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.

learn more… | top users | synonyms

1
vote
1answer
27 views

Criterion for Isometry

Let $X$ be a topological vector space, with $d$ an invariant metric compatible with the metric. Let $f:X\to X$ be an involutive linear isomorphism. How do you show that $f$ is an isometry? I ...
-1
votes
2answers
73 views

Prove $\ell^5$ is contained in $\ell^6$.

I am struggling with the proof to show that, for any $p$, $r$ such that $1 \le p <r < \infty$, that $\ell^ p\subset\ell ^r$. Could somebody please give a helpful nudge by showing how this ...
0
votes
0answers
8 views

Decomposing continuous linear functionals on a locally convex space with 2 seminorms

Let $X$ be a locally convex topological vector space whose topology is defined by the seminorms $\rho_1$ and $\rho_2$. (Let us require that topological vector spaces be Hausdorff by definition.) If ...
4
votes
1answer
101 views

How to check some topological concepts in product and direct sum spaces

Given $a=(a_i)_{i=1}^\infty$ with $a_i \geq 0$ and $b=(b_i)_{i=1}^\infty$ with $b_i \in \mathbb{R}$, let $$E_i = \lbrace (x_n)_{n=1}^\infty : n^{b_i}|x_n|\leq a_i, \forall n\in \mathbb{N} \rbrace$$ ...
1
vote
0answers
54 views

a question of my real analysis class,can someone help me solve this question?

Let $n \geq 1$ be an integer, and $C: \mathbb{N}^n \to \mathbb{R}$ be a function. Prove that there exits a infinitely differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ whose value and partial ...
0
votes
0answers
20 views

Banach-Alaoglu theorem for dual pairs

The versions of the Banach-Alaoglu theorem that I know of always concern some space $X$ and its topological dual $X^*$. Can the theorem be restated for arbitrary (nondegenerate) dual pairings $(X,Y)$ ...
0
votes
2answers
34 views

a question about functional analysis conclusion,and I am not sure whether it is true or not?

we have $R^n$,$R^m$ spaces, suppose open set $O_{1}\subset R^n $ and $O_{2}\subset R^m$, $f:O_{1}->O_{2} $ is k-times differentiable$(1<=k<=\infty)$,then at $x_{0}\in O_{1}$,$rank(f)(x_{0})$ ...
5
votes
0answers
50 views

Why do we give $C_c^\infty(\mathbb{R}^d)$ the topology induced by all good seminorms?

Briefly, my question boils down to the following: What benefits do we gain from considering the space of test functions in the topology induced by all "good" seminorms, as opposed to other topologies ...
1
vote
1answer
45 views

Bounded linear map on topological vector spaces is continuous

Let $X$ and $Y$ be topological vector spaces and $T\colon X\to Y$ linear. Suppose that $T$ sends bounded sets to bounded sets and that $X$ is first countable. The claim is that $T$ is continuous. ...
1
vote
2answers
19 views

Does every LCS has a convex balanced local base?

Does every LCS--locally convex (topological vector) space has a convex balanced local base? Then it implies every LCS is topologized with a countable number of separating seminorms. So there seems ...
2
votes
2answers
39 views

Can every (Hausdorf) topological space be homeomorphically embedded in a topological vector space?

It's true that for any metric space, we can isometrically embed it in a Banach space, so that the image of the metric space by that embedding is a linearly independent set. Is the analogous theorem ...
10
votes
1answer
198 views

Does the vector space of compactly-supported continuous functions $X \rightarrow \mathbb{R}$ satisfy an interesting universal property?

Let $S$ denote a set. Then the vector space $FS$ freely generated by $S$ can be identified with the set of all finitely-supported functions $S \rightarrow \mathbb{R}$. This gave me the following idea; ...
1
vote
1answer
70 views

properties of a Köthe space s

Could you please help me answering the following question? Consider the Köthe space $K_ {\infty}(n^p) = \{ x= (x_n)_1^{\infty}: |x|_p := \sup_n|x_n|n^p<\infty, \forall p \in \mathbb{N} \}$ with ...
2
votes
0answers
12 views

Is the unit ball of $H^\infty(\mathbb{D})$ a metrizable topological semigroup under multiplication?

The space $H^\infty(\mathbb{D})$ of all bounded holomorphic functions on the open unit disc carries many different topologies. One such topology is given by uniform convergence on compact subsets; ...
0
votes
0answers
14 views

In the first countable TVS, if every Cauchy sequence convergence then every Cauchy net convergent

Let $X$ be a topological vector space with the first countable topology(that is, every point has a countable neighborhood basis).If every Cauchy sequence convergence, we want to show that every Cauchy ...
2
votes
1answer
43 views

Why convergence implies cauchy in topological vector space?

