Questions tagged [locally-convex-spaces]

For questions about topological vector spaces whose topology is locally convex, that is, there is a basis of neighborhoods of the origin which consists of convex open sets. This tag has to be used with (topological-vector-spaces) and often with (functional-analysis).

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3
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2answers
141 views

Is there a simple proof that every compact convex $K\subset \Bbb R^n$ has at least one extreme point?

Can someone give me a simple proof that every compact convex $K\subset \Bbb R^n$ has at least one extreme point? I'd found this one but couln't understand it: Convex compact set must have extreme ...
3
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1answer
478 views

How can this theorem about weakly measurable functions on $\sigma$-finite measure spaces be deduced from the finite measure space case?

I am reading a theorem about measurability of vector-valued functions in a note on functional analysis: Theorem 3.6.1. If $X$ is a separable, metrizable locally convex space, $(\Omega, \Sigma, \mu)$...
3
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2answers
699 views

What is Convex about Locally Convex Spaces?

This might be a silly question, but what motivates the name "locally convex" for locally convex spaces? The definition in terms of semi-norms seems to have nothing to do with convexity or with the ...
3
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5answers
650 views

(Hausdorff ) Locally convex spaces and their “natural” metric

Today we were introduced to locally convex spaces, defined thusly: A vector space is locally convex iff it has a family of semi-norms $(p_i)$ such that $x=0$ if and only if $p_i(x)=0$ for all $i$. ...
3
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2answers
224 views

Closedness in the proof of the Alaoglu Theorem

I'm reading a proof the Bourbaki-Alaoglu Theorem: Could someone explain how the closedness of $\Phi(V^\circ)$ (namely, $\Phi(V^\circ)$ contains all of its limit points) is done in the proof? I ...
3
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1answer
488 views

linear subspace of dual space

$X$ is a locally convex space and $X^*$ is its dual space with weak* topology or uniform topology. If $H$ is a linear subspace of $X^*$ such that $\bar H \ne X^*$, then is there a non-zero $x \in X$ ...
3
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1answer
158 views

Dual topology and Mackey–Arens theorem

I read only by wikipedia the Mackey–Arens theorem, that is: Given dual pair $(X, X')$ with $X$ a locally convex space and $X'$ its continuous dual, then $\mathcal{T}$ is a dual topology on $X$ if ...
3
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1answer
232 views

In the Krein-Milman theorem, can the weak closure of the convex hull be replaced by norm-closure?

I have a question on the following formulation of the Krein-Milman theorem: Consider a vector space $X$ equipped with the weak topology induced by a separating space $X^*$ of functionals on $X$. ...
3
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2answers
619 views

Can a smooth convex functions be non-differentiable?

Consider the definition of the $\beta$-smoothness (for some constant $\beta$): $$ \|\left. \nabla f \right|_{ y } - \left. \nabla f \right|_{ x } \| \leq \beta \| x - y \| $$ And convexity: $$ f(x)...
3
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1answer
461 views

Extending continuous, densely-defined linear maps between locally convex spaces

Let $X$ and $Y$ be locally convex topological vector spaces, say over $\mathbb{C}$. To set the stage a bit, I'll say that the topology on $X$ is given by a separating family of semi-norms $(p_i)_{i \...
3
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1answer
709 views

Questions regarding internal and interior points for a convex subset of a topological vector space

Suppose that $X$ is a topological vector space, with a convex subset $A$. How do we show that if the vector $u$ is in the interior of $A$, then $u$ is an internal point of $A$ and if the interior of $...
3
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1answer
395 views

Hahn Banach theorem for locally convex topological vector space.

Given a locally convex topological vector space $X$, and a closed proper subspace $Y \subset X$. Take $x \in X \setminus Y$. Is it true we can find a continuous linear functional $f : X \to \mathbb R$,...
3
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1answer
335 views

Mackey–Arens theorem and the weak topology

I have read by Wikipedia about the Mackey–Arens theorem, that is: Let $X$ be a topological vector space and let $\mathcal {T}$ be a locally convex Hausdorff topological vector space topology on $...
3
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1answer
303 views

Convex open neighborhood of compact convex subset

I'm stuck on what ought to be a straightforward topology problem. Say $X$ is a compact convex subset of a locally convex space (everything in sight is assumed Hausdorff). Say $Y\subseteq X$ is a ...
3
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1answer
1k 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 ...
3
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2answers
231 views

About the convexity of Ky Fan's norm

As we know, the Ky Fan norm is convex, and so is the Ky Fan k-norm. My question is, does this imply that the difference between them is a non-convex function, since it results from "difference between ...
3
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1answer
105 views

How to show $X := \bigcup_{n \in \mathbb{N}}X_n$ is a locally convex space.

