Questions tagged [topological-vector-spaces]

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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Proof: $S$ a subset of $l^2(\mathbb{N})$ is a closed subset

I am doing exercice on a book and sometimes or i haven't the solution to the question or i didn't understand their solution. Question: Proove that the subset $S$ that countain all of the sequences of ...
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Circled (balanced) subsets of $\mathbb{K}$

I'm reading the proof of Lemma 3 in Jarchow, H. Locally Convex Spaces. A subset of a vector space $E$ is called circled if $\mathbb{D} \cdot A \subset A$, where $\mathbb{D}=\{\rho \in \mathbb{C}:|\rho|...
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Compare weak and finest locally convex topology on $\mathbb{R} [x]$

Let $V = \mathbb{R} [x] \cong \bigoplus_{\mathbb{N}} \mathbb{R}$ be the vector space of univariate polynomials, or the space of real sequences that have all but finitely many elements equal to zero. ...
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Why is a compact linear operator also a bounded linear operator?

I know that a linear operator is bounded iff continuous, with these definitions of "bounded" and "continuous": a linear operator T (on H Hilbert space) is bounded if ∃M>0: ||Tf|...
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The role of sub-multiplicative property (∥AB∥≤∥A∥∥B∥) and ∥I∥≥1 in the definition of 'matrix norm'

A normed linear space $(X, \lVert \cdot \lVert)$ can form a metric space induced by the norm and thereby a topological vector space. Thus the analytical concepts can be taken along with $X$. The ...
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Is the space of $n\times n$ real symmetric matrices with strictly positive determinant connected within the vector space of $n\times n$ real matrices?

I want to make clear that I am aware of the connectedness in the case of general real matrices. But here I ask about the subspace of symmetric ones. If it is not the case, which are the connected ...
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Necessary and sufficient condition for metrizability of topological group , module , ring

We know that a topological vector space is metrizable iff it has a countable local base and in general a topological space is metrizable iff it is $T_3$ and has a countably locally finite basis. Now ...
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Intersection of compact sets in different spaces

Let $A$ be a compact set in $L^1$ and $B$ a compact set in $L^2$. Determine if $A \cap B$ is compact in $L^1$ or $L^2$ or both. My idea: Since $L^2 \subset L^1$ and $\Vert \cdot \Vert_2$ is stronger ...
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A proof of Krein-Milman theorem

Disclaimer This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) Let $X$ be a locally convex Hausdorff ...
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The converse of the Heine–Borel property

I have been reading on the Heine–Borel theorem and Heine–Borel property and their relation to topological vector spaces. The Heine–Borel theorem states each subset of Euclidean space $\mathbb{R}^n$, ...
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Is there a sense in which a function converges to its total derivative?

In short: Is there a way to rigorously define the total derivative (of $F$ at $x$) as a function $dF_x$ which is (1) linear and (2) a usual topological limit of some function $\mu:X\to Y$, where $X$ ...
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What is the constant of hyperbolicity?

I am studying dynamical systems of discrete time, and I am having some trouble in understanding what is the constant of hyperbolicity for a closed hyperbolic set $\Lambda \in M$ of a diffeomorphism $f:...
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Let (X, τ ) be a topological space. Show that ${\{x\}}= \bigcap_{G \in τ }G $

Let (X, τ ) be a topological space. Suppose that for any x ∈ X one has that {x} is a closed set. Show that: It is known that $\bar{\{x\}}=\{x\}$ by theorem $${\{x\}}= \bigcap_{\{x\} \subset F}F $$ ...
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Compatibility of the topology with linear nature of the set

I see that for analysis (functional analysis) in a vector space $V$ over $\mathbb F$ ($\mathbb R$ or $\mathbb C$ ) , we need some topology $\tau$ on $V$ compatible with the linear nature. The so ...
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Are these two definitions of sublinear functionals equivalent?

Let $E$ be a t.v.s. In Brezis's book Functional Analysis, a sublinear functional $p:E \to \mathbb R$ is defined as a map that satisfies $p(\lambda x)=\lambda p(x)$ for all $\lambda > 0$ and $x\in ...
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Let $E$ be a t.v.s. and $f$ a discontinuous linear functional on $E$. There is a net $(x_d)$ such that $x_d \to 0$ and $f(x_d) = 1$ for all $d$

In a previous post, I proved that Let $E$ be a locally convex t.v.s. and $f \not \equiv 0$ a linear functional on $E$. Then $\ker f$ is either closed or dense in $E$. Then I have found a proof ...
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Let $E$ be a locally convex t.v.s. and $f \not \equiv 0$ a linear functional on $E$. Then $\ker f$ is either closed or dense in $E$

I'm trying to prove below result. Let $E$ be a locally convex t.v.s. and $f \not \equiv 0$ a linear functional on $E$. Then $\ker f$ is either closed or dense in $E$. The space $E$ is assumed to be ...
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Prove that the function $φ:C(I)\rightarrow\Bbb R$ defined as $φ(f):=f(1)$ for any $f\in C(I)$ is not continous with respect the $L_2$ topology.

