The study of vector space with a topology which makes the maps which sums two vectors and which multiply a vector by a scalar continuous. It's a natural generalization of normed spaces.
Questions tagged [topological-vector-spaces]
1748 questions
12
votes
2 answers
Topological vector space with discrete topology is the zero space
Hello I have a question about topological vector spaces. To remind the definition of such a space:
A topological vector space is a pair $(X,\tau)$ with $X$ a vector space and $\tau$ a topology on $X$ such that all singeltons are closed sets and the…
Local Base
- 121
11
votes
1 answer
Is the standard structure of a topological vector space on reals unique?
The standard stucture of a topological vector space on reals is this given by the metric d(x,y)=|x-y| on the vector space $\mathbb{R},$ with the field of scalars $\mathbb R$ with standard topology.
I would like to know if we can change the topology…
Bartek
- 6,265
8
votes
3 answers
A surjective linear map into a finite dimensional space is open
I'm in search of different proofs of the following proposition:
$\bf{Proposition}$: Suppose $X$ and $Y$ be topological vector spaces, $\text{dim }Y<\infty$, and $\Lambda:X\to Y$ is a surjective linear map. Then $\Lambda$ is open.
Any and all…
Benji
- 5,880
5
votes
2 answers
Question about Topological Vector Spaces
Let $E$ be a Topological Vector Space and $U$ a bounded set of $E$ with $0\in U$, i.e. given any neighborhood $W$ of the origin, there exist $\alpha>0$ such that $\alpha U\subset W$. Is it true that $$\bigcap_{n=1}^\infty2^{-n}U=\{0\}$$
Note: I have…
Tomás
- 22,559
5
votes
3 answers
Metric linear space and locally convex topological vector space
I was wondering which one is more general, metric linear spaces or locally convex topological vector spaces?
Is a metric linear space a locally convex
topological vector space? Vice versa?
In terms of the number of books and websites that have…
Tim
- 47,382
4
votes
1 answer
Why does the continuity of $(x,y) \rightarrow x-y$ mean the commutativity of a topological group?
The following is a part from P.13 in Topological Vector Spaces (Third Printing Corrected 1971) by H.H.Schaefer
Given a vector space $L$ over a (not necessarily commutative) non-discrete
valuated field $K$ and a topology ${\mathfrak T}$ on $L$,…
Aki
- 1,299
4
votes
1 answer
Continuity of Minkowski functional
I'm working through a proof and the setting is that I have a convex, balanced, open subset $U$ of a topological vector space $V$.
The claim that I can't verify is made briefly: the Minkowski functional, $p_{C} := inf\{t\geq 0 : x\in tC\}$ is…
roo
- 5,598
3
votes
3 answers
Is there a $\gamma>1$ such that $\frac{\binom{4p}{2p-1}}{\binom{4p}{0}+\binom{4p}{1}+\cdots+\binom{4p}{p-1}}>\gamma^{p}$
In a proof of the Larman-Rogers conjecture (there is $\gamma>1$ such that $\chi(\mathbb{R}^{d})>\gamma^d) $ they used that there is a $\gamma>1$ such that $\frac{\binom{4p}{2p-1}}{\binom{4p}{0}+\binom{4p}{1}+\cdots+\binom{4p}{p-1}}>\gamma^{p}$. I do…
fjlkjlkjlkj
- 33
3
votes
1 answer
Translate of a closed set is closed
We assume that $X$ is a topological vector space with a topology $\tau$. I want to show that if $F$ is a closed subset of $X$, then its translate $a+F$ is also closed in $X$. Am I right to say that
$$ (a+F)^c=a+F^c?$$
If this is so, then how can we…
Juniven Acapulco
- 9,654
3
votes
1 answer
Does an indefinite inner product induce a seminorm?
I know that an inner product defined for a vector space induces a norm and the norm induces a metric and the metric will eventually induce a topology. I'm also aware that a seminorm induces a pseudometric and that again induces a pseudometric…
sanaz mat
- 135
2
votes
1 answer
Definition of boundedness in topological vector spaces
From Wikipedia:
Given a topological vector space $(X,τ)$ over a field $F$, $S$ is
called bounded if for every neighborhood $N$ of the zero vector there
exists a scalar $α$ so that $$
S \subseteq \alpha N $$ with $$
\alpha N := \{…
Tim
- 47,382
2
votes
1 answer
Does the projective tensor product obey a tensor-hom adjunction?
Let $X, Y, Z$ be three lctvs. Then https://ncatlab.org/nlab/show/inductive+tensor+product , the first theorem in section 3, tells us that
$$\operatorname{Hom}(X\otimes_{\iota} Y, Z) = \operatorname{Hom}(X, L(Y, Z)_{\sigma}),$$
where $L(Y,…
oggledog
- 300
2
votes
0 answers
A closed set and compact set in topological vector space
I have read the post in the link: A and B disjoint, A compact, and B closed implies there is positive distance between both sets
I wonder that the property in the link above is still correct in topological vector space or not? It means: If $A$ is…
Xiuwei Lee
- 121
2
votes
1 answer
Finite-dimensional subspaces of TVSes
The following result is presented in many sources on topological vector spaces (TVSes):
Any finite-dimensional subspace of a Hausdorff TVS is closed.
However, having had a look at various sources, I've yet to come across one that doesn't seem to…
kahen
- 15,760
2
votes
1 answer
Metrics vs. Norms (Fréchet spaces, Banach spaces, etc.)
While I understand on paper the fundamental differences between Fréchet spaces and Banach spaces, I'm struggling to truly internalize/digest what their respective differences mean in layman's terms.
Fréchet Spaces:
These are locally convex spaces…
Patch
- 4,245