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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1answer
75 views

There exists function sequence $\{f_{n}\}$ converges to $0$ such that $\{a_{n}f_{n}\}$ not converges to $0$

Let $X$ be the vector space of all complex functions on the unit interval $[0,1]$, topologized by the family of seminorms $$p_{x}(f) = |f(x)|, \quad (0 \le x \le 1).$$ Show that there exists a ...
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1answer
54 views

Tempered fundamental solutions

According to the Malgrange–Ehrenpreis theorem every nontrivial linear constant coefficient PDO $P(\partial)$ admits a fundamental solution $E\in\mathscr{D}'$; I wonder whether $P(\partial)$ admits a ...
2
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1answer
73 views

Measure-valued maps equal a.e. if their integrals equal over any set

Let $X,Y$ be two standard Borel spaces and let $p,q:X\to\mathcal P(Y)$ be two stochastic kernels, which can be alternatively seen as measure-valued maps. Suppose that for some measure $\mu\in \mathcal ...
0
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2answers
79 views

Example of a sequence of functions

Construct an example of a sequence of functions $(f_n)$ defined on $[0,1]$ such that $f_n$ converges pointwise to $0$ and for every sequence of numbers $(a_n)$ that tends to $\infty$, sequence ...
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1answer
80 views

Question on the proof that $C(\Omega)$ is a Frechet space

I am using Rudin's book on Functional Analysis. I am studying the proof that the space $C(\Omega)$ of continuous functions on an open set $\Omega \subseteq \mathbb{C}$ is a Frechet space. I ...
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1answer
108 views

is the vector space $\mathbb{R}^\mathbb{N}$ locally compact?

is the vector space $\mathbb{R}^\mathbb{N}$ locally compact? for example, let $x=(x_1,x_2,....)$ any point of $\mathbb{R}^\mathbb{N}$ and let $V=[x_1-\epsilon,x_1+\epsilon] \times ...
0
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1answer
60 views

Proof details for the first theorem in functional analysis

The Proof details for the first theorem in functional analysis is given in this website: I have one questions for the proof of lemma 5 given here. (1) Why this claim hold? from equality $0+0=0$ ...
3
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1answer
71 views

In a topological vector space, show if $A$ and $B$ are bounded, then $A + B$ is bounded?

I get as far as this before I am stuck: Pick any neighbourhood of $0$ and call it $U$. Then there exists $a, b$ such that $A \subseteq aU$ and $B \subseteq bU$. So hence $ A + B \subseteq aU+bU$. ...
3
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2answers
111 views

How to prove in a topological vector space: cl(A) + cl(B) is a subset of cl(A+B), where cl denotes closure?

I'm not sure where to really proceed. My process is as follows. Take any $x \in cl(A)+cl(B)$. Assume for a contradiction that $x \notin cl(A+B)$. Then there exists an open set $U$ such that $ x \in ...
3
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1answer
64 views

Constructing a closed, convex subset of $X^{\ast}$ that is not weakly-* closed

I'm asked to show that if $X$ is a non-reflexive Banach space, there exists (norm) closed and convex subsets of $X^\ast$ that are not $w^{\ast}$-closed. In other words, there's no analogue of Mazur's ...
3
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1answer
736 views

Sum of closed and compact set in a TVS

I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin's Functional Analysis), where $A$ and $B$ are subsets of a topological vector space ...
3
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0answers
54 views

How general is the convergence of the exponential function's power series?

Let $\mathbf{V}$ be a Fréchet space whose underlying set is $V$. Let $\;\; \beta \: : \: V\times V \: \to \: V \;\;$ be a continuous bilinear map that has an identity element and is ...
0
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1answer
64 views

How can we ensure that a space is a subset of locally convex topological space?

I am looking for fast ways to ensure that a given set is a subset of topologically locally convex space. I have already read the posts post1:seminorms-in-locally-convex-spaces, ...
2
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1answer
40 views

Notions of topological spaces and matrices

How can I show that: For all $r>0$, exists $A$ nonsingular matrix, such that $B_r(A)\subseteq GL_n(\mathbb{C})$ For all $A\in GL_n(\mathbb{C})$ and for all $r>0$, there exists $\alpha>0$ ...
3
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0answers
48 views

Every locally connected non separating plane continuum is an AR

A continuum is a compact connected metric space. A plane continuum is a continuum contained in $R^2$. We say that a plane continuum does not separate the plane provided that $R^2\setminus X$ is ...
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2answers
20 views

if $v$ is a member of $H$ and $v$ is not a member of $M$ then $u$ is member of $K$. How is this possible?

