Questions tagged [topological-vector-spaces]
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.
1,749
questions
3
votes
2
answers
291
views
Motivation behind locally convex spaces, seminorms, and Frechet spaces
I am looking for some motivation behind the definition of locally convex spaces, seminorms, and Frechet spaces. Since all three concepts are related I have grouped them as one question. I am familiar ...
- 46.1k
0
votes
0
answers
62
views
Countable family of norms and seminorms
In volume 4 of Gelfand and Vilenkin they study countably normed spaces. I assume a definition for a countably normed space was given in an earlier volume, but I do not have access to them and I could ...
- 5,975
1
vote
2
answers
39
views
seminorm continuous at zero
Let $X$ be a topological vector space over $\mathbb{R}$ or $\mathbb{C}$ and $p$ be a seminorm on $X$. I suppose that $p$ is continuous at $0$ and I want to prove that $p$ is continuous.
The proof ...
1
vote
0
answers
42
views
Nuclear modules over a topological ring
Let $A$ be an $\mathbb R$ or $\mathbb C$ algebra. (You may assume $A$ is nuclear and Frechet)
Is there a notion of Nuclear modules over $A$, which extends the notion of a nuclear TVS over $\mathbb R$ ...
- 921
0
votes
0
answers
49
views
On a theorem that weak convergence implies strong convergence in "Ordered Topological Vector Spaces" by Peressini
I have a question regarding Proposition 3.12 of Chapter 2 of "Ordered Topological Vector Spaces" by Peressini, which states the following:
Suppose that $E$ is a locally convex space ordered
...
- 327
1
vote
1
answer
63
views
Inductive limit and denseness
Let $\{E_n, n\in \mathbb{N}\}$ be an increasing sequence of linear subspaces of a vector space $E$, i.e. $E_n \subset E_{n+1}$ for all $n\in \mathbb{N}$ such that
$E = \bigcup_{n\in \mathbb{N}} E_n$. ...
- 126
0
votes
0
answers
16
views
limit of sum of sequence in topological vector space
Here is the statement I want to prove:
Let $X$ be a topological vector space over $\mathbb{K}$. Let $(x_k)$ and $(y_k)$ be sequences in $X$. I assume that $x$ is a limit of $(x_k)$ and $y$ is a limit ...
0
votes
0
answers
20
views
Proof verification: if $\Omega$ is a convex subset of a topological vector space $X$, then $\overline{\Omega}$ is convex.
I'm working on a proof that if $\Omega$ is a convex subset of a topological vector space $X$, then so is its closure, $\overline{\Omega}$. My proof was different than the proof in the book, so I want ...
- 989
7
votes
3
answers
1k
views
Proving that scalar multiplication is continuous
Let $\mathbb{K} \in \{ \mathbb{R} , \mathbb{C} \}$ and $s= \mathbb{K}^{\omega}$ be the usual sequence set with entries on $\mathbb{K}$. I proved that $\mathbb{K}$ induces a $\mathbb{K}$-vector space ...
- 3,257
2
votes
1
answer
107
views
What kind of a set do decomposable tensors form in the tensor product $V\otimes W$? Are decomposable tensors dense in finite dimensions?
Q
If $V,W$ are finite dimensional over $\mathbb R$ or $\mathbb C$, how does the set of decomposable tensors $$S=\{v\otimes w:v\in V,w\in W\}\subset V\otimes W$$ "fit" in its ambient space $V\...
- 1,478
0
votes
1
answer
67
views
Topology from seminorms
Let $X$ be a vector space, and $\{p_i\}_{i\in I}$ be seminorms on $X$.
The usual definition of the induced topology to say that it is the coarsest topology for which the maps $p_i$ are continuous.
(...
- 2,578
0
votes
3
answers
70
views
A line segment in the closure of a convex subset of a topological vector space
I am trying to solve the following problem. Let $C$ be a convex subset of a topological vector space $X$. Let x be in the interior $C^\circ$ and let y be in the closure $\bar{C}$. I am asked to prove ...
