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

13 votes
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Does every Banach space admit a continuous (not necessarily equivalent) strictly convex norm?

If $\|\cdot\|_1$ is a continuous strictly convex norm on $(X,\|\cdot\|_0)$, then $\|x\|_2=\|x\|_0 + \|x\|_1$ defines an strictly convex norm on $X$ equivalent to $\|\cdot\|_0$. Therefore, a space like …
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12 votes
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When is the closed unit ball in a smaller Banach space closed in a larger Banach space?

Suppose that you have two Banach spaces $X$ and $Y$, and a (bounded) operator $TX:\to Y$. The operator $T$ is called semi-embedding if $T$ is injective and $T(B_X)$ is closed in $Y$. So your are aski …
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10 votes

Ideal of strictly singular operators

[J. Lindenstrauss and L. Tzafriri. Classical Banach spaces I. Sequence spaces. Springer 1977]. In page 76, after Prop. 2.c.3, it says that the proof of 2.c.3 shows that an operator $T:\ell_p\to\ell_p$ …
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7 votes

reflexive banach space

Several geometric properties equivalent to non-reflexivity for a Banach space were studied by R.C. James in "Some self-dual properties of normed linear spaces". Ann. of Math. Studies 69 (1972), 159-17 …
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7 votes
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Are nuclear operators closed under extensions?

The answer is no: you can even have $T_1=T_3=0$ and $T_2$ equal to the identity $id$ on an infinite dimensional Banach space. Indeed, consider the following commutative diagram with exact rows: $$\ …
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On hereditarily reflexive Banach spaces

The question has a negative answer: Following the idea in Bill Johnson's comment, I looked at the work of Argyros. In this paper (see the reference below), there are several examples of hereditarily i …
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6 votes

When is it $C(X)$?

This is a detailed version of a previous comment that gives a partial answer. If $K$ is a compact space and $C(K)$ is isometrically isomorphic to $X^{**}$, then $X$ is a $\mathcal{L}_{\infty}$-space. …
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4 votes

Extracting subsequences in Banach spaces, along an ultrafilter?

A (non-trivial) ultrafilter $\mathcal{U}$ on $\mathbb{N}$ is a p-point if and only if every bounded sequence $(a_n)$ of real numbers contains a convergent subsequence $(a_{n_k})$ such that $\{n_k\}\in …
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4 votes
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Sum of subspaces is closed iff inclination is positive

You can find this result in the book of T. Kato. Perturbation theory for linear operators. Springer 1980, 1995. In Theorem IV.4.2, page 219.
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3 votes

Embedding of a Banach space in $\ell_\infty(\Gamma)$ with a subspace embedded in a copy of $...

In this reference (Definition 2.47) a Banach space $X$ is called separably automorphic if given a separable space $Y$ and isomorphic copies $A$ and $B$ of $Y$ in $X$, every bijective operator $T:A\to …
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2 votes

Duals of ideals of operators between Banach spaces

An operator $T:X\to Y$ is Banach-Saks if for every bounded sequence $(x_n)$ in $X$ there is a subsequence $(Tx_{n_k})$ such that the Cesàro means $N^{-1}(\sum_{k=1}^N Tx_k)$ from a norm-convergent sub …
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2 votes
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Duals of ideals of operators between Banach spaces

Let $\mathcal{K}$, $\mathcal{W}$ and $\mathcal{C}$ denote the compact, weakly compact and completely continuous operators, respectively, and let $\mathcal{W}^{-1}\circ \mathcal{K}$ denote the operator …
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2 votes

What is the structure of a Banach space $X$ when $Y$ and $X/Y$ are hereditarily indecomposable?

There exists a separable reflexive Banach space $\hat X^*$ which is indecomposable but not HI, and admits a subspace $Y$ such that both $Y$ and $\hat X^*/Y$ are HI. This example provides a negative an …
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1 vote
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Two measures of noncompactness of operators

The answer is yes. Indeed, let us denote $Y_h=\ell_\infty(B_{Y^*})$. For every $L\in\mathcal{K}(X,Y_h)$, $\|T\|_m=\|JT\|_m=\|JT-L\|_m\leq \|JT-L\|$. Hence $\|T\|_m\leq \|JT\|_e$. Conversely, since …
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0 votes
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Tauberian operators

You can proceed as follows. Let $(x_n^{**})\in \ell_2(X^{**})$. Then $$(x_n^{**})\in T^{**-1}(\ell_2(X))\Rightarrow T^{**}(x_n^{**})= (\frac{x_n^{**}}{n})\in \ell_2(X),$$ hence $(x_n^{**})\in \ell_2 …
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