Let $(V,K)$ and $u,v$ is a member of $V$. Suppose that $M$ is a subset of $V$ is a subspace of $V$ with basis $B_m=\{m_1,...,m_r\}$ with $r$ less than and equal to $n$. Let $H$ be a subspace spanned ...
5
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1answer
94 views

Dual of a topological vector space. Is it nontrivial?

In the case of normed spaces we know their duals are nonempty using a quick application of the Hahn Banach Theorem. If we step back to the larger class of locally convex spaces, an enthralling ...
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0answers
15 views

Seminorm topology on $C^{\infty}(\Omega)$ and proof of continuity [duplicate]

Let $\Omega\subset\mathbb{R}^n$ be open and $\Omega\neq\varnothing$ and suppose we have the Fréchet topology on $C^{\infty}(\Omega)$ (this can be obtained by the topology construction from out ...
0
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0answers
31 views

On function of bounded semi-variation

The following ideas were taken from the article of DUCHON, entitled "On Vector Measures and Distributions." Definition. Let $X$ be a locally convex space whose topology is generated by a family $P$ ...
3
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1answer
339 views

proving that $SO(n)$ is path connected

Our professor gave us exercise to show that $G=SO(n,\mathbb R)$ is path connected. He gave some hints, using them I have come upto this far: I have shown that $SO(n)$ acts on $S^{n-1}$ transitively ...
2
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1answer
176 views

Why are the Differential- and multiplication mapping on $C^{\infty}(\Omega)$ continuous?

Let $\Omega\subset\mathbb{R}^n$ be open and $\Omega\neq\varnothing$ and suppose we have the Fréchet topology on $C^{\infty}(\Omega)$ (this can be obtained by the topology construction from out ...
0
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0answers
59 views

An Extended Definition of Function of Bounded Variation

I am trying to formulate a definition of function of bounded variation in the setting of locally convex spaces. This is what I've tried. Definition. Let $X$ be a locally convex space whose topology ...
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1answer
110 views

Equicontinuous sequence of linear maps and a closed subspace

I'm having some difficulty with a homework problem regarding Exercise #14 from Chapter 2 of Rudin's $\textit{Functional Analysis}$. (a) Suppose $X,Y$ are topological vector spaces, $\{f_n\}$ is an ...
2
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0answers
52 views

Differential Operators over the space of Analytic Functions

Let $\mathcal{A}(-a,a)$ be the vector space of functions that are analytic on the interval $(-a,a)$ Is there a common topology to place on this space, if yes what is the topology and is it induced ...
3
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1answer
75 views

What norm makes $C^\infty[a,b]$ a complete space?

I have been searching for some common norms used on vector spaces of functions but I am not having any luck finding what the most common norm is to use on $C^\infty[a,b]$ More specifically I would ...
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1answer
56 views

Are all metric space as a euclidean space?

I believe that all euclidean space is a metric space? But I need to know about inverse? I mean: are all metric space as a euclidean space? Is there any kind of metric space which is not euclidean ...
3
votes
1answer
113 views

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 ...
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1answer
131 views

Exercise involving topological vector spaces, linear maps, and the quotient map

I'm doing a homework problem out of Rudin's $\textit{Functional Analysis}$ which is basically a proof of which I have completed some of it, but I'm not sure about the rest of it. Without further ado, ...
1
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1answer
57 views

Some properties about $L^p$ with $0<p<1$

We are coming across many Banach spaces $L^p$ with $1\leq p\leq\infty$. But how about $0<p<1$? Can it be normed? How about its metric induced by the norm? And how about its ...
0
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1answer
105 views

How can this linear map be injective?

Studying the Peter-Weyl theorem, I've come across the following linear maps: $\theta_E: E'\otimes E \rightarrow$ Hom$(E,E)$ Where $ E'\otimes E$ denotes the tensor product of a finite-dimensional ...
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1answer
155 views

The topological vector space that is not metrizable.

Let $C_0(\mathbb{R})$ denote the vector space of continuous functions on the real line with compact support. For any positive function $\rho$ let $$||f||_{\rho}:=\sup_x\rho(x)|f(x)| \ \ .$$ 1) I could ...
0
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1answer
98 views

How to prove that complete implies sequentially complete?

Let us start with a topological vector space (TVS) $X$. We say that $X$ is a complete (resp., sequentially complete) TVS if each Cauchy net (resp., Cauchy sequence) in $X$ converges to a point of $X$. ...
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1answer
61 views

Does $H(\operatorname{div})$ have a Schauder basis?

