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4
votes
1answer
550 views

Uniform boundedness of an $L^2[0,1]$-ONB in $C[0,1]$

Assume that we have an orthonormal basis of smooth functions in $L^2[0,1]$. Are there useful practical criteria to determine whether the sup-norm of the basis functions has a uniform bound? I am sure ...
9
votes
1answer
308 views

What (classes of) Banach spaces are known to have Schauder basis?

Motivation: I am trying to see for what class of Banach spaces the following result is true: There exists an increasing sequence of finite dimensional subspace {$V_n$} of a Banach space X (with ...
53
votes
2answers
4k views

Norms of Commutators

If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant ...
16
votes
5answers
2k views

Brownian motion and spheres

Consider a Brownian motion on $[0;1]$. A (finite) discrete approximation of this Brownian motion consists of $N$ iid Gaussian random variables $\Delta W_i$ of variance $\frac{1}{N}$: $$ ...
6
votes
1answer
544 views

“Vector bundle” with non-smoothly varying transition functions

I'm working my way through Lang's Fundamentals of Differential Geometry, and when he introduces vector bundles, he states that for finite dimensional bundles, the third axiom is redundant. I'm hoping ...
12
votes
0answers
1k views

Infinite convex combinations in a Banach space

Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds: For any sequence $(x_k)_{k\ge0}$ in $C$, and for any sequence of non-negative real numbers ...
6
votes
2answers
624 views

Direct proof of injectivity of $L_\infty$

I would like to know a simple proof of isometric injectivity of $L_\infty$. The proof I've found in Topics in Banach space theory. F. Albiac, N. Kalton uses two deep result. $L_\infty$ as ...
13
votes
2answers
437 views

A natural center of a convex weakly compact set in Banach space

Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$. Motivation: A lot! For example, in game theory $S$ can be a set of ...
10
votes
1answer
406 views

Quotients of l^infty

Let $M$ be a closed subspace of $l^\infty$. Suppose that the quotient $l^{\infty}/M$ is isomorphic to $l^\infty$. Is it true that $M$ is complemented in $l^\infty$?
4
votes
0answers
254 views

Self-dual finite-dimensional complex normed spaces

Suppose $X$ is a complex normed space of dimension 2 or 3 and $X$ is isometrically isomorphic to its dual. Is $X$ a Hilbert space? Remarks: There are easy counterexamples in the real case, and in ...
5
votes
1answer
258 views

Info about Elton–Odell theorem

Hello everyone, could anyone please tell me where can I find information about the Elton–Odell theorem? It states: For any infinite dimensional Banach space $X$ there is a $q > 1$ so that $X$ ...
4
votes
1answer
183 views

series representation in injective tensor products

All books on tensor products of Banach spaces contain the well-known theorem of Grothendieck that every element of the completed projective tensor product $X \tilde{\otimes}_ \pi Y$ has a ...
1
vote
0answers
258 views

Unambiguous “weak” vector valued $L^{+\infty}$ spaces?

For some time, I have been stuck to the problem to be described as follows. The (perhaps not so commonly known) facts given here are taken from R. E. Edwards' Functional Analysis (Holt, Rinehart and ...
0
votes
0answers
121 views

On ultrafilters

Let $\mathcal{U}$ be a free ultrafilter of $\mathbb{N}$. Is true or false what $(C([0,1],\ell_p)^*)_{\mathcal{U}} = L_1(\mu, \ell_q)$, where $\mu$ is a measure, $C([0,1],\ell_p)^*$ is the dual of ...
0
votes
1answer
109 views

Constructing a continuous matrix valued function

Given $d<k$. Let ${\cal M}_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}_{d\times k}(\mathbb{R})$ is a continuous ...
0
votes
1answer
222 views

Integral in a σ−convex set.

Having had no (proper) answer to this question, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be ...