# 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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### 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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### 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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### 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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### 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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### $X$ 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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### 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 ...