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
171 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 ...
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0answers
56 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
107 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 ...
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0answers
51 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
73 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
55 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
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1answer
108 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
120 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, ...
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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 ...
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1answer
100 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
146 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 ...
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1answer
89 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
59 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
368 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
56 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 ...
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1answer
266 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.
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1answer
105 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
46 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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46 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
31 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
90 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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38 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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177 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 ...
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2answers
134 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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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 ...
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1answer
140 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
175 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 ...
3
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1answer
95 views

How Does A Precessing Sphere Precess?

In particular, consider a perfectly spherical object or radius $r$, with a certain axis $L$, and this axis titled relative to a vertical axis by some amount $\theta$. Say that it's "wobbling" in a ...
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70 views

Pseudo norm-exercice

Let $f$ be a measurable function with finite values almost everywhere. We put $$N_0(f) = \displaystyle\int \dfrac{|f|}{1 + |f|} d \mu.$$ We denoted by $L^0$ the set of measurable functions $f$ such ...
2
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1answer
109 views

Smooth function which is not continuous

I have seen it mentioned that in certain infinite dimensional topological vector spaces it is possible to have a smooth curve which is not continuous, but I've never seen an explicit example. Can ...
3
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1answer
117 views

An counterexample of Hahn-Banach theorem in a topological vector space

Problem : Give an example of a TVS $\mathcal{X}$ that is not locally convex and a subspace $\mathcal{Y}$ of $\mathcal{X}$ such that there is a continuous linear functional $f$ on $\mathcal{Y}$ with no ...
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1answer
48 views

On Polar Sets with respect to Continuous Seminorms

In the following, $X$ is a Hausdorff locally convex topological vector space and $X'$ is the topological dual of $X$. If $p$ is a continuous seminorm on $X$ then we shall designate by $U_p$ the ...
2
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1answer
600 views

Question about proof that finite-dimensional subspaces of normed vector spaces are direct summands

I am reading a proof that finite-dimensional subspaces of normed vector spaces have closed direct sum complements. This is the proof: Let $\{e_1, ..., e_n\}$ be a basis for $\mathcal M$. Every $x ...
2
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1answer
124 views

On Absolutely Continuous Functions

I would like to know if we can extend the concept of absolute continuity to functions $f:[a,b]\to X$, where $X$ is a topological vector space. I browsed some books on Topological Vector Spaces but ...
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1answer
34 views

About Regulated Functions

Definition. Let $X$ be a Banach space. A mapping $f:[a,b]\to X$ is called regulated if it has one sided limits. In the setting of a Hausdorff topological vector space $X$, can we still define ...
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53 views

A property involving a polar of a set

Let $X$ be LCTVS and let $X'$ be its topological dual. Let $A\subseteq X$. Suppose that $x\in X$ and satisfies the following property: $$|x'(x)|<1 \mbox{ for all } x'\in A^0$$ where $A^0$ is a ...
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2answers
83 views

A separable, regular space which has cardinality of the continuum but is not first countable?

Actually the title says it all. Is there such a topological space which is separable, regular, has cardinality of the continuum but is not first countable? If so, is there also an example of a ...
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1answer
46 views

About the induced vector measure of a Pettis integrable function(part 2)

Notations: In what follows, $X$ stands for a Hausdorff LCTVS and $X'$ its topological dual. Let $(T,\mathcal{M},\mu)$ be a finite measure space, i.e., $T$ is a nonempty set, $\mathcal{M}$ a ...
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1answer
171 views

About Lusin's condition (N)

We say that $f:[0,1]\to \mathbb{R}$ satisfies Lusin's condition (N) provided $$m(f(B))=0 \quad\mbox{whenever}\quad B\subseteq [0,1] \mbox{ with }m(B)=0$$ where $m$ stands for the Lebesgue measure on ...
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1answer
59 views

Checking if the induced mapping is well-defined

Let $\mathcal{M}$ be a $\sigma$-algebra of subsets of a nonempty set $T$, $X$ is Hausdorff LCTVS, $X'$ the topological dual of $X$, and $m: \mathcal{M}\to X$ a countably additive vector measure on ...
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1answer
52 views

About the induced vector measure of a Pettis integrable function

In what follows, $X$ stands for a Hausdorff LCTVS and $X'$ its topological dual. Let $(T,\mathcal{M},\mu)$ be a finite measure space, i.e., $T$ is a nonempty set, $\mathcal{M}$ a $\sigma$-algebra of ...
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2answers
58 views

Composition of a vector measure and a linear functional

Assume that $X$ is topological vector space over the field $\mathbb{R}$. Let $\mathcal{M}$ be a $\sigma$-algebra of subsets of a nonempty set $T$. We say that $$m: \mathcal{M}\to X$$ is a vector ...
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1answer
219 views

About Henstock integrable vector-valued function

In what follows, $X$ is a Hausdorff locally convex topological vector space over the reals whose topology is generated by a family $P$ of all continuous seminorms on $X$. We consider the following ...
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1answer
76 views

About Equicontinuous and Boundedness

Let $X$ be a TVS and $X'$ denotes the space of all continuous linear functionals on $X$. Let us denote the $weak^*$-topology on $X'$ by $\sigma(X',X).$ My question is this. Why does every ...
2
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1answer
85 views

About a Weak Topology of a Vector Space

Let $X$ be a real vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each $p\in P$ ...
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1answer
48 views

About a Weak Topology on TVS(part 2)

Let $(X,\tau)$ be a topological vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each ...
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1answer
55 views

Homeomorphism via Minkowski functional?

Suppose $E$ is an infinite dimensional topological vector space and $\Omega\subset E$ is open, convex and $0\in \Omega$. The Minkowski-functional of $\Omega$ is defined by: $$ p_\Omega:E\to ...