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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1answer
117 views

The space of measurable functions is Frechet?

Take a bounded set $S\subseteq \mathbb R^n$ with non-zero measure, and $M_S$ the set of measurable complex functions over $S$. We know that the convergence in measure is metrizable and complete. ...
0
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1answer
16 views

Why is $e \in C$ (a commutative subalgebra of $A$)?

I am reading the following proposition: Here $e$ represents the identity element of $A$ and $\sigma_C(x)$ and $\sigma_A(x)$ denote the spectrum of an element $x$ in $C$ and $A$ respectively. ...
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0answers
26 views

How do I prove the Local intersection property in the example(Economics)

$T$ is said to have local intersection property if for each $x\in X$ with $T(x)\neq\emptyset$, there exists an open neighborhood $N_{x}$ of $x$ such that $\cap_{z\in N_{x}}T(z)\neq\emptyset$. ...
4
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1answer
92 views

It is always possible to define a topology in a vector space endow with a semi-norm?

If $(X,\|\cdot\|)$ is a semi-normed vector space. It is always possible to define a topology on $X$? If it is true What is the definition of a closed subspace of $X$ with respect to $\|\cdot\|$? ...
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2answers
55 views

Topological isomorphism between $C^{\infty}(\mathbb{R}) = \lim_{\leftarrow}{C^{k}([-k, k])}$

Let $X = C^{\infty}(\mathbb{R})$ be the space of smooth functions on a real line considered as a locally convex topological vector space, endowed with the family of seminorms $$p_{m}(\varphi) = \max \...
1
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1answer
76 views

Non-Hausdorff topology on the germs of holomorphic functions

Let $\mathcal{O}_{z}$ be the space of germs of holomorphic functions at $z$, defined as a direct limit of a system $$\mathcal{O}_{z} = \lim_{\rightarrow}{\mathcal{O}(U_{\frac{1}{n}}(z))}$$ where $U_{\...
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1answer
203 views

The finest locally convex topology is not metrizable

Let $E$ be a vector space. Recall that the inductive topology in $E$ taken w.r.t to the family of spaces $\{ E_{\alpha} \}$ and mappings $g_{\alpha}: E_{\alpha} \rightarrow E$ is the finest topology ...
2
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2answers
59 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 ...
3
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1answer
143 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
353 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 $f : X \to \mathbb R$,...
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1answer
27 views

Separating the closed convex envelope of a compact set in Frechet space by a countable family of continuous linear functionals

A space is called Frechet if it is complete metrizable locally convex space. Suppose $Y$ is a Frechet space, and $K$ is a compact subset of $Y$. We let $V$ denote the closed convex envelope of $K$. Is ...
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0answers
115 views

Closed and convex subset of a locally convex space

For the following problem concerning functional analysis I cannot seem to find a solution. It states as follows. Let $(E, \tau)$ be a locally convex vector space and $S \subseteq E$ with $S \ni 0$. ...
1
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1answer
27 views

Is this map continuous?

Let $(E,\langle\cdot\;,\;\cdot\rangle)$ be a complex Hilbert space. For $M\in\mathcal{L}(E)^+$ (i.e. $M^*=M$ and $\langle M x\; |\;x\rangle\geq0$, for all $x\in E$). We define the semi-inner product $...
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0answers
34 views

locally convex vector space, functional analysis, seperation theorem

Let $(E,\tau)$ be a locally convex $\mathbb{K}$-vector-space and $A\subseteq E$ with $0\in A$. The following statements are equivalent: (1) $A$ is closed and convex (2) It exists a subset $B\...
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0answers
74 views

Approximation property for the quotient spaces of $C(T)$

As is known, for any compact space $T$ the Banach space $C(T)$ of all continuous functions on $T$ has the approximation property (see e.g. Albrecht Pietsch, Operator ideals). Is the same true for the (...
3
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2answers
105 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 \...
5
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1answer
85 views

Locally convex subspace

While studying functional analysis the following question came up. Let $(E, P)$ be a locally convex space where $P$ is a family of seminorms. Also, let $F \subseteq E$ be a linear subspace endowed ...
1
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1answer
44 views

Basis locally convex topological linear spaces

Let $X$ be a Hausdorff locally convex linear space, and denote by $\mathcal{N}_{0}$ the class of its (say, closed and absolutely convex) basis neighborhood of zero. Then, can we construct a sequence $...
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0answers
63 views

A convex function with a non-empty domain interior in a non-barreled locally convex space

Given a (Hausdorff separated) locally convex space $X$ what can we say about a proper convex function $f:X\to\mathbb{R}$ whose domain $\emptyset\neq D(f):=\{x\in X\mid f(x)<+\infty\}$ has a non-...
0
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1answer
60 views

how to prove the following equation is convex funcion?

prove that $$ f(x,t)=\frac{\Vert x\Vert _p^p} {t^{p-1}} $$ is convex on $$ \{ (x,t)|x\in R^n ,t\ge 0 \} $$
1
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1answer
129 views

Convex sets in Linear topological spaces

A (real) linear topological space is a real linear space (vector space) $Ε$ with a Hausdorff topology such that: I) vector addition is continuous II) scalar multiplication is continuous For $x$ ...
7
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2answers
296 views

Is the topology that has the same sequential convergence with a metrizable topology equivalent as that topology?