The following definition is from Janich's Topology book : Definition (Topological Vector Space). A $\mathbb{R}$-Vector space $(E,\tau)$ with a topological space structure is called a Topological ...
3
votes
1answer
40 views

Hahn-Banach theorem and complex linear functional

I find this exercise but I cannot prove it. If $ X $ is a complex topological vector space and $ f \colon X \to \mathbb C $ is nonzero continuous linear function, show that $ X \setminus \ker f ...
1
vote
0answers
22 views

Isomorphism of finite dimensional topological vector space with $(\mathbf{R}^k,\mathcal{R})$

Let $(T,\mathcal{T})$ be a topological vector space over $\mathbf{R}$ with finite positive dimension. Is it true that there exists an isomorphism between $(T,\mathcal{T})$ and ...
1
vote
0answers
22 views

Generate a mesh from unsorted points (eight points)

I'm trying to generate a mesh from eight points. The challenge is that I don't know the order/label of the points, and I want it to work regardless of variations in the shape (see example below). The ...
0
votes
0answers
38 views

Weak* topology on Hilbert space

I am a little confused about the weak* topology on Hilbert space $H$. Beyond doubt, the weak* topology on $H^{**}$ is $\sigma(H^{**},H^*)$. Suppose $\tau$ is the natural embedding from $H$ onto ...
1
vote
4answers
42 views

Embedding vs continuous injection (in topological vector spaces)

When working with topological vector spaces (say $X,Y$), the term “embedding” is often used for a continuous injection $f:X\rightarrow Y$. Now, $f$ is of course a bijection onto its image, but it's ...
5
votes
1answer
141 views

Generalization of inner product spaces (analogue to uniform spaces/locally convex spaces)

In the following I am going to devise a chart of topological spaces that contains inner product spaces, normed vector spaces, metric spaces and other related spaces. In the end there will be a gap in ...
6
votes
1answer
60 views

Open neighbourhoods in topological vector spaces

It is well known that each open ball in a Banach space is homeomorphic to the whole space. Can we extend this to topological vector spaces? In other words, does every non-void open set in a ...
0
votes
0answers
17 views

A question involving normed spaces and strictly convex spaces

Let $(X, \| \cdot \|_X)$ be a normed space and let $\| \cdot \|$ be a norm on $X$ such that $(X, \| \cdot \|)$ is strictly convex. How can I find a strictly convex space $(Y, \| \cdot \|_Y)$ and a ...
0
votes
0answers
34 views

Is any closed and bounded subset of a reflexive Banach space compact in the weak topology?

It seems to me that Alaoglu's theorem implies that any closed and bounded subset of a reflexive Banach space is compact in the weak topology. Is convexity of the set also needed?
3
votes
1answer
46 views

Why doesn't Alaoglu's theorem imply that $X^{*}$ is locally compact in the weak* topology?

I must be missing something basic and simple: If $X$ is a normed vector space and the closed unit ball in $X^{*}$ is weak* compact, and translations and dilations are homeomorphisms, why isn't $X^{*}$ ...
2
votes
2answers
23 views

If $E\subset X^{*}$ is bounded, then so is its weak* closure

If $X$ is a Banach space and $E\subset X^{*}$ is norm-bounded, I've shown that its weak* closure is also norm-bounded using Alaoglu's theorem. But perhaps using Alaoglu's theorem is not necessary? ...
1
vote
1answer
27 views

Trying to show that $(c_0, \| \cdot \|_s)$ is strictly convex, where $\| x \|_s = \underset{i = 1}{\overset{\infty}{\sum}} \frac{1}{2^i} | x_i |$

I'm trying to show that $ (c_0, \| \cdot \|_s) $ is a strictly convex space, where $$ \| x \|_s = \underset{i = 1}{\overset{\infty}{\sum}} \frac{1}{2^i} | x_i |,$$ $ x = (x_1, x_2, ..., x_i, ...) \in ...
2
votes
1answer
85 views

Dual Pairs, topology of weak convergence and weak* topology

Edit for Bounty: I decided to put a bounty on this question because I would really like to get it properly. Thus, I would like to get feedbacks on my basic questions, and a detailed answer on my ...
0
votes
0answers
21 views

Inductive/Projective Limits of Topological Algebras

It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance, For $k \ge 0$ and $K_n$ compact ...
2
votes
1answer
73 views

Dual Spaces and Topological Vector Spaces

I have a question regarding dual spaces. Before, let me write that this all issue looks really problematic to me, and I already touched it quickly in another question. However, in that occasion, the ...
3
votes
2answers
44 views

How to show that the topology is compatible with the metric?

This is in the contest of Toplogical-Vector-Spaces, but can be interepreted as a simply topology question. For my matter, assume $\|\cdot\|_n$ is a countable family of seminorms, and define ...
3
votes
1answer
42 views

Is the 3d Schwartz space isomorphic to a subspace of the 1d Schwartz space?

