3
votes
1answer
153 views

When is the bound in Riesz-Thorin Interpolation Theorem attained?

Let me recall the statement of Riesz-Thorin theorem (see also http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem). Theorem (Riesz-Thorin): Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite ...
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 ...
2
votes
0answers
100 views

Is reflexive Banach space valued scalarwise Lebesgue space isomorphic to the Bochner space?

I first specify the setting and then formulate the question precisely. (A very long post follows.) Definitions 1. For $E$ a (real Hausdorff) locally convex space, say that $E$ is suitable iff there ...
4
votes
1answer
220 views

Approximation of an integral over the unit ball of L_1

For every $\varepsilon>0$ find a piecewise continuous function $q:[0,1]\rightarrow \mathbb{R}$ such that $\int_0^1 q(x)dx=1$ and $$\int_{0}^1 \int_{0}^{s} \left|\frac{q(s)q(t/s)}{s}- ...
7
votes
0answers
242 views

Uniform closure of subspaces of Baire class 1

Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...
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 ...
3
votes
2answers
677 views

How to prove the Hahn-Banach constructively

I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space. Thanks in advance for any helpful answers.
4
votes
2answers
1k views

Baire Category Theorem Application

In Antoine Henrot Michel Pierre - Variation et optimisation de formes, Une analyse geometrique, a book I'm studying I found an interesting problem. The problem is listed below. The first 3 points of ...
5
votes
2answers
271 views

Embeddings of Weighted Banach Spaces

Let be $d$ a positive integer, $\Omega=\mathbb{R}^{\mathbb{Z}^d}$ and fix $R\geq 2$. We define weighted Banach spaces $$ \Omega_p:=\left\{ x\in \Omega\left| ...