Let $\Omega$ an open bounded subset of $\mathbb{R}^n$, $n\in\{2,3\}$, and let $$H(\operatorname{div};\Omega):=\{v\in L^2(\Omega):\operatorname{div}v \in L^2(\Omega)\}.$$ My question is: does ...
4
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1answer
472 views

Interior of a convex set is convex [duplicate]

A set $S$ in $\mathbb{R}^n$ is convex if for every pair of points $x,y$ in $S$ and every real $\theta$ where $0 < \theta < 1$, we have $\theta x + (1- \theta) y \in S$. I'm trying to show that ...
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0answers
58 views

On Some Locally Convex Topologies of a Vector Space(Update)

This is an update of my previous question in here. Suppose that $(X,\tau)$ is already a locally convex TVS. Let us denote by $X'$, the space of all $\tau$-continuous linear functionals on $X$, the ...
0
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1answer
280 views

closed subspace of normed vector space

Is every finite dimensional subspace of a normed vector space closed? If yes, please prove it or else give a counter example.
2
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1answer
111 views

On Some Locally Convex Topologies of a Vector Space

Suppose that $(X,\tau)$ is already a locally convex TVS. Let us denote by $X'$, the space of all $\tau$-continuous linear functionals on $X$, the topological dual of $X$. For each $f\in X'$, define ...
3
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1answer
47 views

A question on the sets $V(p,\epsilon)$ in the book of Rudin

I am reading the book of Rudin's functional analysis. Let us start with a vector space $X$ over the reals and we let $P$ be a separating family of seminorms on $X$. For each $p\in P$ and $\epsilon ...
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47 views

A separation lemma in a real vector space

A lattice $N$ is a free $\mathbb{Z}$-module of finite rank. Let $V$ be the real vector space $N\otimes_\mathbb{Z} \mathbb{R}.$ A cone is a set $\sigma = \{ r_1 v_1 + \ldots + r_k v_k \in V : r_i\geq 0 ...
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1answer
32 views

On the set $U_p=\{x\in X: p(x)\le 1\}$

Let $X$ be a Hausdorff locally convex topological vector space whose topology is generated by a family of continuous seminorms on $X$. For each continuous seminorm $p$ on $X$, let $$U_p=\{x\in X: ...
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0answers
51 views

Is a single nontrivial convex set in a topological vector space enough to make it locally convex?

This is sort of a definition question. While a tvs is locally bounded if it contains a bounded neighborhood of the origin, a tvs is called locally convex if it contains a fundamental system of ...
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99 views

Functional analysis exercise

I would really appreciate it if you could give me some advice on the following exercise in Rudin. Put $K=[-1,1]$; define $\mathcal{D}_K$ as the set of all smooth functionals supported in $K$. ...
2
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2answers
93 views

On the Space of Continuous Linear Operators on LCTVS

Suppose that $X$ is a locally convex topological vector space (LCTVS) and that $L(X)$ denotes the space of all continuous linear operators on $X$. Question. How can we construct a topology on $L(X)$ ...
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0answers
39 views

Bounded Semivariation and Bounded Variation in locally convex TVS

Let $(X,\tau)$ be a Hausdorff locally convex TVS and let $P(X)$ be a family of seminorms on $X$ that generates $\tau$. We consider the following definitions. Definition 1. A function $f:[a,b]\to X$ ...
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1answer
182 views

Is there finest topology which makes given vector space into a topological vector space?

I we are given a vector space $(V,+,\cdot)$ over a field $\mathbb K$ (where $\mathbb K=\mathbb R$ or $\mathbb K=\mathbb C$), is there the finest topology $\mathcal T$, such that $(V,\mathcal T)$ is ...
3
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2answers
144 views

How to endow topology on a finite dimensional topological vector space?

This post may be coincide with some of the contents here. From Conway, A course in functional analysis, page 104. If $H$ is a finite dimensional vector space and $F_{1},F_{2}$ are two topologies ...
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0answers
92 views

Compactness in a new convexity

Let $V$ be a topological vector space. For any $m,n\in\mathbb{N}$, denote by $M_{m,n}(V)$ the vector space of all $m\times n$ matrices with entries in $V$. In particular, we denote ...
5
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1answer
145 views

Taking a convex hull does not increase a supremum of a linear function

Let $X$ be a topological vector space, let $f:X\to\Bbb R$ be a continuous linear function and let $P(X)$ denote the set of all Borel probability measures on $X$. For any $M\subseteq X$ we define the ...
2
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1answer
180 views

About functions of bounded variation

I got the following the following idea in one of the articles that I'm reading. It goes this way. Let $X$ be a Hausdorff topological vector space and let $\mathcal{D}$ be the family of all divisions ...
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2answers
53 views

Countable subsets of TVSs

This is something which is not clear to me. Take any countable subset $C$ of a compact set $K$ in a locally convex topological vector space $X$. Can we conclude that there is a point $x\in X$ such ...