My textbook on functional analysis says as follows.(The book is written in Japanese, ISBN: 978-4-946552-18-2) Let $\{X_n\}$ be a sequence of locally convex spaces over $K (= \mathbb{R}$ or $\mathbb{C}...
3
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1answer
154 views

Characterization of extreme points of $B_{C_0(X)^*}$

Let $X$ be a compact Hausdorff space and $C(X)$ the Banach space of continuous $\mathbb K$-valued functions equipped with the supremum norm. We denote the dual space of $C(X)$ by $C(X)^*$. A well-...
3
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1answer
74 views

Relations between different definitions of polar topologies

Let $\langle X, Y \rangle$ be a separated dual pair of (real) vector spaces. A non-empty family $\mathcal{A}$ of non-empty subsets of $Y$ is called polar if every $A \in \mathcal{A}$ is bounded (w.r....
3
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1answer
130 views

Proof that a locally convex space $E$ is regular

Where a topological space $E$ is a regular space if, given any closed set $F$ and any point $x$ that does not belong to $F$, there exists a neighbourhood $U$ of $x$ and a neighbourhood $V$ of $F$ that ...
3
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1answer
578 views

Tempered distributions and convergence

It is known that the Schwartz class $\mathcal{S}(\mathbb{R}^n)$ is a Fréchet space and also that the space of test functions $\mathcal{D}(\mathbb{R}^n)$ is dense in $\mathcal{S}(\mathbb{R}^n)$. Let $\...
3
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1answer
303 views

Continuous inclusions in locally convex spaces

Let $(X, \left \| \cdot \right \|_X )$, $(X, \left \| \cdot \right \|_Y)$ two normed vector spaces with $X \subset Y$, by definition we have $X \hookrightarrow Y$ if $\left \| x \right \|_Y \leq C \...
3
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1answer
120 views

Nuclearity of $\mathscr{S}$

I have big problems proving that the Schwartz Space $\mathscr{S}(\mathbb{R}^n)$ together with the topology induced by the family $$ \|\varphi\|_{p}:=\sup_{x\in \mathbb{R^n}}\sup_{|\alpha|\leq p}|(1+|...
3
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0answers
45 views

Is $A = \{ (z_1 , z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2|\}$ an absorbing, balanced and convex set?

Is $A = \{ (z_1 , z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2|\}$ an absorbing, balanced and convex set? I am new in StackExchange I from Colombia, because I don't write English very well. Edited from ...
3
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0answers
228 views

The Minkowski gauge is always bounded by some semi-norm?

My problem sheet asks the following: Let $X$ be a locally convex space generated by the semi-norms $(p_i)_{i\in I}$. Let $A$ be some convex neighborhood of $0$, and $p$ the Minkowski gauge of $p$. ...
3
votes
1answer
87 views

Dense locally convex subspace

$V$ is a locally convex space. I can't manage to prove that a subspace $M$ is necessarily dense if $\forall f\in V^*(f(M)=\lbrace{0\rbrace} \implies f(V)=\lbrace{0\rbrace})$. All I have is $\exists ...
3
votes
2answers
322 views

Product topology on a product space of normed spaces is normable iff the product is finite [duplicate]

Suppose $(X_{i}, \Vert \cdot \Vert_i)_{i\in I}$ are all normed spaces over the same field $\Phi= \mathbb{R}, \mathbb{C}$ and suppose $X= \prod_{i \in I} X_i$ is the product space. I want to show that ...
3
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2answers
529 views

Is the subdifferential always convex and closed set?

Two properties of the subdifferential set are stated as follows: Given a function $f : \mathbb{R}^n → \mathbb{R}$, (i) the subdifferential set $\partial f(x)$ is always convex and closed, even if $f$...
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0answers
99 views

Show that the “folium” is norm closed

A C*-algebra $\mathfrak{A}$ is a Banach algebra with an involution operation $* : \left\lbrace\begin{aligned} \mathfrak{A} &\longrightarrow \mathfrak{A} \\ a &\longmapsto a^* \end{aligned} \...
3
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0answers
73 views

In a proof of “weakly measurable implies measurable”

This is a follow-up question to the following ones: How can this theorem about weakly measurable functions on $\sigma$-finite measure spaces be deduced from the finite measure space case? properties ...
3
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0answers
184 views

Are not all neighborhoods of $0$ in a locally convex space absorbent?

