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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).

18
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1answer
852 views

Is $[0,1]^\omega$ homeomorphic to $D^\omega$?

Let $n\in \mathbb N$ and let $D^n$ be the closed $1$-ball in $\left(\mathbb R^n, \|\,\cdot\,\|_1\right)$. It is not too hard to show that $[0,1]^n \cong D^n$ in this case. This observation leads to ...
12
votes
1answer
2k views

The dual of a Fréchet space.

Let $\mathcal{F}$ be a Fréchet space (locally convex, Hausdorff, metrizable, with a family of seminorms $\{\|~\|_n\}$). I've read that the dual $\mathcal{F}^*$ is never a Fréchet space, unless $\...
12
votes
1answer
226 views

Supremum of Banach Spaces

Let $X$ be a linear space with a family of complete norms $(\| \circ \|_I)_{I \in \mathcal{I}}$ on $X$, i.e. for every $I \in \mathcal{I}$ the tuple $(X,\|\circ\|_I)$ is a Banach space. Now define $$\|...
12
votes
1answer
1k views

Elementary applications of Krein-Milman

Recall that the Krein-Milman theorem asserts that a compact convex set in a LCTVS is the closed convex hull of its extreme points. This has lots of applications to areas of mathematics that use ...
9
votes
2answers
875 views

Definition of the convolution with tempered distributions and Schwartz function

In the book where I'm studying there is the following exercise. If $x \in \mathbb{R}^n$, $\varphi \in \mathcal{S}(\mathbb{R}^n)$ and $u \in \mathcal{S}'(\mathbb{R}^n)$ we define $(u \ast \varphi)(x)=...
8
votes
1answer
349 views

Strong topology vs Natural topology

Let $X$ be a locally convex space and $\left< X, X^{\prime} \right>$ stands for the dual pair. The bidual of $X$ is denoted by $X^{\prime \prime}$ and this is a dual of $X^{\prime}$ with a ...
8
votes
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 $...
8
votes
1answer
852 views

Dual space of space of all smooth function

On the space $C^\infty(S^1,\mathbb R)$, for each $n\in \mathbb N$, define $$p_N(\gamma)= \max\{|f^{(k)}(t): t\in S^1, k\leq N\}$$ Topology of all norms above define a metrizable locally convex ...
7
votes
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 ...
7
votes
1answer
890 views

Constructing a countable family of seminorms in a metrizable LCS.

Here's some context before my question. Let $\mathbb{V}$ be a topological vector space, which is Hausdorff and such that its topology is generated by some arbitrary family of seminorms $\{\rho_{\...
7
votes
1answer
553 views

Uniqueness of the derivative in locally convex topological vector space

I need a hint of proof of uniqueness of the derivative in locally convex topological vector space (it's asserted in Lang's "Introduction to differentiable manifolds"). Define derivative of a function ...
7
votes
1answer
159 views

Does the operation of completion preserve injectivity?

It seems to me I saw a counterexample somewhere, but I can't find it, can anybody help me? Let $\varphi:X\to Y$ be a linear continuous map of locally convex spaces, and $\widetilde{\varphi}:\...
7
votes
1answer
267 views

Reference request: infinite-dimensional manifolds

The following books and/or notes develop various aspects of the theory of infinite-dimensional manifolds: Lang, Fundamentals of Differential Geometry. Kriegl & Michor, The Convenient Setting of ...
7
votes
0answers
261 views

If locally convex topologies exhibit the same dual spaces, do they exhibit the same continuous linear operators?

Consider the following setting: Let $X, Y$ be vector spaces over the field $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$. Furthermore, let $\tau_1, \tau_2$ be locally convex topologies on $X$ and $\...
6
votes
6answers
3k views

Is there such thing as an unnormed vector space?

