# 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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### Strong differentiability and the inverse function theorem in Banach spaces

I am trying to prove the strong differentiability version of the Inverse Function Theorem for Banach spaces, but I am not sure if it is true. I am interested in this because it is a kind of punctual ...
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### Banach spaces whose biduals are $L_{1}$

Let $X$ be a Banach space. If $X^{**}$ is linearly isometric to $L_{1}(\mu)$ for some $\sigma$-finite measue $\mu$, we shall say that $X$ is an $L_{1}$-pre-bidual. Question 1. What are the examples of ...
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### Characterization of differentiability

For a normed space $(V, \lVert\cdot\rVert_V)$ let us define: \begin{equation} \forall x, y \in V \quad [0,1] \mapsto \gamma_x^y (t) = (1-t)x + ty. \end{equation} I would like to ask whether the ...
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### Banach spaces whose second conjugates are separable

It was known that the James space $J$ has separable second conjugate, is non-reflexive and isometric to its second conjugate. I want to know whether there are Banach spaces $X$ with separable second ...
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### On weakly precompact sets

Recall that a bounded subset $A$ of a Banach space $X$ is said to be weakly precompact if every sequence in $A$ admits a weakly Cauchy subsequence. Rosenthal's $l_{1}$-theorem states that a bounded ...
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### A characterization of the Dunford-Pettis property

A Banach space $X$ is said to have the Dunford-Pettis property if for any Banach space $Y$ every weakly compact operator $T:X\rightarrow Y$ is completely continuous. Recall that $T$ is completely ...
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### Leray-Schauder degree in Banach manifolds

The so called Leray-Schauder degree is usually defined for maps of the form $I - f$, where $f: X \to X$ is a compact map defined on a Banach space. Is there an extended definition for the setting of ...
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### Surjective linear isometries on $\ell_\infty(\mathbb{N})$

In Volume 1 of "Classical Banach Spaces" Lindenstrauss and Tzafriri note that all surjective linear isometries on $\ell_\infty$ are of the from $(a_i) \mapsto (\varepsilon_i a_{\pi(i)})$ ...
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### Trying to understand construction of $C^*$-algebra corresponding to a ternary $C^*$-ring from a paper

Recall that a ternary $C^*$-ring is a complex Banach space $X$, equipped with a associative ternary product $[.,.,.]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle ...
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### Uniform boundedness principle for almost surely converging sequence of operators

I'd like to do the following: I consider a separable Banach space $X$ with a probability measure $\mu$ on the Borel $\sigma$-algebra $\mathcal B(X)$. Additionally, I have a sequence of measurable, ...
A Banach space $X$ has Gordon-Lewis local unconditional structure (G.L. l. u. st.) if for every finite dimensional subspace $E$ of $X$, the inclusion operator $i:E\to X$ factors through a finite ...
### $K$-convex Banach spaces
Let $X$ be a Banach space. We say that $X$ contains $\ell_1^n$'s uniformly iff for all $n\in\mathbb N$ there exist subspaces $X_n\subseteq X$ with $d(X_n,\ell_1^n)\leq \lambda$ for some $\lambda\geq 1$...