If $C(I)$ is the set of continous function on $I:=[0,1]$ then it is a well know result that the positions $d_\infty(f,g):=\sup\{|f(x)-g(x)|:x\in I\}$ $\langle f,g\rangle:=\int_0^1fg$ for any $f,g\in ...
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Does dense inclusion of dual space implies reflexive

If $X$ and $Y$ are Banach spaces that $Y \subsetneq X$, suppose $i : Y \hookrightarrow X$ and $i^{\star} : X^{\star} \hookrightarrow Y^{\star}$ are both continuous and norm dense. Must $Y$ be ...
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Completion of Linear Topological Spaces

I would like to read the proof of the theorem stating that every linear topological space has a completion. Is it the same as arguments in metric spaces? Do you have a resource suggestion?
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Is singleton set of interior of a convex set an extreme point?

Let $K$ be a convex set in topological vector space and $a$ an interior point of $K$. Can it be an extreme point of $K$? Suppose $a$ is an extreme point of $K$. Then $a=(1-t)x + ty$ for $x\neq y \in K$...
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The weak topology on a dual pair $(X,Y)$ is metrizable iff the dimension of $Y$ is at most countable.

Here $X,Y$ are assumed to be vector spaces, and $Y$ a subset of the algebraic dual of $X$. The weak topology if of course generated by the family of seminorms $\{ |y(x)| < \epsilon, y \in Y, \...
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Showing some set is open in a normed vector space.

Given $A,B$ two convex subsets of a normed vector space $E$, such that $A\cap B = \emptyset$. I’m trying to show that if $A$ is open then the set $A-B = \left\lbrace a-b ,\; a\in A, \; b\in B\right\...
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Equivalence of topologies on the Schwartz space

I am dealing with the Schwartz space: set of all inifinite differentiable functions on $\mathbb{R}$ such that for all $a, b\geq 0$, $x^{a}f^{(b)}(x)$ is a bounded function. The topology is generated ...
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Existence of continuous norm in topological vector spaces

I am being asked to check whether at least one continuous norm exists in some locally convex Hausdorff topological vector spaces. I am not sure what is an efficient way (or useful theorem) that would ...
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$[x,y)$ belongs to interior of convex set $K$.

$X$ is a topological vector space. $K$ is convex subset of $X$. if $x\in int(K)$ and $y \in K$, then $$[x,y)=\{xt+(1-t)y| t\in (0,1]\} \in int(K)$$ . for $t=1$ and if $K=\phi$, proof is trivial. I ...
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If intersection of closed convex hull is singleton that implies weak convergence

$X$ be a locally convex topological vector space. $\{x_n\}$ converge weakly to $x$ iff $x$ in the closed convex hull of every subsequence of $\{x_n\}$. I'm able to show the only if part that is if $...
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Showing $L^{\infty}(\mathbb R)^*\neq L^1(\mathbb R)$ using the weak star compactness of unit ball in $L^{\infty}(\mathbb R)^*$.

$\lambda_n(f)=\frac{1}{2n}\int_{-n}^{n}f$ defines a dual element of $ L^{\infty}(\mathbb R)$. It is easy to see that $\lambda_n\in L^{\infty}(\mathbb R)^*_1.$ By using the weak-star compactness of ...
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Set of all probability measure is not compact

I am trying to find an example of the following fact : If $X$ is not a compact space, then set of all probability measure $\mathcal M(X)$ may not be compact in weak star topology. As my intuition ...
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A Hausdorff locally convex space is normable iff it can be generated by a finite subfamily of seminorms.

Suppose $X$ is a Hausdorff locally convex space, then we know its topology can be generated by a family of seminorms. Let us call $P$ to this defining family of seminorms on $X$. I need to show that X ...
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Prove T is continuous and open map

$X$ and $Y$ is topological vector spaces. $T$ is linear map between this two spaces. Also given that $ker(T)$ is closed and $Y$ is finite dimensional. We have to show that $T$ is continuous and open ...
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Prove that $A-A = X$, where A is subset of Banach Space and dense $G_{\delta}$ set.

$X$ is Banach Space and $A\subseteq X$. $A$ is dense $G_{\delta}$ set. We have to show that $A-A=X$. Since $A$ is dense we can write $\bar{A}= X$. So, it is enough to show that $A-A= \bar{A}$. Here $A-...
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Confusion about the proof that every convex proper l.s.c. function is bounded below by an affine function

I'm reading a proof of Theorem 2.20 in Barbu's textbook Convexity and Optimization in Banach Spaces. Proposition 2.20 Any convex, proper and lower-semicontinuous function is bounded from below by an ...
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Is the infinite product of bases of coordinates the basis of product topology

Let $X_1,X_2,\ldots, X_n$ be finitely many topological spaces. For $i=1,2,\ldots, n$, if $\mathscr B_i$ is the base of $X_i$, then it is well know that the family $$\mathscr B=\{B_1\times\cdots \times ...
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Can I find the topology according to certain definition of convergence?

