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
24 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
20 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
36 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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0answers
62 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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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?
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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 ...
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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: ...
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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: $||D^nf||_1$ $||D^nf||_2$ ...
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1answer
25 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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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 ...
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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 ...
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1answer
65 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
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 ...
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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) $ ...
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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 ...
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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 ...
2
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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 ...
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 ...
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. ...
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
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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 ...
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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, ...
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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 ...
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0answers
35 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
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?
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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 ...
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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
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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
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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 ...
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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 ...
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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
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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 ...
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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, ...
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0answers
21 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}) ...
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1answer
30 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 ...
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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
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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 ...
2
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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 ...
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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 ...
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0answers
17 views

Let $K$ be a closed, convex absorbing subset of $X$. why $K$ is everywhere dense?

In this proof I dont understand why $K$ is everywhere dense in its self? $- $ Proof : Let $X$ be a topoligical vector space which is of second category in itself. Let $K$ be a closed, convex ...
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0answers
20 views

Let A⊆B, If B has second category itself then A has second category itself

Let $X$ be topological vector space and $A⊆B$. If $B$ has second category itself then $A$ has second category itself. Is this true?( $A$ is not empty)
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For a normed vector space $ E $ and an element $ x \in E $, prove that if $ L(x) = 0 $ for every continuous linear functional $ L $, then $ x = 0 $.

Question. Let $ E $ be a normed vector space. Is it true that for a given $ x \in E $, if $ L(x) = 0 $ for every $ L \in E' $, then $ x = 0_{E} $? One way to prove this is to find an $ L \in E' ...
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0answers
19 views

$A \subseteq B$, $B$ has second category itself and $A$ is convex. Why dose $A$ has second catergory itself?

Let $X$ be topological vector space and $A \subseteq B$. $B$ has second category itself and $A$ is convex. Why dose $A$ has second catergory itself?
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1answer
28 views

Criterion for Isometry

Let $X$ be a topological vector space, with $d$ an invariant metric compatible with the metric. Let $f:X\to X$ be an involutive linear isomorphism. How do you show that $f$ is an isometry? I ...
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2answers
85 views

Prove $\ell^5$ is contained in $\ell^6$.

I am struggling with the proof to show that, for any $p$, $r$ such that $1 \le p <r < \infty$, that $\ell^ p\subset\ell ^r$. Could somebody please give a helpful nudge by showing how this ...
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0answers
10 views

Decomposing continuous linear functionals on a locally convex space with 2 seminorms

Let $X$ be a locally convex topological vector space whose topology is defined by the seminorms $\rho_1$ and $\rho_2$. (Let us require that topological vector spaces be Hausdorff by definition.) If ...
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1answer
107 views

How to check some topological concepts in product and direct sum spaces

Given $a=(a_i)_{i=1}^\infty$ with $a_i \geq 0$ and $b=(b_i)_{i=1}^\infty$ with $b_i \in \mathbb{R}$, let $$E_i = \lbrace (x_n)_{n=1}^\infty : n^{b_i}|x_n|\leq a_i, \forall n\in \mathbb{N} \rbrace$$ ...
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69 views

a question of my real analysis class,can someone help me solve this question?

Let $n \geq 1$ be an integer, and $C: \mathbb{N}^n \to \mathbb{R}$ be a function. Prove that there exits a infinitely differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ whose value and partial ...
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24 views

Banach-Alaoglu theorem for dual pairs

The versions of the Banach-Alaoglu theorem that I know of always concern some space $X$ and its topological dual $X^*$. Can the theorem be restated for arbitrary (nondegenerate) dual pairings $(X,Y)$ ...