# Tagged Questions

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 convergence on Dual of Reflexive Banach Space

Let $\mathcal{H}$ be a separable Hilbert space and let $\mathfrak{S}_{p}(\mathcal{H})$, $1<p<\infty$, denote the $p$-th Schatten class of compact operators acting on it. Suppose we have a net ...
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### An inequality in product space $V$ [on hold]

I found an inequality as following: Let $x, y, z$ be three complex numbers then: \begin{equation*} \frac{1}{2}(|y+z-x|+|x+z-y| + |y+x-z|) \le |x| + |y|+|z|+\frac{1}{2}|x+y+z| \end{equation*} (1) The ...
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### Extracting subsequences in Banach spaces, along an ultrafilter?

There are various principles in Banach space theory that allow one to pass from a given sequence of vectors $(x_n)$, to a subsequence $(x_{n_k})$ with some desired property. I'm thinking here, in ...
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### Is the $L^1$-space dual to a Banach space

Let $(\Omega,\mu)$ be a measure space. It is well known that for $1<p\leq \infty$ one has the duality $$L^p=(L^{p*})^*,$$ where $1/p+1/p^*=1$. Question. Is it known that the Banach space $L^1$ is ...
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### Weak Topology and Domain Theory: Which topology on the function domain restricts to the weak topology on C([0,1])?

Let $\mathbb{IR}$ be the interval domain over the set $\mathbb{R}$ of real numbers, defined by: $$\mathbb{IR} := \{ [a,b] \mid a, b \in \mathbb{R}, a \leq b\} \cup \{ \mathbb{R}\},$$ and ordered by ...
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### Sequences in $L_{p}(1<p<\infty)$ that is equivalent to the unit vector basis of $l_{p}$ or $l_{2}$

Let $1<p<\infty$. Johnson and Schechtman (Multiplication operators on $L(L_{p})$ and $l_{p}$-strictly singular operators, 2008, DOI: 10.4171/JEMS/141, eudml, arxiv) observed that if $(x_{n})_{n}$...
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### A universal operator between separable Banach spaces

The Banach space $C[0,1]$ is universal for all separable Banach spaces in the sense that for a separable Banach space $X$ there is an isometric isomorphism from $X$ into $C[0,1]$. My question is ...
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### Subspaces of $L_{p}(2<p<\infty)$

Let $p>2$ and $X$ a subspace of $L_{p}$. Then Kadec and Pelczynski proved that either $X$ is isomorphic to $l_{2}$ or $X$ contains a subspace isomorphic to $l_{p}$. Question: if $X$ is ...
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### Weakly null sequences in $L_{p}(1<p<2)$

Let $1<p<2$. Let $(f_{n})_{n}$ be a normalized weakly null sequence in $L_{p}$ such that the sequence $(f_{n})_{n}$ contains no subsequence that is equivalent to the unit vector basis of $l_{p}$....
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### Equivalence of questions regarding restrictions of pure states

In Davidson and Szarek's article "Local Operator Theory, Random Matrices and Banach Spaces" in the Handbook of the Geometry of Banach Spaces, the authors discuss the (now solved) Kadison-Singer ...
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### Set of w*-continuous operators closed for the weak* topology or not?

Let $X$ be a dual Banach space, i.e. $X=(X_*)^*$ for some Banach space $X_*$. Consider the weak* topology of $B(X)$, i.e. the topology of pointwise convergence on $X$ endowed with the $\sigma(X,X_*)$-...
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### A possible norm on a subspace of $C^\infty([0,1])$?

I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again. Take the vector space of infinitely ...
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### Compact non-nuclear operators

I am not sure if this question makes sense, or if it is trivial, but does there exists an infinite dimensional Banach space (necessarily without the approximation property) such that no compact, non-...
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### Two questions on the James $p$-space $J_{p}(1<p<\infty)$

Let $1<p<\infty$. The James $p$-space $J_{p}$ is the Banach space of all sequences of real numbers $(a_{i})_{i}\in c_{0}$ such that \|(a_{i})_{i}\|=\sup\{(\sum_{j=1}^{n}|a_{p_{j-1}}-a_{p_{j}}|...
A theorem of Lindenstrauss and Phelps states that if $X$ is a separable reflexive Banach space then the unit ball of $X$, $Ba(X)$, has uncountably many extreme points. The proof goes by contradiction ...
In a different thread, we stumbled upon the following question: Given a continuous finite measure $w(x)dx$ on some interval $(a,b)$, $-\infty \leq a <b \leq \infty$, and the set of respective ...