linear map on tensor product spaces with different norm properties

I originally asked this question on StackExchange, but I think that it may be more suitable to here. Let $V$ and $U$ be Banach spaces. I'm considering a linear map $\phi: V \rightarrow U$, and ...
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### Infinite dimensional homology theory for submanifolds of Hilbert and Banach spaces

Is there a version of homology theory for spaces for which explicitly infinite dimensional "cells" are allowed? The spaces in question include e.g. X = (x: x \in l_2: p_i(x) ...
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### Is there a version of dominated convergence theorem for local $L^p$ spaces?

Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of ...
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### Understanding $k$-rotundity

We know that the definitions of $k$-Uniform rotundity ($k$-UR) or locally $k$- uniform rotundity (L$k$-UR) act to find the volume in higher k-dimensional spaces by defining $V(x_1, ..., x_{k+1})$ (For ...
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### Injective envelopes of 1-extensible spaces

Please read this post as a naive follow up on a previous question. Let $X$ be a Banach space and let $(I(X),\alpha)$ denote its injective envelope (e.g., CohenLacey1969). A low hanging fruit is the ...
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### For $\mu$-a.e. $x \in X$, the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$. Then $(f_n)$ is Cauchy in $L^1 (X \times Y)$

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces, $(E, | \cdot |)$ a Banach space, $S (X)$ the ...
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### A notion of restricted injectivity for Banach spaces

I apologize in advance if this is well-known. Let $X$ be a Banach space. Let's call only for this post that $X$ is self-injective if for every closed subspaces A\subseteq B\subseteq X ...
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