All Questions
Tagged with fa.functional-analysis topological-vector-spaces
213 questions
23
votes
8
answers
8k
views
Grothendieck on topological vector spaces
In a short biography article on Alexander Grothendieck, it is mentioned that after Grothendieck submitted his first thesis on topological vector spaces (TVS), apparently, he told Bernard Malgrange ...
6
votes
3
answers
1k
views
Topological vector spaces that are isomorphic to their duals
After reviewing the (locally convex)
topological vector spaces that I know,
the only examples I could find where there is an isomorphism from the
space to its (anti)dual, are Hilbert spaces.
So my ...
- 357
3
votes
2
answers
1k
views
Sequences of linear combinations of measures
Let $X$ be a Polish space. Let $J\in\mathbb{N}$.
Let $\lbrace a^n_1\rbrace_n,\dots,\lbrace a^n_J\rbrace_n$ be $J$ sequences of reals.
Let $\lbrace \mu^n_1\rbrace_n,\dots,\lbrace \mu^n_J\rbrace_n$ be ...
user12713
12
votes
1
answer
468
views
Status of the compact AR problem?
The so-called "compact AR Problem" reads:
Is every compact convex set in a metrizable topological vector space an absolute retract?
It is open according to the chapter by T. Banakh, R. Cauty and ...
- 5,575
5
votes
1
answer
510
views
The space $H(D)$ of holomorphic functions.
A very natural example of a nuclear Montel space is the space $H(D)$ of all holomorphic functions on the open disc topologized by the family of seminorms
$$p_n(f)=\sup\{|f(z)|\colon |z|\leq 1-\tfrac{...
- 51
1
vote
1
answer
961
views
Strong topology
Let $E$ and $F$ be a locally convex topological vector spaces (LCS) and let $E^{\star}$ and $F^{\star}$ denote the strong duals of $E$ and $F$, respectively.
A dual of $E^{\star}$ given by the $\beta(...
- 145
-3
votes
2
answers
768
views
Question on Linear Operators
Let $V$ be a normed infinite dimensional vector space. Let $L: V \longrightarrow V$ be a bounded linear operator. Moreover assume that $L$ is 'locally nilpotent' that is:
$$ \forall v \in V \quad \...
- 741
1
vote
1
answer
342
views
Compact sets in TVS
Let $K$ be a compact subset of a Hausdorff topological vector space. Is it true that
$\bigcap_{n\in \mathbb{N}}\frac{1}{n}K$ is either empty of or is a set consisting of the origin only?
- 11
2
votes
3
answers
538
views
General theory for p-normed spaces
Hello,
in functional analysis and operator theory, you encounter several (at first glance, at least) similar constructions of normed spaces that can be indexed with some $ p \in [1,\infty]$, and ...
- 5,327
21
votes
3
answers
3k
views
Can you tell whether a space is Banach from the unit ball?
Let $V$ be a real vector space. It is well known that a subset $B\subset V$ is the unit ball for some norm on $V$ if and only if $B$ satisfies the following conditions:
$B$ is convex, i.e. if $v,w\...
- 8,493
5
votes
3
answers
843
views
Topology on the set of linear subspaces
Hello,
let $X$ be a seperable Hilbert space. Let $(e_i)_i$ be a Hilbert basis, and for each index let $E_i = \langle e_1,\dots,e_i \rangle \subset X$ the span of the first $i$ basis vectors. For any $...
- 5,327
4
votes
4
answers
1k
views
An example of a non-paracompact tvs (over the reals, say)
What is an example of a non-paracompact topological vector space?
I'm aware of this question, but I don't care if my tvs is locally convex. In fact the wilder the better. The only criterion is that ...
- 35.5k
6
votes
3
answers
2k
views
Sequential topological vector spaces
Since I'm dealing with the distinction between sequential continuous and continuous maps at the moment I came to ask myself once again what can be said about spaces where these two notions agree (...
- 9,650