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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1answer
86 views

characterization of topological vector spaces

It is known that if $X$ is a topological vector space (TVS), then all the translations and nontrivial scalar multiplications are homeomorphisms. I'm curious about the following question about which ...
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1answer
88 views

Linear span of general set in topological linear spaces

I'm studying Functional Analysis and I'm in doubt with the definition of linear span. The book states that: Let $\mathscr{L}$ be a topological linear space and let $\mathscr{M}$ be a linear ...
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37 views

“Algebraic isomorphism $\implies$ Homeomorphism” in the topological vector space context

This question just pop up when I was trying to solve another problem. Let $X,Y$ be vector topological spaces over the field $\mathbb{K}$ ( with $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ ) and let $f:X\...
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1answer
55 views

How does the definition of compactness imply that all continuous operators are compact in finite dimensional spaces?

Let $S \subset X, Y$ be normed spaces over $K$. An operator $A:S \to Y$ is called compact if: $A$ is continuous $A$ transforms bounded set into relatively compact sets i.e. if $(c_n)$ is ...
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1answer
66 views

Is $C^{\infty}(\mathbb{T})$ dense in $C(\mathbb{T})$?

For a topological space $X$, the space of smooth functions with compact support (denoted by $C^{\infty}_0(X)$) is dense in the space of continuous functions vanishing at infinity (denoted $C_{\infty}(...
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1answer
57 views

How strong is the operator norm topology?

Let $(V,\tau_V), (W,\tau_W)$ be normable topological vector spaces. Let $||\cdot||_V, ||\cdot||_W$ be norms on $V,W$ inducing $\tau_V, \tau_W$ respectively. Let $||\cdot||_{op}$ be the operator norm ...
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35 views

Norms on $L(V,W)$

Let $V,W$ be normable topological vector spaces over $\mathbb{F}$. Let $C(V,W)$ be the set of continuous linear transformations $T:V\rightarrow W$. Let $||\cdot||_V, ||\cdot||_W$ be norms on $V,W$ ...
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1answer
29 views

Does every Hamel basis for an infinite dimensional topological vector space have a maximal countable spanning subset?

‎‎‎‎Let ‎$ X $ ‎be an infinite dimensional topological vector space and ‎$\{ e‎_{i}: i ‎\in I ‎\}$‎ be a Hamel basis for $X$. Does there exist a maximal countable subset ‎$J‎\subset ‎I$ ‎such ‎...
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1answer
69 views

Sequence of topological spaces

A friend of mine did an exercise where a part of the text was: In $\mathbb{R}^3$, with euclidian topology, we consider $X=\mathbb{S}^2 \setminus \{ N \}$, where $N= (0,0,1)$ and $E=\{(x,y,z) \in \...
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35 views

The restriction of a discontinuous linear functional to any open set is surjective.

Problem. Let $X$ be a topological vector space and $f:X\to\mathbb{K}$ a linear mapping. Prove that if $f$ is discontinuous, then $f(A)=\mathbb{K}$ for all nonempty open set $A\subset X$. I'd like to ...
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2answers
71 views

Question about vector spaces with the discrete topology

Is it true that every vector space with the discrete topology is a topological vector space? (That is, a topological space with continuous addition and scalar multiplication whose singletons are ...
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2answers
25 views

Is multiplication by scalars a homeomorphism?

Let V be a complex topological vector space. For any nonzero v in V, consider the map x -> x * v from C to V. This map is continuous, but is it a homeomorphism onto its image? Thank you.
4
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1answer
118 views

Example of Topological Vector Space

Is there a topological vector space such that, for every $x\in X$, there is a proper neighbourhood $V$ of $x$ in $X$ which is convex, but the whole space is not locally convex (i.e. $X$ has a local ...
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0answers
162 views

Hilbert Cube and Metric Space

Given that $d(x,y)=\sum_{n=1}^{\infty}2^{-n}|x_{n}-y_{n}|$ defines a metric on $H^{\infty}$ where $H^{\infty}$ is the Hilbert Cube, a collection of all real sequence $x=(x_{n})$ with $|x_{n}|\leq 1$ ...
2
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1answer
59 views

Topology question with closed sets.

