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.

# Questions tagged [topological-vector-spaces]

1544 questions

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### Differential forms on fuzzy manifolds

This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me.
Question
In the usual setting of open subsets of $\mathbb{R}^n$, differential forms are defined as…

Isaac Sledge

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### "Every linear mapping on a finite dimensional space is continuous"

From Wiki
Every linear function on a finite-dimensional space is continuous.
I was wondering what the domain and codomain of such linear function are?
Are they any two topological vector spaces (not necessarily the same), as along as the domain…

Tim

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### Hahn-Banach From Systems of Linear Equations

In this paper1 on the history of functional analysis, the author mentions the following example of an infinite system of linear equations in an infinite number of variables $c_i = A_{ij} x_j$:
\begin{align*}
\begin{array}{ccccccccc}
1 & = & x_1 & +…

bolbteppa

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### When do weak and original topology coincide?

Let $X$ be a topological vector space with topology $T$.
When is the weak topology on $X$ the same as $T$? Of course we always have $T_{weak} \subset T$ by definition but when is $T \subset T_{weak}$?
Assume that $X$ is any topological space, not…

Rudy the Reindeer

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### Semi-Norms and the Definition of the Weak Topology

When I was searching for the definition of the weak topology, I found two different definitions. One defines the weak topology in terms of a family of semi-norms, while the other defines it in terms of the inverse image of open subsets of the base…

Math

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### When is a notion of convergence induced by a topology?

I'm interested in sufficient conditions for a notion of sequential convergence to be induced by a topology. More precisely: Let $V$ be a vector space over $\mathbb{C}$ endowed with a notion $\tau$ of sequential convergence. When is there a topology…

lvb

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### Learning Aid for Basic Theorems of Topological Vector Spaces in Functional Analysis

I am self-teaching myself the basics of functional analysis (e.g. topological vector spaces), and frankly I am starting to get a migraine sorting out/organizing in my head all of the details/equivalances/implications of the basic theorems involving…

Sargera

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### Hahn-Banach theorem: 2 versions

I have a question regarding the Hahn-Banach Theorem. Let the analytical version be defined as:
Let $E$ be a vector space, $p: E \rightarrow \mathbb{R}$ be a sublinear function and $F$ be a subspace of E. Let $f: F\rightarrow \mathbb{R}$ be a linear…

KevinDL

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### Intuition for separable spaces?

What should I think of when I read that a space is separable? Is it a 'nice' property? Why would I prefer a separable space over a non-separable space or vice versa?
When I think of compact spaces I think of spaces that are in some sense finite. Is…

csss

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### Question about proof that finite-dimensional subspaces of normed vector spaces are direct summands

I am reading a proof that finite-dimensional subspaces of normed vector spaces have closed direct sum complements. This is the proof:
Let $\{e_1, ..., e_n\}$ be a basis for $\mathcal M$. Every $x \in
\mathcal M$ has then a unique representation…

Potato

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### Does every $\mathbb{R},\mathbb{C}$ vector space have a norm?

Is there a canonical way to define on any vector space over $\mathbb{K}=\mathbb{R},\mathbb{C}$ a norm ?
(Or, if there isn't, can someone give me an example of a vector space over $\mathbb{K}$ that is not normable ?)
I have now looked through several…

resu

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### Sum of closed and compact set in a TVS

I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin's Functional Analysis), where $A$ and $B$ are subsets of a topological vector space $X$. In case $X=\mathbb{R}$ this is easy, using…

ScroogeMcDuck

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### The kernel of a continuous linear operator is a closed subspace?

If $V$ and $W$ are topological vector spaces (and $W$ is finite-dimensional) then a linear operator $L\colon V\to W$ is continuous if and only if the kernel of $L$ is a closed subspace of $V$.
Why is this so?

rk101

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### Interior of a convex set is convex

A set $S$ in $\mathbb{R}^n$ is convex if for every pair of points $x,y$ in $S$ and every real $\theta$ where $0 < \theta < 1$, we have $\theta x + (1- \theta) y \in S$.
I'm trying to show that the interior of a convex set is convex.
If $x, y \in$…

Student

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### Examples of the difference between Topological Spaces and Condensed Sets

There is apparently cutting-edge research by Dustin Clausen & Peter Scholze (and probably others) under the name Condensed Mathematics, which is meant to show that the notion of Topological Space is not so well-chosen, and that Condensed Sets lead…

Archie

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