Let $\mathscr T_1$ and $\mathscr T_2$ be two topologies on space $X$. Assume that $(X,\mathscr T_1)$ is metrizable, and any sequence in $X$ that converges in one of the two topologies must also ...
0
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1answer
95 views

Why ${^\perp}(F^\perp)$ is separable in this proof? (I couldn't find the countable set.)

In this proof line -3: "${^\perp}(F^\perp)$ is closed linear span of $F$ and this subspace of $X$ is separable." I know ${^\perp}(F^\perp)=\overline{\operatorname{span}(F)}$, but we only know $F$ is ...
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0answers
75 views

About continuity on the space of measures.

Consider a separable, metrizable and locally compact space $\mathrm{X},$ that is not compact. Define $\mathrm{M}_{\mathbf{C}}(\mathrm{X})$ to be the set of complex measures on $\mathrm{X}$ and regard ...
0
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1answer
158 views

Why ${^\perp}(F^\perp)$ is the closed linear span of $F$?

Let $X$ be a normed space and $F\subseteq X$. How to prove ${^\perp}(F^\perp)$ is the closed linear span of $F$? Here is some definitions: If $A\subseteq X$, then we denote $$A^{\perp}=\{x^*\in X^*:\...
2
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1answer
115 views

Elementary proof that every compact convex subset of a normed vector space has an extreme point

I've shown that the following result is valid for $V=\Bbb R^v$: If $K\subset V$ is compact and convex and $V$ is a $v$-dimensional Banach space, then $K$ has at least one extreme points. The proof ...
2
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2answers
125 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 ...
0
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1answer
21 views

Showing that $\text{int}(\text{ch}(x_1, …, x_k))\subset\text{int}(\text{ch}(x_1, …, x_n))$

I am studying the book Convexity: An Analytic Viewpoint by B. Simon and I am not sure if understood a passage in the following proof (page 124): Proposition 8.7 $\text{ch}(e_1, ..., e_n)$ always ...
0
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1answer
58 views

Help on a passage

I am trying to understand the following proof, which seems to be simple, but I think I am missing something. I couldn't find the open interval about $L(y)$. Lemma 4.2: Let $A$ be an open convex set ...
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0answers
102 views

Over the separation of convex sets in a Banach space

I am working with Banach spaces and aim to prove the following property: If $A$ and $B$ are disjoint convex sets of a Banach space $X$ with $A$ open, then $A$ and $B$ can be separated, that is, ...
0
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2answers
90 views

Convex hull of extremal points

Let $E$ be a locally convex Hausdorff space and $X$ a compact convex set. Is it true that $\overline{conv}(\overline{ex(X)})=\overline{conv}(ex(X))$? Here $conv(X)$ is the convex hull of the subset $X$...
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1answer
141 views

Convex Sets and Linear Subspace

Consider the set $\{(x,y) \in \Bbb R_+ \times \Bbb R \text{ s.t. } y\leq \ln x - e^x\}$. This set is: A) A linear subspace of $\mathbb R^2$ B) Convex C) Convex & a linear subspace of $\mathbb ...
4
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0answers
220 views

A weakly closed set in $X$ that remains weakly-star closed in $X^{**}$

I have $X$ a (non-reflexive) Banach space and $B\subset X$ a weakly closed convex subset. I wonder under what additional conditions (other than weak compactness) $B$ remains weakly-star closed in $X^{...
2
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1answer
72 views

Barreled topology finer than the weak-star topology

I have an infinite-dimensional (Hausdorff separated, non-metrizable) locally convex space $(X,\tau)$ with topological dual $X^*$. My questions are: Under what conditions is there a barreled ...
1
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1answer
218 views

Quotient topology defined by seminorms

Please Prove this: If $p$ is a seminorm on $X$, where $M$ is a linear manifold in $X$ and ${p}_M:X/M \rightarrow[0,\infty)$ is defined by $p_M(x+M)=\inf\{p(x+y): y \in M\}$ then $p_M$ is a seminorm ...
6
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2answers
272 views

Is every seminorm induced by a linear operator into a normed space?