- 1,083
1
vote
1
answer
108
views
Extension of closed operator densely defined
Let $X$ be a Banach space. Suppose $A:D(A)\subset X \longrightarrow X$ is a closed linear operator and $D(A)$ is dense in $X$. Prove that $A$ cannot be extended as a closed linear operator to any ...
- 5,172
2
votes
0
answers
52
views
Is the dual of the space of semi-regular distributions complete
Consider $C^\infty(X)$, equipped with its Fr'echet structure, and $\mathcal D'(Y)$, equipped with the strong topology, where $X,Y$ are compact manifolds. Is the strong dual of $C^\infty(X)\hat\otimes \...
- 921
1
vote
1
answer
59
views
Semi-regular kernels and distributions on open subset of $\mathbb R^n$
I am reading Francois Treves book Topological Vector Spaces, Distributions and Kernels, page 532. Given open subsets $X\subset \mathbb R^m$ and $Y\subset \mathbb R^n$, the author defines a kernel $K(x,...
- 19.7k
0
votes
0
answers
35
views
Weak and strong topologies on a compact set
Let $X$ be a Banach space and let $K$ be a (strongly) compact subset of $X$.
First $K$ is also weakly-compact because the weak topology is weaker than the strong topology.
Both trace topologies on $K$ ...
- 1,473
3
votes
0
answers
30
views
Characterization by nets of relatively countable compact set on TVS
Let $E$ be a topological vector space and $A \subset E$. How can I prove that $A$ is relatively countable compact if, and only if, every sequence in $A$ has a subnet that converges in $\bar A$.
I know ...
- 1,275
1
vote
1
answer
46
views
Prove that integration is continuous under weak star topology
Let $\mathcal{B}(0,1)$ be the Borel sigma algebra on $(0,1)$ and $m$ be the Lebesgue measure. Consider $L^\infty((0,1),\mathcal{B}(0,1),m)$ equipped with weak-star topology and $B=\{f\in L^\infty(0,1):...
- 31.4k
2
votes
0
answers
96
views
TVS Completion of Banach space wrt weak topology
We're talking about weak topologies in my FA courses and I thought of the following question, that I don't know the answer to.
Say we have a normed vector space $X$ which we view as a TVS with its ...
- 469
8
votes
0
answers
1k
views
$j(X)$ weak*-dense in $X^{**}$, $ j$ is the canonical embedding
Let $X$ be a Banach space and consider the canonical embedding in it's bidual $X^{**}$, namely
$j:X\to X^{**}, \; x\mapsto j(x)$, where $j(x)(x^*)=x^*(x)$ for $x^*\in X^*$.
My question: Why is $j(X)$...
1
vote
0
answers
28
views
Different definitions of locally boundedness of TVS
A susbset $A$ of a topological vector space $E$ is called bounded if for any neighborhood $U$ of the origin there exists a number $\alpha > 0$ such that $\lambda U \supset A$ for any $\lambda \geq \...
0
votes
0
answers
70
views
Is it possible for a sequence in $L^p$ to converge to an element of $L^q$ ($q>p$) w.r.t. the $q$-norm?
Suppose that $1 \leq p < q \leq \infty$ and that $f_n$ is a sequence in $L^p(\mathbb{R})$.
If $f_n$ is a Cauchy sequence with respect to the metric induced by the $L^p$ norm, then (I believe that) ...
- 6,219
0
votes
3
answers
1k
views
$X^*$ with its weak*-topology is of the first category in itself
Let $X$ be an infinite-dimensional Fréchet space. Prove that $X^*$,with its weak*-topology is of the first category in itself.
- 1
0
votes
0
answers
20
views
For any (not necessarily continuous) linear form $f$ and any convex set $C$, $\overline C\subseteq f^{-1}(\overline{f(C)})$
Let $C$ be a convex subset of a l.s.c space $E$. If $f$ is a continuous linear form on $E$, then $\overline C\subseteq f^{-1}(\overline{f(C)})$, indeed $\overline C$ is the intersection of all closed ...