I learned about Banach spaces a few weeks ago. A Banach space is a complete normed vector space. This of course made me wonder: are there unnormed vector spaces? If there are, can anyone please ...
6
votes
1answer
210 views

Conceptual question about Locally Convex Spaces

Suppose we have a locally convex space $(V,P)$, where $V$ is a topological vector space and $P$ is a family of seminorms defined on $V$ such that for each nonzero $x\in V$ there is a $p\in P$ such ...
6
votes
2answers
270 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 ...
6
votes
1answer
681 views

Rainwater theorem, convergence of nets, initial topology

I've stumbled upon a result called Rainwater's theorem a few times, it seems to be a very useful result in connection with weak convergence in Banach spaces. Rainwater's theorem. Let $X$ be a ...
6
votes
1answer
8k views

Is $\sqrt{x}$ concave?

I have function $f(x)= \sqrt(x)$. To check is it concave or convex i am checkin $f''(x). $ Which is $ -\frac{1}{4x^{\frac{3}{2}}} < 0$ So the $f(x)$ is concave. Is it correct ? And is is the same ...
6
votes
1answer
552 views

Evaluation map is not continuous always.

Let $E$ be a not normable locally convex space, define $$F: E'\times E\to \mathbb R$$ $$(f,e)\to f(e)$$ I have to show that $F$ is not continuous when $E'\times E$ is given product topology. I was ...
6
votes
2answers
887 views

Topology on the space of test functions

I try to read into the theory of distributions and there is one thing which bothers me. I read that a distribution is a linear, continuous functional from the space of test functions, which, depending ...
6
votes
1answer
417 views

Generalization of inner product spaces (analogue to uniform spaces/locally convex spaces)

In the following I am going to devise a chart of topological spaces that contains inner product spaces, normed vector spaces, metric spaces and other related spaces. In the end there will be a gap in ...
5
votes
4answers
640 views

Books on locally convex topological vector spaces

My friend asked me for a good book about locally convex topological vector space. I'm not familar with this. Could you give me some good references on it?
5
votes
1answer
2k views

Definition of locally convex topological vector space

From Wiki A locally convex topological vector space is a topological vector space in which the origin has a local base of absolutely convex *absorbent* sets. Also from Wiki Locally convex ...
5
votes
1answer
539 views

If $L$ is normed and for every hyperplane $M \cap \operatorname{ball} L^*$ weak*-closed implies $M$ weak*-closed then $L$ is a Banach space

If $L$ is a normed space with the property that if $M$ is a hyperplane in $L^*$ and $M \cap \operatorname{ball} L^*$ is weak-star closed $\implies$ $M$ itself is weak star closed, then how do I show $...
5
votes
1answer
2k views

How is the weak-star topology useful?

Today I learnt something about the weak-star topology, but I don't know what the use of weak-star topology is. I hope someone can tell me what we can do with the weak-star topology. Thanks in advance!
5
votes
1answer
207 views

Example of Topological Vector Space

Is there a topological vector space such that, for every $x\in X$, there is a proper neighbourhood $V$ of $x$ in $X$ which is convex, but the whole space is not locally convex (i.e. $X$ has a local ...
5
votes
1answer
83 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 ...
5
votes
1answer
163 views

Caught in the net

I'm reading through some notes one locally convex spaces ("lcs" from now on) analysis and there the following version of the Banach-Steinhaus theorem is given Theorem (Banach-Steinhaus) $\quad$ The ...
5
votes
1answer
294 views

Generalization of Hahn-Banach Theorem

Let $Y$ be a subspace of a locally convex topological vector space $X$. Suppose $T:Y\longrightarrow l^\infty$ is a continuous linear operator. Prove that $T$ can be extended to a continuous linear ...
5
votes
3answers
533 views

Example of a topological vector space which is not locally convex

I'm currently studying Functional Analysis and the professor gave an example for a TVS (which we have defined to be a vector-space $X$ in which addition $X \times X \rightarrow X, (x, y) \mapsto x + y$...
5
votes
0answers
216 views