I'm reading an analysis textbook chapter on convex topological vector spaces, and there is this statement that (one of) the most common way(s) to define a topology on a vector space $X$ is by ...
8
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1answer
258 views

Krein-Milman theorem and dividing pizza with toppings

In this question the OP mentions the following problem as an exercise on Krein-Milman theorem: You have a great circular pizza with $n$ toppings. Show that you can divide the pizza equitably among $...
0
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1answer
290 views

Can a strictly increasing convex function $F$ meet a line segment in 3 places, without being linear?

$$(1) F:[0,1] \rightarrow\, [0,1]\,$$ Where a $F$ is a $(C)$ continuous convex function $$\forall t\in [0,1];\forall(x,y)\in[0,1];\,F(tx+(1-t)y)\leq tF(x)+(1-t)\times F(y)$$. where $F$ ...
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0answers
40 views

What are weakest conditions on a vector space that allow defining dual (adjoint) operator?

I have an infinite dimensional topological vector space $V$ and its dual $V^*$ endowed with a sesquilinear form $$\phi:V^*\times V\to \mathbb{C}: (u,v)\to \phi(u,v) , \qquad u\in V^* , v\in V$$ I've ...
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0answers
86 views

Is this true? A strict monotonic function F satisfying Jensen equation with F(0)=0, only expresses dyadic rationals multiples?

. Is is really true that a function satisfying jensen equation F(x/2+y/2)=F(x)/2+F(y)/2, F(0)=0, and F strictly monotonic, where F is a function from F:[0,1] to [0,1] can only express dyadic fraction ...
1
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1answer
90 views

The notation of weak-star topology on the second dual of a locally convex space

I don't understand some notation. I know that if $\mathscr{X}$ is a LCS, then $(\mathscr{X}^*,\text{wk}^*)^* = \mathscr{X}$, where $\text{wk}^*$ denotes the weak-star topology on $\mathscr{X}^*$. ...
2
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0answers
36 views

Minimization of regularized functional

Background: Let $x,y$ be points in $\mathbb{R}^D$. Let $C$ be functionals from $\mathbb{R}^D$ to $\mathbb{R}$, which are strictly convex and convex respectively. Moreover, assume that $F$ is ...
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0answers
81 views

Closed graph theorem for operator topologies - Do operator topologies yield Fréchet spaces?

Consider the SOT and WOT operator topologies on $B(H)$, the bounded operators on a hilbert space $H$. I'm interested in the properties of the topological vector spaces induces on $B(H)$. Are they ...
0
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1answer
50 views

$f|_{(a, b)}$ convex. Is $f$ convex on $[a, b]$? [closed]

Suppose $f : [a, b] \rightarrow R$ is continuous on $[a, b]$ and convex on the open interval $(a, b).$ Show that $f$ is convex on the closed interval $[a, b].$
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0answers
29 views

Showing $\forall q\in\mathcal{P}_Y$ there exists $ p\in\mathcal{P}_X:q\circ A\le p\implies A:X\to Y$ continuous

Let $(X,\mathcal{P}_X)$ and $(Y,\mathcal{P}_Y)$ be locally convex topological vector spaces with topologies induced by the families of continuous seminorms $\mathcal{P}_X$ and $\mathcal{P}_Y$ ...
0
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1answer
45 views

Showing that the balls given by seminorms form a local base

Let $X$ be Hausdorff and let $\mathcal{P}$ be a (countable) family of seminorms on $X$ which generate a topology on $X$ via the neighbourhoods $B_{p}(x,r):=\{x\in X:p(x)<r\}$, for $p\in\mathcal{P}$ ...
0
votes
1answer
44 views

Construct a sequence $(x_{n})$ such that $\bigcap_{n=1}^{\infty} K_{n} = \{x\}$, and $(x_{n})$ is not bounded.

Let $(x_{n})$ be a sequence in $\ell^{p}$ for some $1 < p < \infty$ and let $x \in \ell^{p}$ be given. Set $$ K_{n} = \overline{\text{conv}\left( \bigcup_{i=n}^{\infty} \{x_{i}\} \right)}. $$ ...
3
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1answer
309 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 $...
1
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0answers
28 views

Why is $\bigcup\limits_{|c| \leq 1} cU$ convex?

I have a question about proposition in this set of notes (https://www.math.ubc.ca/~cass/research/pdf/TVS.pdf) on topological vector spaces. We are in a complex vector space $V$. A subset $S$ is said ...
1
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0answers
74 views

Prove limit exists for an almost monotonic bounded function, involves convexity and square integrability

Let us consider the dynamics $\dot{x}(t)=-\gamma(t)y(t)\left(y(t)+m(t)\right)$ where $\gamma(t)\in [0,1]$ is a convex function that gaurantees the bound $\vert x(t)\vert \leq x_{\max}<\infty$ ...