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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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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128 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
169 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
109 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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63 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 ...
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588 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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62 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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362 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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115 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 ...
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1answer
49 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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48 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
34 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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52 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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105 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 $K$. ...
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2answers
97 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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41 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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186 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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165 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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93 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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154 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 ...
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1answer
188 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 ...
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104 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 ...
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115 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 ...
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132 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
51 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 ...
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1answer
749 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 ...
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1answer
135 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
42 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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73 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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91 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
47 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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194 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
63 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
55 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
220 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
77 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 ...
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1answer
88 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
52 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
58 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 ...
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72 views

About a weak topology on TVS

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
117 views

About Closed Unit Balls

If $(X, \|\cdot \|_X)$ is a normed vector space over $\mathbb R$, then the closed unit ball of $X$ is given by $$B(X)=\{x\in X: \|x \|_X \le 1\}.$$ If $X^{*}$ is the set of all bounded linear ...
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91 views

$V$ is finite dimensional iff $V'$ with the weak topology is normable

Why is the following statement valid? Note, $V$ is locally convex Hausdorff topological vector space over $\mathbb{C}$ and $V'$ is the space of all continuous linear maps from $V \to \mathbb{C}$. ...
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1answer
117 views

The weak topology of a product

Let $E$ and $F$ be normed vector spaces and let $E^{\sigma}$, resp. $F^{\sigma}$ be $E$, resp. $F$ with the weak topology associated with the elements of the duals $E^*$, resp. $F^*$. Then, for ...
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474 views

Contractibility of convex set

Suppose that $\Omega$ is a convex open subset of an infinite dimensional vector space $E$ such that $\Omega$ is not contained in any finite dimensional subspace of $E$. Let $Q_m\subset \Omega$ denote ...
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0answers
104 views

Which are nontrivial examples of analytical functions on Frechet spaces?

Let $X$ be a real linear topological space, which is a (separable) Frechet space, such that the topology on $X$ is generated by the countable family $\{p_n:n\in\omega\}$ of norms . A real-valued ...
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2answers
136 views

Hahn-Banach Separation Theorem and Bishop's Theorem

I am looking at the proof of Bishop's Theorem on pages 122 and 123 of Rudin's Functional Analysis. The following quote is from the the last two sentences of the proof on pg. 123. "Every continuous ...
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328 views

Topology of the space of hermitian positive definite matrices

Let $\mathcal{H}_n \mathbb{C}$ be the set of hermitian $n \times n$ complex matrices. This set carries the structure of a vector space over $\mathbb{R}$ under usual addition. It also inherits the ...