A locally convex space (LCS) can be defined as a topological vector space (i.e. scalar product and sum are continuous) whose topology is generated by translation of a family of balanced and absorbent ...
3
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0answers
130 views

Continuity of a function in a locally convex topological space

I endow the space of bounded sequences with a locally convex topology $\tau$ such that $\tau$ is strictly finer than the product topology (the topology of pointwise convergence), $\tau_p$, and ...
3
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0answers
89 views

Closedness of convex sets in a locally convex space

Let $C$ be a convex subset of a locally convex topological vector space. Consider the properties: a) $C$ is closed. b) $C$ is weakly closed. c) $C$ is weakly sequentially closed. d) $C$ is ...
3
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0answers
121 views

Motivation for the notion of locally convex topological vector space

Is the only motivation for the notion of locally convex topological vector space that the local bases have some nice property i.e. convex, balanced, absorbing ?
3
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0answers
102 views

Strong Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
3
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0answers
121 views

Understanding bornologic spaces

The definition as it appears in Yosida's book: A locally convex space $X$ is called bornologic if it satisfies the condition: If a balanced convex set $M$ of $X$ absorbs every bounded set of $X$, ...
2
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2answers
169 views

Are closed subspaces of reflexive locally convex spaces reflexive?

We know that if $X$ is a Banach space which is reflexive, then any closed subspace of $X$ is reflexive, could we extend the conclusion to any locally convex topological vector space, where $X^{'},X^{''...
2
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3answers
68 views

Is a linear functional on a locally convex space $X$ that continuous on a dense subset of X continuous?

Let $X$ be a locally convex space and let $M$ be a dense subset of $X$. Let $f$ be a linear functional on $X$ such that $f$ is continuous on $M$. Then is $f$ continuous on $X$? Thank you in advance!
2
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2answers
732 views

Barrelled space

A locally convex space is called Barrelled if each closed absorbing convex set is 0-neighborhood See. But i doubt that every absorbing set contains zero. Then is every LCV is barreled. I think, ...
2
votes
1answer
406 views

Characterize polytopes resulting from cutting a convex polytope by a hyperplane

We have a convex polytope $P$ for which we know its set of vertices. Using this set we characterize the H-representation of $P$ as: $\mathbf{A}\mathbf{x} < \mathbf{b}$. If a hyperplane defined by ...
2
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2answers
61 views

Locally convex space with a *algebra structure

Consider a $*$-algebra $A$ (over the complex numbers $\mathbb{C}$), which is also a locally convex space, say by the separated family of norms $\{ n_i\}_{ i \in I }$. Assume that the $*$ automorphism ...
2
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1answer
160 views

Continuous semi-norms on subspace

Suppose $X$ is a locally convex topological vector space, let $P$ be the set of all continuous semi-norms on $X$. Suppose $M$ is a subspace of $X$, denote $P|_M$ as the set of semi-norms in $P$ ...
2
votes
2answers
362 views

A characterization of tempered distributions

The Schwartz space on $\mathbb{R}^n$ is the function space $$ S \left(\mathbf{R}^n\right) = \left \{ f \in C^\infty(\mathbf{R}^n) : \|f\|_{\alpha,\beta} < \infty,\, \forall \alpha, \beta\in\mathbb{...
2
votes
1answer
164 views

Please verify this definition of locally convex vector space

A locally convex vector space is a vector space $V$ equipped with a family $P$ of separating semi-norms. Is this a correct definiton? So to determine whether a norm induces a locally convex topology ...
2
votes
2answers
122 views

Tensor product of two distributions $u \in \mathcal{D}(X)$ and $v \in \mathcal{D}(Y)$

I'm following the proof of the theorem 4.3.3 p. 47 of "Introduction to the distributions theory" by Friedlander and Joshi. We have the following identity \begin{align*} \textbf{(1)} \displaystyle \...
2
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1answer
471 views

Equivalent definitions of locally convex topological vector space

This Wikipedia article gives two equivalent definitions of locally convex space (l.c.s). I don't see clearly the equivalence and I'd like to make it crystal clear. Definition 1 Let $(V,\tau)$ be a ...
2
votes
1answer
139 views

Dual of continuous functions in various topologies

Let $S$ be compact and Hausdorff and $C(S)$ be its space of continuous complex functions. When $C(S)$ is endowed with the $\sup$ norm, its dual is well known. Since this topology is too strong for my ...
2
votes
1answer
112 views

A question Kolmogrov's generalized inequality for projection onto convex sets

Kolmogrov's inequality says that, if $C$ is a convex set, and $P_C(x)$ is an operator for projecting point $x$ into the convex set $C$, if $z = P_C(x)$, then for any $y \in C$ we have $$ (z - y).(x - ...
2
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1answer
224 views

Equivalence of continuity of a functional on a locally convex space

Let $X$ be a locally convex space whose topology is defined by a family of seminorms $\mathcal{P}$. Let $f$ be a linear functional on $X$. Then, I am trying to prove that the following statements are ...
2
votes
1answer
134 views

Mackey topology vs. uniform convergence on weakly compact convex sets

Let $X$ be a locally convex Hausdorff space and $X'$ its dual space. By the Mackey-Arens theorem, there is a finest locally convex topology $\tau$ on $Y$ such that $(Y, \tau)' = X$. $\tau$ is called ...