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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

3 votes
1 answer
174 views

Sufficient condition for asymptotic-$\ell_{p}$ in terms of spreading models?

Let $(X,\|\cdot\|)$ be a Banach space with a Schauder basis and fix $p\in[1,\infty]$. Suppose that $X$ is asymptotic-$\ell_{p}$ with respect to this basis. It is known that the closed linear span of e …
JWP_HTX's user avatar
  • 201
1 vote
1 answer
92 views

Definition question: asymptotic-$\ell_{p}$ versus coordinate-free asymptotic-$\ell_{p}$

Let $(e_{j})_{j=1}^{\infty}$ be a basis for the Banach space $X$. If there exist constants $\zeta_{1},\zeta_{2}>0$ such that for all $N\in\mathbb{N}$, \begin{equation*} \zeta_{1}\left(\sum_{i=1}^{N}\| …
JWP_HTX's user avatar
  • 201
2 votes
0 answers
88 views

Another question about asymptotic models in Banach spaces

The array $(x_{i}^{k})_{i=1,k\in\mathbb{N}}^{\infty}$ of normalized $M$-basic sequences in a Banach space $X$ is itself called $M$-basic if, for every $k\leq i_{1}<i_{2}<\ldots$, the diagonal sequence …
JWP_HTX's user avatar
  • 201
1 vote
1 answer
163 views

Asymptotic models and passing to sub-arrays

If, for all $k\in\mathbb{N}$, $(x_{i}^{k})_{i=1}^{\infty}\in X^{\mathbb{N}}$ is normalized and $M$-basic and if, in addition, for all $k\leq i_{1}<i_{2}<\ldots$ the diagonal sequence $(x_{i_{k} }^{k}) …
JWP_HTX's user avatar
  • 201
6 votes
1 answer
232 views

Radon-Nikodym property in Diestel & Uhl: a definition clarification

I posted the following question on MSE originally because, not being research-level, it seemed more appropriate for that site. However, there was no activity for it on MSE and I feel that it certainly …
JWP_HTX's user avatar
  • 201
4 votes
1 answer
272 views

Name for certain property of equivalent norms on finite-dimensional subspaces of a Banach space

Let $X=(X,\|\cdot\|)$ be a Banach space and suppose that $F\subset X$ is a finite-dimensional subspace. There is then an equivalent norm $|\cdot|$ on $F$ such that $|\cdot|$ is induced by an inner pro …
JWP_HTX's user avatar
  • 201
2 votes
0 answers
126 views

Fourier type of asymptotic-$\ell_{2}$ Banach spaces

A Banach space $X$ is said to have Fourier type $p\in[1,2]$ if the Fourier transform $\hat{f}(s):=\int_{\mathbb{R}}e^{-ist}f(t)dt$ defines a bounded linear operator from $L_{p}(\mathbb{R},X)$ to $L_{p …
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  • 201