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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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Examples of topological spaces that closed+boundness implies compactness

One familiar example of such one is the Euclideans. Recently I learnt that the space of holomorphic functions over an open set $U\subset\mathbb{C}^n$ also has this property. The formal statement is as ...
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empty interior and openness [closed]

In my teacher's lecture notes have these two notes • Every proper subspace of a topological linear space has empty interior. • Every proper subspace of a topological linear space is not open. My ...
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Is the family of pseudometrics defining the topology of a TVS always equivalent to a translation-invariant family?

Rudin's Functional analysis Theorem 1.24 states, for a metrizable TVS, there always exists a translation-invariant metric matching the topology. But since every TVS, not necessarily Hausdorff, is ...
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Compatibility - Topological modules contra vector spaces

So Tréves in his book on topological vector spaces shows that a filter $\mathcal{F}$ on a $\textit{vector space}$ $E$ is the filter of neighbourhoods of zero compatible with the linear structure if ...
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What is the difference between these separation theorems?

Theorem 1: Let $A$ and $B$ be convex sets of real normed $X$. If $A$ is compact in $X$ and $B$ is closed, then there is $f\in X^*$ and $\lambda_1,\lambda_2\in \mathbb{R}$ such that $f(x)\leq\lambda_1&...
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Is every proper subspace of T. V. S. closed?

I know that every proper finite dimension subspace of T. V. S. Is closed. And I also know that every proper subspace of of T. V. S. is not open. Does this imply that every proper subspace is closed?
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Vector space topologies stronger than the strongest locally convex topology

Every real vector space has the strongest locally convex topology, which is the topology generated by all the convex sets whose intersection with every line is an open interval. What about topologies ...
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Linear surjection from Finite-dim non-Hausdorff TVS to real or complex field always continuous?

I have some confusion about whether some well-known results involving finite-dim TVS still valid if not requiring Hausdorffness in the definition of a TVS. For example, if $\mathbb{F}=\mathbb{R}/\...
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Do "halves" of open sets exist in locally convex vector spaces?

Let $V$ be a locally convex Hausdorff topological vector space (over $\mathbb{R}$) and let $U\subseteq V$ be an open neighbourhood of the origin. Does there always exist another open neighbourhood $U'$...
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Complemented subspaces of $s$

Crossposted to MathOverflow Let $\mathcal s$ denote the space of rapidly decreasing sequences. It is well known that a space $X$ is isomorphic to a complemented subspace of $s$ if and only if it is ...
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Show that the given map is continuous in a Hausdorff space

Let $E$ is an $n$-dimensional Hausdorff topological vector space and $F$ is any topological vector space. If, $\{e_1, \cdots,e_n \}$ any basis for $E$ and $T: E \to F$ is linear map, then show that $$...
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Cancellative convex spaces and topological vector spaces

I am looking at the definition of convex space here: https://ncatlab.org/nlab/revision/diff/convex+space/38 . There is a result that links convex spaces to vector (or affine) spaces and I am trying to ...
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Can individual topological space be considered as category?

Of course, I am aware of Top (category of topological spaces). My question is about something different - can any topological space be considered as category? E.g. its objects may be the open sets of ...
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weak topology of isometrically isomorphic spaces

if two vector spaces are isometrically isomorphic, do their weak topologies have to be the same? for example if we consider the Lebesgue spaces on $\sigma$ finite measure spaces, then $L^p$ is ...
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Intersection of Gelfand-Shilov spaces

Recall the following definition of Gelfand-Shilov spaces: the space ${S_{\sigma,h}^s}$ is defined as $$S_{\sigma,h}^s:=\left\{f\in C^\infty(\mathbb R): \sup \frac{\|x^\beta\partial_\alpha f\|_\infty}{...
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topological jargon confuses me - weak/strong topology and convergence

I am confused about certain terms in topology that are used frequently, and that I can not find a precise explanation for. In this post somebody mentioned, that weak and strong are not used ...
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How are polynomials of tempered distributions defined?

