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.

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10
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1answer
227 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
97 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
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0answers
16 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; ...
1
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1answer
74 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
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1answer
63 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
66 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 ...
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0answers
26 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
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0answers
30 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 ...
1
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3answers
154 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 ...
6
votes
1answer
220 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
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1answer
72 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
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0answers
24 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
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0answers
183 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
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1answer
62 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
41 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
30 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
123 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 ...
2
votes
1answer
112 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
61 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
64 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
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1answer
76 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
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1answer
69 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 ...
4
votes
1answer
101 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). ...
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0answers
39 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 ...
3
votes
2answers
69 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
30 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
32 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 ...
1
vote
1answer
162 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 ...
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0answers
41 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
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1answer
298 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
156 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: ...
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0answers
49 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
97 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
66 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
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0answers
50 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
53 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 ...
1
vote
1answer
19 views

Existence of translated nbhds of zero in a TPVS

Let $X$ be a topological vector space and let $W$ be an open set which contains $0$ (a nbhd of $0$). How do you prove that $0$ has nbhds $V_1$ and $V_2$ such that $V_1 + V_2 \subseteq W$? (This was ...
1
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0answers
70 views

Weak operator topology is the smallest topology on $B(H)$

Show that weak operator topology is the weakest locally convex topology on $B(H)$ such that every $\phi\in F(H)$ is continuous. (F(H) means finite rank operators on $H$). To show it , let $\tau$ ...
2
votes
1answer
90 views

Closed unit ball of $B(H)$ with wot topology is compact

The following is a Theorem of Conway's operator theory: I can not understand how he proves it. I think $\phi(\text{ ball B(H)})$ is compact if $\phi(\text{ ball B(H)})$ is closed subset of compact ...
0
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1answer
62 views

Does $B(H)$ satisfy in Heine-Borel property?

Based on here, I know that every bounded and closed subset of a space is not compact. I really want to know that $B(H)$, the space of bounded linear operators, satisfies in Heine - Borel property. ...
2
votes
1answer
152 views

On separable Hilbert space $H$, weak operator topology is metrizable on bounded parts of $B(H)$

The following is a theorem of Takesaki's operator theory: In this proof, weak topology means weak operator topology. I'm wonder why the theorem holds just for bounded parts of $B(H)$ and also ...
0
votes
1answer
57 views

Proving linearity implies (or can imply under opportune conditions) lower semicontinuity

A function $f:X\to\mathbb{R}$, with $X$ being a topological space, is termed as lower semicontinuous (lsc) at $x_0\in X$ if: $$\forall\epsilon>0\,\,\exists V\text{ an open neighborhood of }x_0:x\in ...
0
votes
1answer
61 views

Proof that difference of compact and closed sets is also closed [duplicate]

I am trying to prove that for any $A$ compact, $B$ closed sets $\Rightarrow A-B = \{a-b | a\in A, b\in B\}$ is also closed, where A and B are subsets of a topological vector space $X$.
1
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1answer
43 views

Is this extension a Hilbert space?

Let $V$ be an inner product space over $\mathbb{F}$. Let $H$ be a complete subspace of of $V$ and $x\in V\setminus H$ Define $K= span(H\cup \{x\})$. Is $K$ a Hilbert space? How do I prove it?
3
votes
2answers
113 views

Metrizability of the unit ball $B_{X^*}$.

I am trying to prove the assertion: If $X$ is a separable normed space, then the unit ball in $X^*$ with the weak* topology, $(B_{X^*},\sigma(X^*,X))$, is metrizable. Firstly, I took ...
1
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1answer
34 views

Regarding direct sums in topological vector spaces

If $E=E_0\oplus E_1$ is a topological vector space and $A\subseteq E_0$ open in the induced topology on $E_0$, can I conclude that $A+E_1$ is open in $E$? Possibly if I assume $E$ to be locally ...
0
votes
1answer
22 views

Regarding embeddings of locally convex spaces

If $f:E\rightarrow E'$ is a linear embedding of locally convex topological vector spaces, and $A\subseteq E$ open and convex, can we always find $A'\subseteq E'$ open and convex sucht that ...
4
votes
2answers
103 views

A typical example of Homeomorphism

The set $\mathbb{R}^2-\{(0,0)\}$ with the usual topology is: (A) Homeomorphic to the open unit disc in $\mathbb{R}^2$ (B) the cylinder $\{(x,y,z)\in \mathbb{R}^3/ x^2+y^2=1 \}$ (C) the ...
0
votes
1answer
108 views

Topological supremum of family of linear topologies

In Hans Jarchow - Locally Convex Spaces 2.4.4 (c) it says: The topological supremum of any family of linear topologies on a fixed vector space is linear. I couldn't find a proof in the book and ...
2
votes
0answers
200 views

Continuity of implicit function

I find a proof of the following theorem in A.N. Kolmogorov and S.V. Fomin's Элементы теории функций и функционального анализа (pp. 492-493 here): Let $X,Y,Z$ be Banach spaces, $U$ a neighbourhood ...