- 6,031
3
votes
0
answers
69
views
Is it true that the intersection of two dense subspace of a linear normed space is also dense provided that one of them is of finite codimension?
Let $X$ be a normed vector space and $X_1,X_2$ be two dense subspace of $X,$ in Is is true that the intersection of two dense subspaces of a linear normed space is also dense? it was showed by an ...
- 1,322
2
votes
1
answer
102
views
A topological vector space with cardinality smaller than the weight of topology
A topological vector space is a vector space endowed with a topology such that the sum of vectors and the multiplication by scalars are continuous.
The weight of the topology τ is the smallest ...
- 3,912
9
votes
1
answer
162
views
Is there a infinite dimensional topological vector space with the weight of topology larger than the algebraical dimension?
A topological vector space is a vector space endowed with a topology such that the sum of vectors and the multiplication by scalars are continuous.
The weight of the topology $\tau$ is the smallest ...
- 331k
4
votes
1
answer
211
views
Why is a certain projective limit of weighted symmetric Fock space, namely $\bigcap\limits_{\tau \in T, p\ge 1 } \mathcal{F}(H_\tau,p)$, separable
I have a question regarding separability of a certain locally convex space.
Let $H_{\tau}:=H^{\tau_1}(\mathbb{R}^n,\tau_2(x)dx)$ the weighted Sobolev Hilbert space with $\tau_1 \in \mathbb{N}, \tau_2(...
- 1,322
2
votes
1
answer
78
views
Rudin functional analysis theorem 1.14 part a
I'm trying to understand. this theorem
Theorem 1.14. In a topological vector space 𝑋,
(a) every neighborhood of 0 contains a balanced neighborhood of 0.
The proof asserts that as scalar ...
- 121
0
votes
0
answers
44
views
Aggregate first-order stochastic dominance
A random variable $X$ dominates another random variable $Y$ in first order stochastic dominance, denoted as $X\geq_1 Y$, if ti holds that $\mathbb{E}\left(\phi(X)\right)\geq \mathbb{E}\left(\phi(Y)\...
- 60.7k
9
votes
2
answers
873
views
Schwartz space of functions versus Schwartz space in a more general sense?
Part of me is afraid that this isn't a well-formed question, but try as I might, I can't seem to figure out anything reasonable on this topic. I'm hoping someone here can help.
In functional analysis,...
- 52.5k
1
vote
0
answers
45
views
Is this condition redundant (neighborhood filter for TVS in Trèves)?
In Trèves's Topological Vector Spaces, Distributions and Kernels, Theorem 3.1 is as follows.
A filter $\mathscr F$ on a vector space $E$ is the filter of neighborhoods of the origin in a topology ...
- 6,585
0
votes
0
answers
60
views
The closed ball is contained in the closure of open ball
$Bf(a,r) = \{ x \in \mathbb R \mid d(a,x) \le r \}$
I want to show that the the closed ball is contained in the closure of the open ball.
Am guided to take an $x$ from the sphere
Here is what I did,
...
- 76.5k
3
votes
1
answer
102
views
Unique vector space topology on $F^n$?
Every finite dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$ has a unique topology that makes addition and scalar multiplication continuous. Is the same true of finite dimensional vector ...
- 15.2k
3
votes
1
answer
103
views
Is multiplication by a scalar an open map in topological groups?
If n is any non-zero integer and G is a topological abelian group, under what conditions is the continuous homomorphism $g \mapsto ng$ open or weakly open?
If this map happens to be open for all n and ...
- 331k
0
votes
1
answer
95
views
Counterexample concerning Rudin's definition of open sets in $\mathscr D(\Omega)$
In Rudin's Functional Analysis, in definition 6.3, he defines a family $\beta$ subsets of $\mathscr D(\Omega)$ as follows.
$\beta$ is the collection of all convex balanced sets $W \subset \mathscr D(\...