Isomorphism between $C^\infty_0(B_1)$ and $\mathscr{S}(\mathbb{R}^n)$

Background: Related question I am trying to prove, that the countably-normed spaces $C^\infty_0(B_1)$ on the open unit ball (i.e. function and all derivatives vanish at the boundary) in $\mathbb{R}^n$...
4
votes
1answer
1k views

Convex Hull of Precompact Subset is Precompact

I'm trying to prove that, if $K$ is a precompact (I've also heard the phrase totally bounded used for this) subset of a Banach Space $X$, then its convex hull is also precompact. I've come across a ...
4
votes
2answers
84 views

Convergence on locally convex spaces

I'm new on the locally convex spaces. I know that if $X$ is a vector space and $S$ an irreducible set of seminorms defined in $X$, $(X,S)$ is a locally convex vector space. The first question is, how ...
4
votes
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 $\|\cdot\|$? ...
4
votes
1answer
151 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 ...
4
votes
1answer
521 views

Uniqueness of projection in a Banach space

Let $X$ be a Banach space, $M$ be a subspace of $X$ and $x \in X$ be any vector in $X$. Consider $\displaystyle \hat{x}_M=\arg \inf_{m\in M}\|x - m\|$. Under what conditions for $l_p$ norms $p = 1,...,...
4
votes
1answer
504 views

Local convexity of the topology of weak convergence of probability measures

Let $X$ be a Polish or standard Borel space, and $\mathcal P(X)$ be the space of all Borel probability measures on $X$ endowed with the topology of weak convergence. I am thinking of using Choquet-...
4
votes
1answer
772 views

Strict topology - what is the “importance” of the concept?

From Conway, A course in functional analysis, page 105. Problem 21. Let $X$ be a locally compact space and for each $\phi\in C_{0}(X)$, define $p_{\phi}(f)=|\phi f|_{\infty}$ for $f\in C_{b}(X)$. ...
4
votes
1answer
164 views

Openness of linear mapping 2

I quote a previously asked question : Let $X$ be a topological vector space over the field $K$, where $K=\mathbb{R}$ or $K=\mathbb{C}$, and let $\mathbb\{f\colon X\rightarrow K^n\}$ ($n \in \mathbb{...
4
votes
1answer
242 views

How are the assumptions used in the proof of Bourbaki-Alaoglu Theorem?

This is a follow up question to a previous one. In the proof of the following theorem, where are the assumptions "Hausdorff" and "locally convex" used?
4
votes
1answer
105 views

Topology Book including specific aspects

I am looking for a basic book about Topology (maybe also a bit of Functional analysis but basically Topology) including the following points (in addition to the basic points): $\bullet$ Seminorms $\...
4
votes
2answers
129 views

On the convexity of a particular form of integrals

EDIT: I made some critical corrections below. Let $\mathcal{H}\colon\mathbf{w}\cdot\mathbf{x}+c=0$ be a hypeplane in $\mathbb{R}^n$. Also, let $g\colon\mathbb{R}^n\to\mathbb{R}_+$, be a non-negative, ...
4
votes
1answer
434 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 ...
4
votes
1answer
78 views

convexity of a relatively open subset of a compact set

I'm struggling with the following problem: it seems to be true but I'm not able to prove it! Let $C$ be a compact convex subset of a locally convex metric vector space and $\hat{C}$ be a relatively ...
4
votes
1answer
173 views

Universal property of topology of uniform convergence

What kind of universal property does the strong dual topology on $X'$ have, for $X$ being a locally convex space. Is it possible to define $X'$ as the projective limit of the normed spaces $\mathcal{L}...
4
votes
0answers
218 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^{...
4
votes
0answers
115 views

State space of $C([0,1])$

Consider the $C^*$-algebra $A := C(X)$, where $X$ is a compact Hausdorff space. Denote by $S(A)$ the state space of $A$. In the weak$^*$ topology this is a convex compact set such that $S(A)$ is the ...
3
votes
1answer
473 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
votes
5answers
606 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 $i$. ...