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Results tagged with fa.functional-analysis
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user 11081
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
6
votes
Accepted
separability of a certain space of continuous functions, II
No. Ady's construction still works, I think. Here's another, easier one:
Choose an orthonormal basis $\{e_{n}\}$ of $H$. Since $\|e_{i} - e_{j}\| = \sqrt{2}$, the continuous functions $f_{n}(x) = \ma …
- 5,743
34
votes
Accepted
Does there exist an isometry between $L^p$ and $l^p$?
Variants of this question show up often enough here on MO and over at math.SE that it seems worthwhile to collect some facts and links. I say isomorphic for linearly homeomorphic and isometric for iso …
- 5,743
3
votes
Accepted
Projective Banach spaces
You essentially answered your first question yourself: the ground field is a (contractive) retract of any nonzero Banach space by Hahn-Banach. If there were a non-zero projective Banach space in your …
- 5,743
44
votes
1
answer
4k
views
Example of a compact set that isn't the spectrum of an operator
This question is somewhat ill-posed (due to the word easy) and is triggered by idle curiosity:
Is there an easy example of a (separable, infinite-dimensional) Banach space $X$ and a nonempty compa …
- 5,743
5
votes
Accepted
On a decomposition of L^1(G)
I'm answering Yemon's version of the question.
The answer is trivially yes for discrete $G$ since $\ell^1(G) \subset \ell^2(G)$, so let me focus on the non-discrete case.
The first observation to ma …
- 5,743
3
votes
Does a log-concave function on a convex set extend continuously to the boundary?
I do not see how $\log$-concavity should imply any form of continuity. For instance, if $\|\cdot\|$ is any semi-norm on the locally convex space $X$ then $f(x) = e^{-\|x\|}$ will be bounded and $\log$ …
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41
votes
Accepted
Why the name 'separable' space?
As far as I know the word separable was introduced by M. Fréchet in Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo 22 (1906), 1-74. The paper can be obtained via this link (Spring …
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6
votes
Accepted
Characterizations of amenable groups which use the space $\ell_1(G)$ and convolution
I like the following characterization, due to Kaimanovich and Vershik (conjectured by Furstenberg):
A (countable) group is amenable if and only if there is an everywhere positive $\ell^{1}$-functi …