# 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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### Must a linear functional on a TVS with a sequentially closed kernel be sequentially continuous?

Let $(X, \tau)$ be a Hausdorff topological vector space, $\mathbb{K}$ the scalar field with its usual topology and $\Lambda : X \to \mathbb{K}$ a linear functional. There is this general criterion ...
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### A characterization of closure of a certain class of sets in $\mathbb{R}^n$

Consider a set $K\subset \mathbb{R}^n$ that is symmetric ($B = -B$) and verifies $aK\subset bK$ if $|a|<|b|$. Can I conclude that $\overline{K} = \cap_{a>1}aK$? If not, does the result hold ...
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### Theorem 1.36 in Rudin's Functional Analysis

I have a few questions regarding the proof of Theorem 1.36 in Rudin's Functional Analysis: Why does $V$ being open imply $x/t\in V$ for some $t<1$. How does the inequality $\mu_V(x-y)<r$ come ...
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### Is multiplication by a scalar an open map in topological groups?

If n is any non-zero integer and G is a topological abelian group, under what conditions is the continuous homomorphism $g \mapsto ng$ open or weakly open? If this map happens to be open for all n and ...
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### Proof that every $T_0$ Topological Vector Space is regular

I'm currently reading through Convex Analysis and Beyond by Mordukhovich and Nam where the following preposition and proof are given (note that the authors define TVSs to be $T_0$). Proposition 1.92 ...
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### Is the topology essential for local fields?

References to local fields tend to define local fields in some topological way. For example, a field $K$ is a local field if it is complete with respect to a topology induced by a discrete valuation, ...
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### Finding a compact set to apply the Hahn-Banach separation theorem in a locally convex topological vector space

I am trying to justify how the Hahn-Banach theorem was applied in the proof below. It looks like the proof is using the case for locally convex space (because the inequalities are strict). that ...
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### Local basis criterion for topological group / vector space?

Let $X$ be a topological vector space (TVS), and let $\mathcal B$ be nonempty family of subsets of $X$ that each contain $0$. Are there simple conditions that guarantee $\mathcal B$ is a local basis ...
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### Dual space of a topological vector space that doesn't separate points?

While I am studying FUNCTIONAL ANALYSIS by Walter Rudin, I found the following corollary. Now, I wonder how the dual space(the set of all continuous linear functionals on $X$) of some pathetic ...
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### Can I define boundedness in a topological vector space w.r.t. arbitrary point?

I have started reading Rudin's Functional Analysis, and in Section 1.6, he makes the following definition: A subset $E$ of a topological vector space is said to be bounded iff for every open ...
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### What is the relationship between convex conjugate and polar set?

Given a duality $\left<X^*,X\right>$ over field $\mathbb R$, and any set $A \subseteq X$, the polar set of $A$ is defined as \begin{align}A^\circ = \{x^* \in X^* | \left<x^*,x\right> \leq ...
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### For non second-countable TVS, is the sum of measurable functions again measurable? [duplicate]

Let $(\Omega, \mathcal A)$ be a measurable space and $E$ a topological vector space. Let $f,g:\Omega \to E$ be measurable. I already proved that Theorem $E$ is second-countable, then $f+g$ is ...
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### Closure of balanced convex set and its dilation

While reading a proof, one of the steps was showing that for a (specific) balanced and convex set in a Banach space we had the inclusion $\overline{B} \subseteq 2B$. The proof continued using specific ...
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### Why does $C^\infty(\Omega)$ have the Heine-Borel property?

$\let\uto\rightrightarrows \let\ii\infty \let\W\Omega \let\a\alpha \let\b\beta \let\e\varepsilon \let\d\delta \let\sbe\subseteq$ I'm struggling to understand the examples at the end of the first ...
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### Why is $C^\infty(\Omega)$ complete?

$\let\uto\rightrightarrows \let\ii\infty \let\W\Omega \let\a\alpha \let\b\beta \let\e\varepsilon \let\d\delta \let\sbe\subseteq$ I'm struggling to understand the examples at the end of the first ...
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### If $\overline A$ is compact, then $\overline A=\overline A^w$

Let $X$ be a Hausdorff topological vector space such that $X'$,it topological dual separate points of $X$, and $A\subset X$. I want to know if the following sytatement is true or not? If $\overline A$ ...
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### Scales of a bounded neighborhood of $0$ forms a neighborhood basis of $0$.

Exercise: Let $B$ be an open bounded neighborhood of $0$ in a topological vector space. Show that every neighborhood of $0$ contains a set of the form $\{sB : s\in(0,\infty)\}$. Hint: One can use the ...
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In topological vector spaces, translation and dilation are homeomorphism. From this one can proof that for any $x\in X$, where $X$ is the topological vector space, the neighbourhood filter $\mathcal{U}... 1 vote 1 answer 51 views ### Homeomorphism in topological vector spaces So on the german wikipedia page of topological vector spaces it is written that, if a topological vector space is a Hausdorff space, then translation by a vector and dilation by a scalar are ... 1 vote 0 answers 90 views ### Showing that entire analytic functions are dense in Schwartz space The following theorem is presented as question 15.6 in Treves (1967, 1995).$\mathscr{S}$is the Schwartz space of functions on$\mathbb{R}^n$, and when one says functions on$\mathbb{C}^n$are 'dense'... 0 votes 0 answers 32 views ### A locally convex space has a topology given by a single norm if the topology is generated by finitely many seminorms This is a problem 1a from section V in Reed & Simon's book on functional analysis. It states: 1a. Prove that a locally convex space has a topology given by a single norm if the topology is ... 0 votes 1 answer 58 views ### The Mackey topology$\tau(X^*, X)$Let$X$be a Banach space and$X^*$its dual. I would like to better understand the Mackey topology$\tau(X^*, X)$on$X^*$. The Mackey topology on$X$,$\tau(X,X^*)$is defined as the topology of ... 1 vote 0 answers 16 views ### Is there any way to show that the algebraic tensor product$A[[h]] \otimes A[[h]] \subsetneq (A \otimes A) [[h]]\ $? Let$A$be a Hopf algebra over$k.$Consider the formal power series$A[[h]]$in$h$over$A$endowed with the$h$-adic topology. Then how do we show that$A[[h]] \otimes_{\text {alg}} A[[h]] \...
Let $\{E_n, n\in \mathbb{N}\}$ be an increasing sequence of linear subspaces of a vector space $E$, i.e. $E_n \subset E_{n+1}$ for all $n\in \mathbb{N}$ such that $E = \bigcup_{n\in \mathbb{N}} E_n$. ...