Let $ K\subseteq \mathbb{R}^n$ be a compact set and let $E\subseteq \mathbb{R}^n$ be a closed set. ***Its also given that $ \inf \{d(x,y)|x\in K, y\in E\}=0$. $ d(x,y)=\sqrt{\sum_j (x_j-y_j)^2}$ ...
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0answers
62 views

A consequence of “every norm is equivalent to the sup-norm” in a finite dimensional normed vector space

So, I am reading the following proposition in Neukirch's Algebraic Number Theory: Proposition. Let $K$ be a complete field with respect to the valuation $|\:\: |$ and let V be an $n$-dimensional ...
2
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1answer
34 views

Is the product of two complete TVS a complete TVS?

I'm reading about Topological Vector Spaces (using Treves's book) and I failed to answer the question on title. I'll be more precise: 1) We say that a TVS $E$ is complete if every Cauchy filter in $E$...
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23 views

Topological vector space with separating dual space

Let ‎$ X $ ‎‎‎be a topological vector space with dual ‎$ X‎^{‎\ast‎}‎ $. Has ‎$ X $ Hahn-Banach Extension Property, ‎i‎f ‎$ X‎^{‎\ast‎} $‎ ‎‎ separates points of ‎$ X $ ? (Hahn-Banach Extension ...
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1answer
48 views

Locally bounded topological vector spaces

Let ‎$ X $ ‎‎‎be a topological vector space ‎such that every neighborhood of zero contains an infinite dimensional subspace. Then ‎$ X $ is not locally bounded. I do not know, why? We know ‎$ X $ is ...
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0answers
71 views

Weak (operator) null sequence is bounded and pointwise convergent to zero

I was reading Diestel book (Absolutely Summing Operators) and it says: "(...) let $(f_n)$ be any weak null sequence in $\mathcal{C}(K)$. Then $(f_n)$ is bounded and converges pointwise to zero." I ...
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1answer
35 views

Cocountable topology and topological vector spaces

I am trying to understand topological vector spaces. Is $(\mathbb R,+,0)$ a TVS when equipped with the cocountable topology ? I mainly have problems to understand continuity of $+$. I think it is ...
2
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1answer
45 views

Dual of continuous functions in various topologies

Let $S$ be compact and Hausdorff and $C(S)$ be its space of continuous complex functions. When $C(S)$ is endowed with the $\sup$ norm, its dual is well known. Since this topology is too strong for my ...
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1answer
78 views

Topology on compactly supported smooth functions

I'm confused by a set of lecture notes I'm reading and would like help in understanding what's going on. First, there is the following nice theorem. Theorem. The topology of a locally convex space is ...
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80 views

When weakly compactness implies compactness?

Let $A$ be a Banach space. The weak topology on $A$ is a topology which produced by the following family of seminorms: $~~~~~~~~~~~~~~~~~~~~P_f(x)=|f(x)|,\qquad$ where $f\in A^*$ and $A^*$ is dual ...
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1answer
40 views

Topology property of inner product and norm

It is known that the norm can induce an inner product if and only if it satisfies Parallelogram law. I just want to know what topology property the inner product has while the norm doesn't have?
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2answers
80 views

Need example of: Algebraic sum of closed vector subspaces need not be closed

I've read somewhere that given two closed subspaces $V_1,V_2$ in topological vector space $X$, their algebraic span $V_1+V_2=\{x_1+x_2 |x_i \in V_i, i=1,2\}$ need not be closed. I always thought that ...
0
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1answer
44 views

Is $C(\Omega) \cong\prod_{n \in \mathbb{N}} C(K_n)$?

Let $\Omega$ be an open set in a topological space and $C(\Omega)$ be the vector space of continuous complex valued functions with the topology given by the following family of seminorms: $$\rho_n(f)...
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1answer
58 views

Showing equivalence of seminorms

Let $K =[0,1]$ and let $X \subset C^{\infty}(K)$ be the subspace of all functions vanishing on the end points of $K$. Show that the following seminorms are equivalent: $||D^nf||_1$ $||D^nf||_2$ $\...
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1answer
33 views

The Mazur–Ulam Theorem: A generalization to arbitrary topological vector spaces

Is there a corresponding result for arbitrary topological vector spaces? For example, is a surjective isometry between linear metric spaces affine?
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59 views

extremal functions and extreme points

A(D) :analytic functions on the unit disk form a locally convex tvs with the topology of uniform convergence on compact sets.Let F be a convex subset of A(D) which is also compact and J be a complex ...
4
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133 views