For instance, consider the space $C[0,1]$, I hope $f_n \rightarrow f$ means that $\sup_{[0,1]} |f_n - f| \rightarrow 0$. I know the convergence is meaningful only by specifying the topology, and I am ...
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If $f:F^n\rightarrow X$ is linear, then is $f$ continuous? [duplicate]

Let $X$ be a topolgical vector space over field $F$. If $f:F^n\rightarrow X$ is linear, then is $f$ continuous?
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Why $p(x-y) \geq p(y)-p(x)$,implies $p(x-y) \geq |p(y)-p(x) |$ for the seminorm $p$

In this lectureFucntional Analisys II at page 2 the author has Definition 1.1. Let $X$ be a vector space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. A function $p: X \rightarrow \mathbb{R}$ is ...
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Is this (not necessarily compact) convex set the closed convex hull of it's extreme points?

Let $(\mathcal X,\Sigma_X)$ be a measure space, let $\mathcal S_X=\{ q : (\mathcal X, \Sigma_X, q)\text{ is a probability space}\}$, i.e. the simplex associated to $(\mathcal X,\Sigma_X)$. We equip $\...
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The inf-convolution of $2$ convex l.s.c. is convex l.s.c.?

Let $E$ be a t.v.s. and $f, g:E \to (-\infty, +\infty]$ be convex l.s.c. such that $f,g \not \equiv +\infty$. We define the infimal-convolution $f \square g:E \to [-\infty, +\infty]$ by $$ (f \square ...
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$X$ be LCTVS and $K$ be compact, $B$ closed and convex subset of $X$. Let $A\subset X$ such that $A+K\subseteq B+K$, prove that $A\subseteq B$

Here LCTVS stands for locally convex topological vector space. I'm getting no idea how to approach the problem. I have tried using Minkowski functional but couldn't get much. I know a result (...
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An exercise on density of Evaluation maps

The following is an exercise I am trying to solve: "Let $X$ be a locally compact Hausdorff topological space, and $CB(X)$ denote the space of all bounded continuous functions on $X$, with the ...
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What is a symmetric tubular neighborhood?

In ``Anomalies and Invertible Field Theories'' by Dan Freed [hep-th/arXiv:1404.7724], the author refers (in footnote 2) to what he calls a symmetric formal $n$-dimensional tubular neighborhood of $N$ ...
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1 answer
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Can this set be not separable ? Can it be not compact?

Let $(\mathcal X,\Sigma_X)$ be a measure space, let $\mathcal S_X=\{ q : (\mathcal X, \Sigma_X, q)\text{ is a probability space}\}$, i.e. the simplex associated to $(\mathcal X,\Sigma_X)$. We equip $\...
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Vector space translation continuous implies addition continuous?

In here, it is proved if a topology on a vector space makes the addition function continuous, then the translation is also continuous everywhere. My question is whether the inverse is still true: Let $...
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Examples of two different vector space topologies with same continuous functionals

Question: Given a vector space $V$, is it possible to endow it with two different vector space topologies $\mathcal T_1$ and $\mathcal T_2$ such that any linear functional on $V$ is continuous in the ...
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What is the relationship between boundedness, total boundedness and compactness in topological vector space?

In our lecture notes, we have that in a topological vector spaces, every compact set is totally bounded and every totally bounded set is bounded but is the converse true?
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Baire Category Theorem proof confusion

I am not understanding the following proof from Folland. 5.9 The Baire Category Theorem. Let $X$ be a complete metric space. a. If $\left\{U_{n}\right\}_{1}^{\infty}$ is a sequence of open dense ...
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If $E$ is locally convex and Hausdorff, then so is $E^I$

In solving Ex 3.17 in Brezis's book of Functional Analysis, I come across below result. Let $E$ be a locally convex Hausdorff t.v.s. and $I$ a set of indices. We endow $E^I$ with the product topology ...
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Some Questions on Topological vector spaces

I am having really hard time in my Functional Analysis II class. I never had this problem with Functional Analysis I class, which mostly focused on metrics spaces, normed spaces, banach space and ...
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Why the finite sequences space $\mathbb{K}^\infty$ with the final topology is a locally convex topological vector space?

On Wikipedia, it is claimed that the finite sequences space $\mathbb{K}^\infty$ with the final topology is a locally convex topological vector space. But I couldn't figure out how to prove this, both ...
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