I am reading a textbook on rigorous quantum mechanics and quantum field theory in which there often appears statements such as Let $A(\phi)$ be a polynomial function defined on $\mathcal{S}'(\mathbb{...
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Neighborhoods in inductive topology

I have a question about the following post in MathStack Exchange (here or read below). Let $\{X_i\}$ a family of topologcial vector space(equipped with topology $\mathcal{T_i}$). Let $X = \bigcup_{i =...
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Prove the compactness theorem for Radon measures by using Banach-Alaoglu theorem

I was reading the proof of the compactness theorem for Radon measures (Theorem 1.5.15) from Leon Simon's book: Geometric Measure Theory I was confused by the highlighted part. I hadn't learned the ...
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Complete locally convex topological vector spaces are not stable under extension

I've heard that complete locally convex topological vector spaces are not stable under extension. However, I don't know of any example. What would be an example of a complete topological vector space $...
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Every TVS is $T_{3.5}$ (Tychonoff) even if it is not $T_0$

I'm studying the first properties of Topological Vector space, and I'm confused about the separation properties. Is every TVS $T_{3.5}$ even if it is not $T_0$? This is confirmed by this wikipedia ...
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What is the topology of a Clifford algebra?

I'm reading Hamilton's Mathematical Gauge Theory and I'm currently studying the Pin and Spin groups. Hamilton defines them as specific subsets of a Clifford algebra, and it is understood that the ...
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Difference between the weak and weak* topology (using seminorms to define the topologies)

A few days ago, I was interested in the weak topology and the fact that the weak topology is the coarsest topology such that $f:X \rightarrow \mathbb{K}$ is continuous. (How to show that, the weak ...
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$\newcommand{\co}{\overline{\mathrm{co}}}\co(A)-\co(B) \subseteq \co(A-B)$ in a linear topological space $X$

I am trying to prove that $\newcommand{\co}{\overline{\mathrm{co} }} \co(A)-\co(B) \subseteq \co(A-B)$ in a linear topological space (topological vector space) $X$. Here $\co(A)$ is the closed convex ...
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Is the space $C_0^{k}(\Omega)$ a Montel space?

I'm trying to find a reference for the following result: Theorem: Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and $k\geq 0$. Then $C_0^{k}(\Omega)$ it is (not) a (semi-)Montel space. I tried to ...
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Can we make this subspace $\aleph_0$-dimensional?

Let $X$ be a compact Hausdorff space and $A\subseteq X$ a subspace of $X$. Is it possible for the space $\{f\vert_A:f\in\mathcal{C}(X,\mathbb{R})\}$ to be $\aleph_0$-dimensional as a $\mathbb{R}$-...
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Concave conjuguate and weak star topology

I consider a TVS locally convex and separated. I define on it the concave conjuguate of a concave and upper semi continuous function as $$ f^{*}(x^{*}) =\inf_{y\in X}\left\{x^{*}(y) - f(y)\right\},\...
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Any Hausdorff topology on a finite-dimensional vector space is equivalent to the usual one

I am trying to understand the proof of this statement. Any Hausdorff topology on a finite-dimensional vector space with respect to which vector space operations are continuous is equivalent to the ...
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Does metric vector space imply topological vector space?

I do know that every normed vector space is also a topological vector space. Is every metric (vector) space also a topological vector space. I.e. is there a metric d, such that the induced topology is ...
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How to prove that the Gauge is a norm? Brezis's hint 1.8

In Brezis 1.8 I need to show that the gauge (or Minkowski functional) of the set $$ C= \bigg\{ u \in C[0,1]: \int_{0}^{1} \vert {u(t)}^{2}\vert dt < 1 \bigg \} $$ is a norm, where $C[0,1]$ have ...
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Exercise regarding finite dimensional topological vector spaces [duplicate]

I want to show the following: Let $V$ be a finite dimensional topological vector space and Hausdroff. Suppose $f: \mathbb{K}^n \rightarrow V$ to be an vector space isomorphism. Show that $f$ is a ...
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Proving a Proposition about linear functionals on topological vector spaces.

I want to show the following: Let $X$ be a topological vector space (and Hausdorff), $f:X \rightarrow \mathbb{K}$ a linear functional. Then the following are equivalent: 1.$f$ is continuous 2.$f$ is ...
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Prove or disprove: if $X$ is a finite-dim TVS, $F\subset X$ is absorbing and convex, $\Omega\subset X$ is closed, and $0\in\text{int}(F)$...