- 35.4k
4
votes
1
answer
51
views
Must a linear functional on a TVS with a sequentially closed kernel be sequentially continuous?
Let $(X, \tau)$ be a Hausdorff topological vector space, $\mathbb{K}$ the scalar field with its usual topology and $\Lambda : X \to \mathbb{K}$ a linear functional.
There is this general criterion ...
- 5,172
3
votes
2
answers
80
views
A characterization of closure of a certain class of sets in $\mathbb{R}^n$
Consider a set $K\subset \mathbb{R}^n$ that is symmetric ($B = -B$) and verifies $aK\subset bK$ if $|a|<|b|$. Can I conclude that $\overline{K} = \cap_{a>1}aK$? If not, does the result hold ...
- 2,807
6
votes
2
answers
3k
views
Geometric Interpretation of Eigendecomposition
If we have an orthogonal matrix $U$, then $U^Tx$ is essentially a rotation of the vector $x$. If we have a diagonal matrix $\Lambda$, then $\Lambda x$ is scaling the vector $x$ in each direction by ...
- 395
0
votes
0
answers
173
views
Topology on modules
Take $M$ an $R$-module with $R$ a commutative ring with unit. Let's take on $M$ a linear topology in the following way.
Take $M=M_0 \supset ... \supset M_n \supset M_{n+1} \supset ...$ a chain of $R$-...
0
votes
1
answer
66
views
Theorem 1.36 in Rudin's Functional Analysis
I have a few questions regarding the proof of Theorem 1.36 in Rudin's Functional Analysis:
Why does $V$ being open imply $x/t\in V$ for some $t<1$.
How does the inequality $\mu_V(x-y)<r$ come ...
- 35.4k
0
votes
1
answer
40
views
Proof that every $T_0$ Topological Vector Space is regular
I'm currently reading through Convex Analysis and Beyond by Mordukhovich and Nam where the following preposition and proof are given (note that the authors define TVSs to be $T_0$).
Proposition 1.92 ...
2
votes
1
answer
88
views
Finding a compact set to apply the Hahn-Banach separation theorem in a locally convex topological vector space
I am trying to justify how the Hahn-Banach theorem was applied in the proof below. It looks like the proof is using the case for locally convex space (because the inequalities are strict). that ...
- 5,172
0
votes
1
answer
49
views
Hausdorff separation for the definition of Mackey topology
I am reading the definition of the Mackey topology relative to a dual system $(X,Y)$ and the author (Edwards-Functional Analysis) imposes the condition that the dual system must be separated in $Y$. ...
- 778
5
votes
2
answers
122
views
Local basis criterion for topological group / vector space?
Let $X$ be a topological vector space (TVS), and let $\mathcal B$ be nonempty family of subsets of $X$ that each contain $0$. Are there simple conditions that guarantee $\mathcal B$ is a local basis ...
- 6,585
2
votes
1
answer
69
views
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{...
- 778
0
votes
1
answer
65
views
Dual space of a topological vector space that doesn't separate points?
While I am studying FUNCTIONAL ANALYSIS by Walter Rudin, I found the following corollary.
Now, I wonder how the dual space(the set of all continuous linear functionals on $X$) of some pathetic ...
- 46.1k
0
votes
0
answers
36
views
Can I define boundedness in a topological vector space w.r.t. arbitrary point?
I have started reading Rudin's Functional Analysis, and in Section 1.6, he makes the following definition:
A subset $E$ of a topological vector space is said to be bounded iff for every open ...
- 43.7k
3
votes
1
answer
54
views
Topology of convergence in measure is not compatible with the vector space structure of measurable functions
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the ...
- 5,657
1
vote
0
answers
44
views
Meaning of injective tensor product
Let $X, Y$ be two locally convex topological vector spaces. I can tell myself a story to make $X\otimes_{\pi} Y$ and $X\otimes_{\iota} Y$ (the projective and inductive tensor products, respectively) ...
- 300