Tagged Questions

3
votes
1answer
77 views

How to construct an “explicit” element of $(\ell^\infty(\mathbb N))^* \setminus \ell^1(\mathbb N)$? [duplicate]

Possible Duplicate: Nonnegative linear functionals over $l^\infty$ An explicit functional in $(l^\infty)^*$ not induced by an element of $l^1$? Everything is in the title: How to ...
3
votes
0answers
56 views

Application of a result on some bounded functionals on a subspace of $C([0,1])$

The following result was proved in a previous post: Bounded functionals on Banach spaces. Let $(X, \|.\|)$ be a Banach space such that $X \subset C([0,1]) $ For every $r\in \mathbb{Q}\cap[0,1], ...
6
votes
1answer
231 views

Rainwater theorem, convergence of nets, initial topology

I've stumbled upon a result called Rainwater's theorem a few times, it seems to be a very useful result in connection with weak convergence in Banach spaces. Rainwater's theorem. Let $X$ be a ...
9
votes
2answers
174 views

If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?

Let $V,W$ be two normed vector spaces (over a field $K$). Then their product $V \times W$ with the norm $\|(x,y)\| = \|x\|_V + \|y\|_W$ is a normed space. Using this norm it's easy to show that if ...
2
votes
1answer
55 views

A question about weakening the conditions of Schauder's fixed point theorem

I'm currently doing a course on the theory of metric spaces. This is the version of Schauder fixed point theorem from my course: Let $(X,\|\cdot\|)$ be Banach and $C\subset X$ a closed, bounded, ...
2
votes
1answer
86 views

completeness of cones in an ordered normed space

Let $(X,\|\cdot\|,\le)$ be a normed, ordered vector space over $\mathbb{R}$ and let $X^+=\lbrace x\in X:x\ge0\rbrace$ denote the (positive) cone in $X$. with a metric $d$ induced by the norm ...
1
vote
1answer
296 views

Convex hull of the extreme points of the unit ball in $C(K)$ where $K$ is the Cantor set.

Let $K$ denote the usual $1/3$ Cantor set and let $B=B_{C(K)}$ (here $B_{C(K)}$ = {$v \in C(K) : \|v\| \leq 1 $} denotes the closed unit ball of $C(K))$. Then how to prove that $B$ coincides with the ...
5
votes
1answer
338 views

Exhibiting open covers with no finite subcovers.

How do I exhibit an open cover of the closed unit ball of the following: (a) $X = \ell^2$ (b) $X=C[0,1]$ (c) $X= L^2[0,1]$ that has no finite subcover?
4
votes
1answer
113 views

Bidual of a WSC space

Let $E$ be a Banach space which is weakly sequentially complete (i.e. each weak Cauchy sequence converges weakly). Must $E^{**}$ be weakly sequentially complete either? Of course, this question is ...
8
votes
3answers
334 views

Is there an easy example of a vector space which can not be endowed with the structure of a Banach space

Let $V$ be a real vector space. Is there always a norm on $V$ such that $V$ is complete with respect to this norm? If not, is there an easy counterexample?
4
votes
1answer
502 views

Reflexive space which is not uniformly convex

I found this beautiful theorem (Milman-Pettis): Every uniformly convex Banach space is reflexive I think it's a remarkable statement, since uniformly convexity is a geometric property of the norm ...
6
votes
1answer
350 views

Isometric to Dual implies Hilbertable?

Let $X$ be a Banach space and suppose that $X$ is isometric to its continuous dual space $X^*$. Must $X$ be hilbertable in the sense that there exists an inner product which induces the norm on $X$? ...
41
votes
3answers
1k views

Is it possible for a function to be in $L^p$ for only one $p$?

I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain). One can use interpolation to show that ...
11
votes
1answer
435 views

Does separability follow from weak-* sequential separability of dual space?

Let $E$ be a Banach space. Suppose that $E'$ is weakly-* sequentially separable, that is, that there exists a countable $D \subset E'$ s.t. every $x' \in E'$ is a limit point of a ...