Are the Schwartz-spaces $\mathscr{S}(\mathbb{R})$ and $\mathscr{S}(\mathbb{R}^3)$ isomorphic (as topological vector spaces)? Is $\mathscr{S}(\mathbb{R}^3)$ at least isomorphic to a subspace of ...
1
vote
1answer
29 views

Every singelton is bounded in a topological vector space

I'm trying to prove that every singelton (one point set $\{x\}$) is bounded in a topological vector space. I can't see it so easily. It is obvous that given a $V$ neighberhood of $0$, there is ...
0
votes
1answer
64 views

Preserved properties through continuous linear maps

I just looked at the fact (at least according to Definition 2.8.1. in Distribution Theory by Friedlander et al.) that for $K_0\subseteq{\bf R}^{n(0)}$ compact, $\Omega_1\subseteq{\bf R}^{n(1)}$ open ...
3
votes
1answer
67 views

Do sequences fully specify the topology of $\mathcal{D}(\Omega)$ and $\mathcal{D}'(\Omega)$?

It is well known that $\mathcal{D}(\Omega)$ and $\mathcal{D}'(\Omega)$ are not metrizable, and that a topological vector space is metrizable if and only if it is first-countable (Rudin, Thm. 1.24). ...
1
vote
0answers
33 views

Direct limits of locally convex spaces and embeddings

I was thinking about whether this positive result would hold in the category of locally convex spaces also... Here is what I got so far: The direct limit of a locally convex system consists of the ...
0
votes
0answers
19 views

When is the completion of a topological vector space a Frechet space?

Suppose $X$ is a topological vector space with the metric topology. If we take the completion of $X$ with respect to the metric, will we get a Frechet space? Are there any extra conditions needed to ...
3
votes
2answers
54 views

Logic behind a proof in Topological Vector Spaces

I found the following result at the beginning of some notes on topological vector spaces (TVS). This is a quite fundamental result, that apparently is considered the corresponding version of the ...
2
votes
1answer
20 views

Closure of intersection with vector subspace

I am confused with the footnote on page 198 of http://www.ma.huji.ac.il/~razk/iWeb/My_Site/Teaching_files/TVS.pdf Essentially: Let $X$ be a topological vector space and $Y$ a finite-dimensional ...
2
votes
0answers
28 views

Existence of particular functionals in a family of linear functionals

Let $U\subset B$ be a subset of a Banach space $B$, and let $D$ be a complete topological vector space. I have given a family $\mathcal L(U)=\{L_u\ |\ u\in U\}$ of linear functionals $L_u:D\to\mathbb ...
0
votes
1answer
37 views

Topology generated by a Family of Seminorms as a Initial Topology?

Let $X$ be a set and $\{(Y_i, \mathscr{T}_i)\}_{i\in I}$ be a family of topological spaces and $\{f_i\}_{i\in I}$ a family of mappings $$f_i:X\longrightarrow Y_i.$$ The initial topology on $X$ is the ...
1
vote
0answers
34 views

Bornological/Barrelled Operator-Topologies?

I'm looking for results concerning the following questions. If those have been already addressed in the literature, it would be nice to know proper citations: Let $(E, \tau_E)$ and $(F, \tau_F)$ be ...
1
vote
1answer
64 views

Is the dual space of all Radon measures the space of signed measures on a $\delta$-ring?

Consider the Banach space $C_c(\mathbb{R})$ of continuous functions with compact support equipped with the uniform norm $||f||_\infty := \sup_{x \in \mathbb{R}} |f(x)|$. Then it is known (Riesz ...
7
votes
0answers
78 views

Nuclear spaces vs Banach spaces

The Wikipedia article on nuclear spaces say the following: "There are no Banach spaces that are nuclear, except for the finite-dimensional ones. In practice a sort of converse to this is often true: ...
1
vote
0answers
39 views

Continuity of the dual product reloaded

Let $X$ be a Banach space with topological dual $X^*$. Then the dual product $(x,x^*)\in X\times X^*\to x^*(x)\in\mathbb{R}$ is strongly$\times$strongly continuous in $X\times X^*$. That does not ...
2
votes
2answers
78 views

How to prove $ \mathit {X}$ is path connected?

${\mathbb{R}}^{2} $ Euclidean 2-space,Let $\mathit {X} \subset \mathbb{R}^{2} $.$$\mathit {X}=[-2,2]\times[-1,0]\cup[-2,-1]\times[0,1]\cup[1,2]\times[0,1]$$ $\qquad\qquad\qquad\qquad\qquad$ ...
0
votes
1answer
50 views

A Problem on Locally Convex Spaces

In the book A Course in Functional Analysis by Conway, there is the following problem: Problem. Let $ X $ be a completely regular topological space, and let $ C(X) $ denote the set of all ...
0
votes
0answers
40 views

Doubt Concerning the Definition of a Locally Convex Space Structure through Seminorms?

In the book Introduction to Functional Analysis written by A. E. Taylor there are the following theorems: Theorem 1. Suppose that $X$ is a linear space and that $\mathscr{U}$ is a nonempty family ...
0
votes
1answer
29 views

Continuity of the dual product

Let $X$ be a Banach space with topological dual $X^*$. Then the dual product $$ (x,x^*)\in X\times X^*\to x^*(x)\in\mathbb{R} $$ is strongly$\times$strongly continuous on $X\times X^*$, mainly because ...