The topology of $GL(V)$

Let $V$ be a topological vector space (not necessarily finite-dimensional) over a field $K$, and let $GL(V)$ be the group of invertible linear maps $V\to V$ under composition. There are two obvious ...
7
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1answer
145 views

Topological space $\nRightarrow$ Metric space $\nRightarrow$ Normed space $\nRightarrow$ Inner product space (Examples)

If I have an inner product space, the hierarchy goes: Inner product space $\Rightarrow$ normed space $\Rightarrow$ metric space $\Rightarrow$ topological space. The reverse, however, is not always ...
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2answers
96 views

$C_c(X)$ is dense in $C_0(X)$

Let $X$ be a topological space, and let $C_0(X)$ be the $\mathbb{C}$-vector space of continuous functions $g:X \rightarrow \mathbb{C}$ with the property that for any $\epsilon > 0$, there exists a ...
0
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1answer
19 views

vector function derivative help

I need help with finding the derivative of a vector function, we haven't done any examples in class hence I have no idea how to proceed. So we have $\alpha:[a, b] \to R^2, \alpha'(t) \neq (0,0) $ ...
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0answers
76 views

Part (e) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (e) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
2
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1answer
104 views

Part (d) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (d) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
2
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1answer
95 views

Part (c) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (c) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
5
votes
1answer
133 views

Part (a) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (a) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
2
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0answers
100 views

Part (b) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (b) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
2
votes
1answer
94 views

Show that linear finite rank operators are open mappings [duplicate]

Suppose $X$ and $Y$ are topological vector spaces, $dim(Y) < \infty$, $\Lambda : X \rightarrow Y$ is linear, and $\Lambda (X) = Y$. Prove that $\Lambda$ is an open mapping. Thanks in advance. ...
5
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2answers
282 views

The convex hull of every open set is open

Let $X$ be a topological vector space. Prove that the convex hull of every open subset of $X$ is open. I tried using definition of Convex Hull and Open Set, but I couldn't prove the statement.
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1answer
37 views

Compact implies Totally Bounded in a TVS

Definition: If $X$ is a TVS and $E\subseteq X$ then $E$ is totally bounded iff for every nbhd of $0_X$, $V$, there exists some finite set $F\subseteq X$ such that $E\subseteq F+V$. Claim: If $K$ is ...
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28 views

Is the unitary group of a pre Hilbert space contractible?

for a separable Hilbert space $H$ it is known that the unitary group $U(H)$ is contractible, both for the norm topology (Kuiper's theorem) and for the strong operator topology (Dixmier and Douady, ...
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1answer
66 views

weak topology and weak* topology on $L^1, L^{\infty}$

Suppose $L^1(I)$ is the primal space and $L^{\infty}(I)$ is the dual. Could I simultaneously define weak topology on $L^1(I)$ with respect to $L^{\infty}(I)$ and define weak or weak* topology on $L^{\...
5
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1answer
128 views

Are countable strict inductive limits of Fréchet spaces always LF-spaces?

I would like to work with a slightly loser definition of an LF-space but am unsure what niceties I'm throwing away in the process. Let me provide a comparison of the conventional definition and my own ...
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2answers
65 views

Simple example of a regular topological space that is not first countable or not metrisable?

Is there an example of a regular topological space that is not first countable? Is there an example of a regular topological space that is not metrisable?
4
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0answers
72 views

Schwartz space of functions versus Schwartz space in a more general sense?

Part of me is afraid that this isn't a well-formed question, but try as I might, I can't seem to figure out anything reasonable on this topic. I'm hoping someone here can help. In functional analysis,...
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2answers
67 views

Norm of a space - How to show its one

Let $C^1([0,1])$ be the space of all functions having continuous derivative. For each $f\in C^1([0,1])$, set $$\|f\|=\left(\int_0^1 (|f|^2+|f'|^2)dx\right)^{(1/2)}$$ Show that $\|\cdot\|$ is a norm ...
2
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2answers
54 views

Compactness and (global) convergence in measure

Let $B$ denote the unit ball of $L^\infty$. Question: is $B$ sequentially compact for the topology of convergence in measure ? I am not necessarily assuming that the measure is finite (but $\sigma$ ...
0
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1answer
29 views

Elements near the identity of a linear subspace

I am currently trying to understand a proof and ran into the following problem. The proof states (everything takes place in a commutative, unital Banach-Algebra): A linear subspace $X$ with ...