I couldn't fit the whole question in the title, sorry! I'm trying the following bonus question from my Convex Analysis homework: Let $X$ be a vector space over $\mathbb R$, and let $F\subset X$ be an ...
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Open set in LF-space

Assume $E_1\subset E_2\subset ...$ is a sequence of Frechet spaces, each one embedded in the next. Let $E$ be their union, and one puts the LF topology on $E$. I wished one can say the following: If $...
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Extreme points of set of bounded measures satisfying continuity equation

$\textbf{Ordinary Differential Equation:}$ Let $x(\cdot) \in C([0,1];\mathbb{R}^n)$ (C($\cdot$) denotes the set of continuous functions) be the trajectories satisfying the following differential ...
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general Convex set definition

A set is $C$ is convex if $tC+(1-t)C\subset C$ where $0\leq t \leq 1$. how do we manage to extend it for n vectors? I manage to get $t^2 C + t(1-t)C + t(1-t)C + (1-t)^2 C \subset C$ but induction ...
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1 answer
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Dual of topological vector space

Assume $E$ is an infinite dimensional vector space. Assume $\tau_1, \tau_2$ are two topologies that make $E$ into topological vector spaces. Assume $\tau_1$ is strictly finer than $\tau_2$. Is there ...
Yuval's user avatar
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4 votes
1 answer
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Dual of completion and weak$^*$-topology

Let $X$ be a dense subspace of a Banach space $Y$. The restriction map $$r: Y^* \to X^*:\omega \mapsto \omega \vert_X$$ is then an isometric isomorphism that is weak$^*$-continuous. I am wondering if ...
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Example of a point that is not the limit of any sequence in a connected topological space

Question: Let $X$ be a connected space with a topology not necessarily sequential. What is an example where a point in $X$ is not the limit of any not eventually constant sequence? Motivation. ...
user760's user avatar
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Final topology coinduced by topological vector spaces

Let $X$ be a vector space, $(X_i)$ a collection of (not necessarily locally convex) topological vector spaces, and $T_i \colon X_i \to X$ linear maps. Then the $T_i$ coinduce a final topology $\...
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closed convex hull, closure of convex hull and convex hull of closure

Let $X$ a topological vector space and $A\subseteq X$ a subspace. Let $co(A)$ the convex hull of $A$ (the smallest convex subspace containing $A$) and $\overline{co}(A)$ the closed convex hull of $A$...
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A topological vector space is separated if and only if $\{0\}$ is closed

I am new to the study of topological vector space and I would like to show the result stated in the title above but I have some doubt, hence the question. For the first implication, I don’t know how ...
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i want to know if s_0 is strictly less than 1 in the the given inequality of $t^{-1}x=s_0 $?

Let $A$ be a convex absorbing set in the vector space $X$. The sets $C$ and $B$ are defined as follows: $C= \{ x : \mu_A(x) \leq1\}$ and $B=\{x: \mu_A(x) < 1\}$. this also means that $B\subset A \...
Jayaraman adithya's user avatar
3 votes
2 answers
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Are complements of proper subspaces dense?

Consider a topological vector space $V$ over $K\in\{\mathbb{R, C}\}$. I ask a simple innocent question: Is the complement of every proper subspace dense? What if the space is normed? Or has an inner-...
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If $E$ is the strict inductive limit of Frechet spaces $\lbrace E_k\rbrace$, why can't the countable basis of $E_1$ be appropriated for $E$?

Not wishing to dispute that strict LF-spaces are non-metrisable, I am trying to see the flaw in the intuitive sense I have that we could appropriate, for a strict LF space, a countable basis from the ...
demim00nde's user avatar
2 votes
1 answer
75 views

Relative interior of convex set and its closure coincide in infinite dimensional spaces? [closed]

Let $X$ be an infinite dimensional Banach space, and $C\subset X$ a convex set. Is it always true $\text{ri}(C)=\text{ri}\left(\overline{C}\right)$? What about for general topological vector spaces? ...
user760's user avatar
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-1 votes
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The space of finite signed measures and duality

Let $(X,\mathcal{A})$ be a measure space and $\mathcal{X}$ be a linear space of measurable functions from $X$ into the real numbers. So $\mathcal{X}$ could for instance be an $L_p$ space. Now in a ...
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If $X$ is a topological vector space over $\mathbb{R}$ and $f:X\to \mathbb{R}$ is a nonzero continuous linear map, then if $G$ is open so is $f(G)$.

I'm trying to prove that if $X$ is a topological vector space over $\mathbb{R}$ and $f:X\to \mathbb{R}$ is a nonzero continuous linear map, then $f$ is open in the sense that if $G$ is open so is $f(G)...
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2 votes
1 answer
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$\mathcal{O}(U)$ as a projective limit of Hilbert spaces

It is well-known that the space of holomorphic functions $\mathcal{O}(U)$ (with the standard topology of compact-uniform convergence) on an open set $U \subset \mathbb{C}$ is a projective limit of ...
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Example of a LCS with a countably compact barrel

I am self-studying topological vector spaces and I wonder if there is an example of a sequentially complete LCS with a countably compact barrel. I am a complete beginner and really can't think of any. ...
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