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.

learn more… | top users | synonyms

0
votes
1answer
51 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 ...
1
vote
1answer
25 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
votes
1answer
24 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 ...
12
votes
3answers
1k views

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 ...
3
votes
0answers
63 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 ...
0
votes
0answers
23 views

Topology property of inner product and norm

It is known that the norm can induce an inner product unless it satisfies Parallelogram law. I just want to know what topology property the inner product has while the norm doesn't have?
61
votes
2answers
2k views

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 ...
2
votes
2answers
55 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 ...
1
vote
1answer
41 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: $||D^nf||_1$ $||D^nf||_2$ ...
0
votes
1answer
38 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: ...
1
vote
1answer
26 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?
0
votes
0answers
29 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 ...
2
votes
1answer
17 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 ...
3
votes
0answers
84 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
votes
1answer
66 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 ...
8
votes
1answer
115 views

Why do we give $C_c^\infty(\mathbb{R}^d)$ the topology induced by all good seminorms?

Briefly, my question boils down to the following: What benefits do we gain from considering the space of test functions in the topology induced by all "good" seminorms, as opposed to other topologies ...
5
votes
3answers
260 views

Boundedness in a topological space?

I was wondering if there is a concept of boundedness for subsets of a topological space? If yes to 1, is it this one from Wiki Elements of a Bornology B on a set X are called bounded sets and the ...
5
votes
1answer
241 views

Evaluation map is not continuous always.

Let $E$ be a not normable locally convex space, define $$F: E'\times E\to \mathbb R$$ $$(f,e)\to f(e)$$ I have to show that $F$ is not continuous when $E'\times E$ is given product topology. I was ...
2
votes
2answers
34 views

Is a metrizable topological vector space normable?

A tvs is metrizable iff it is $T_2$ and has a countable local base; while a tvs is normable iff it is $T_2$ and $0$ has a bonded convex neighbourhood. So any anybody give me an example of a ...
0
votes
2answers
40 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
votes
1answer
11 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) $ ...
5
votes
2answers
104 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.
2
votes
1answer
41 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 ...
2
votes
1answer
48 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. ...
0
votes
0answers
37 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 ...
0
votes
0answers
25 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 ...
5
votes
1answer
56 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
votes
0answers
52 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 ...
1
vote
1answer
243 views

Understanding Hom-bundle on vector bundles (Atiyah)

I am trying to make sense of the following statment (from Atiyah's K-theory book) Suppose $V$ and $W$ are vector spaces, and that $E=X \times V, F=X \times W$ are the corresponding product ...
2
votes
0answers
36 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 ...
1
vote
0answers
15 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, ...
0
votes
1answer
30 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 ...
-1
votes
2answers
41 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?
2
votes
0answers
20 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 ...
3
votes
2answers
440 views

A surjective linear map into a finite dimensional space is open

I'm in search of different proofs of the following proposition: $\bf{Proposition}$: Suppose $X$ and $Y$ be topological vector spaces, $\text{dim }Y<\infty$, and $\Lambda:X\to Y$ is a surjective ...
1
vote
2answers
51 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
votes
2answers
46 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
votes
1answer
27 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 ...
1
vote
0answers
35 views

Show that continuous linear maps on the space of test functions take $C_K^\infty(\Omega)$ into some $C_{K_N}^\infty(\Omega)$

Let $\Omega$ be a nonempty open subset of $\mathbb{R}^n$, and let $\cup_{n=1}^\infty K_n = \Omega$ be an exhaustion of $\Omega$ by compact sets. Let $\mathcal{D}(\Omega) = \mathcal{D}$ be the standard ...
7
votes
2answers
4k views

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 $V$. ...
-2
votes
1answer
42 views

How to prove this: $\overline{A\cap B}=\overline{A\cap \overline{B}}$?

Let $A$ be an open set of $E$ an normed linear space, and $B\subset E$, then I have to prove that $$\overline{A\cap B}=\overline{A\cap \overline{B}}$$ (I'm stuck in the two $\subset$'s) Any help ...
1
vote
1answer
23 views

familyof seminorms on normed spaces

Let $(X,\|\cdot\|)$ be a normed space. It is known that the norm $\|\cdot\|$ induces a topology, known as the norm topology $\tau$ on $X$. Then the pair $(X,\tau)$ is a locally convex topological ...
0
votes
0answers
22 views

Confused about topological vector space defined by seminorms

I'm reading a book in which it's claimed that for a strongly continuous representation, $U: G\rightarrow Aut(E)$ of a Lie group, G, on a locally convex, complete, Hausdorff topological vector space,E, ...
0
votes
0answers
24 views

Necessity of continuity in Topological Vector Space

In the notion of a topological vector space, we define such as a vector space $X$ (over a field $\mathbb{K}$) with topology $\mathscr{T}$ such that $$\iota_+: (X,\mathscr{T}) \times (X, \mathscr{T}) ...
0
votes
1answer
31 views

Characterization of a quotient space

Given the space $C^n[0,1]$ of all real functions of class $C^n$ in $[0,1]$, let $\tilde{d}^j := d_\infty(f^{(j)},g^{(j)})$ a pseudometric $(j=1,\dots,n)$ on $C^n[0,1]$. Here $f^{(j)}$ mean the ...
1
vote
4answers
474 views

Nonconstant linear functional on a topological vector space is an open mapping

In the middle of another proof (Theorem 3.4, p. 60) in his Functional Analysis book, Rudin says that "every nonconstant linear functional on $X$ (topological vector space) is an open mapping." Is ...
0
votes
0answers
16 views

An example of a reflexive vector space that is not a Banach Space

I found in an article of Marianne Smith a definitiion of reflexive vector space that not need the space to be, even, a normed space. She said that a vector space is reflexive if the natural ...
2
votes
1answer
56 views

a question about proving a normed space is complete

Let [a,b] be an interval in R,and denote by E the Vector space of functions f:[a,b]->R such that f is of bounded variation over [a,b] and f(a)=0.Prove that by setting $||f||=Var|_{a}^{b}(f)$ for each ...
1
vote
1answer
26 views

In the first countable TVS, if every Cauchy sequence convergence then every Cauchy net convergent

Let $X$ be a topological vector space with the first countable topology(that is, every point has a countable neighborhood basis).If every Cauchy sequence convergence, we want to show that every Cauchy ...
0
votes
0answers
24 views

How to check a linear map between topological space is continuous?

I am reading something about distributions, and I have a question. I think it is not hard, but I don't know how to explain it rigorously. Suppose $M$ is a smooth manifold, $V$ is a Fréchet space, and ...