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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22 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 ...
4
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2answers
456 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
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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 ...
7
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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
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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
23 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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25 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
34 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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4answers
493 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 ...
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0answers
19 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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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
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1answer
30 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 ...
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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
20 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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2answers
62 views

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 ...
0
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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 ...
5
votes
1answer
170 views

Generalization of inner product spaces (analogue to uniform spaces/locally convex spaces)

In the following I am going to devise a chart of topological spaces that contains inner product spaces, normed vector spaces, metric spaces and other related spaces. In the end there will be a gap in ...
4
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1answer
108 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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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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0answers
72 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 ...
0
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0answers
25 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 $(X,Y)$ ...
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2answers
43 views

a question about functional analysis conclusion,and I am not sure whether it is true or not?

we have $R^n$,$R^m$ spaces, suppose open set $O_{1}\subset R^n $ and $O_{2}\subset R^m$, $f:O_{1}->O_{2} $ is k-times differentiable$(1<=k<=\infty)$,then at $x_{0}\in O_{1}$,$rank(f)(x_{0})$ ...
10
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1answer
213 views

Does the vector space of compactly-supported continuous functions $X \rightarrow \mathbb{R}$ satisfy an interesting universal property?

Let $S$ denote a set. Then the vector space $FS$ freely generated by $S$ can be identified with the set of all finitely-supported functions $S \rightarrow \mathbb{R}$. This gave me the following idea; ...
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1answer
60 views

Bounded linear map on topological vector spaces is continuous

Let $X$ and $Y$ be topological vector spaces and $T\colon X\to Y$ linear. Suppose that $T$ sends bounded sets to bounded sets and that $X$ is first countable. The claim is that $T$ is continuous. ...
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0answers
99 views

Nuclear spaces vs Banach spaces

The Wikipedia article on nuclear spaces say the following: "There are no Banach spaces that are nuclear, except for the finite-dimensional ones. In practice a sort of converse to this is often true: ...
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2answers
23 views

Does every LCS has a convex balanced local base?

Does every LCS--locally convex (topological vector) space has a convex balanced local base? Then it implies every LCS is topologized with a countable number of separating seminorms. So there seems ...
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2answers
50 views

Can every (Hausdorf) topological space be homeomorphically embedded in a topological vector space?

It's true that for any metric space, we can isometrically embed it in a Banach space, so that the image of the metric space by that embedding is a linearly independent set. Is the analogous theorem ...
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1answer
87 views

properties of a Köthe space s

Could you please help me answering the following question? Consider the Köthe space $K_ {\infty}(n^p) = \{ x= (x_n)_1^{\infty}: |x|_p := \sup_n|x_n|n^p<\infty, \forall p \in \mathbb{N} \}$ with ...
2
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0answers
15 views

Is the unit ball of $H^\infty(\mathbb{D})$ a metrizable topological semigroup under multiplication?

The space $H^\infty(\mathbb{D})$ of all bounded holomorphic functions on the open unit disc carries many different topologies. One such topology is given by uniform convergence on compact subsets; ...
2
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1answer
49 views

Why convergence implies cauchy in topological vector space?

The following definition is from Janich's Topology book : Definition (Topological Vector Space). A $\mathbb{R}$-Vector space $(E,\tau)$ with a topological space structure is called a Topological ...
3
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1answer
47 views

Hahn-Banach theorem and complex linear functional

I find this exercise but I cannot prove it. If $ X $ is a complex topological vector space and $ f \colon X \to \mathbb C $ is nonzero continuous linear function, show that $ X \setminus \ker f ...
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0answers
24 views

Isomorphism of finite dimensional topological vector space with $(\mathbf{R}^k,\mathcal{R})$

Let $(T,\mathcal{T})$ be a topological vector space over $\mathbf{R}$ with finite positive dimension. Is it true that there exists an isomorphism between $(T,\mathcal{T})$ and ...
2
votes
1answer
108 views

Dual Pairs, topology of weak convergence and weak* topology

Edit for Bounty: I decided to put a bounty on this question because I would really like to get it properly. Thus, I would like to get feedbacks on my basic questions, and a detailed answer on my ...
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4answers
80 views

Embedding vs continuous injection (in topological vector spaces)

When working with topological vector spaces (say $X,Y$), the term “embedding” is often used for a continuous injection $f:X\rightarrow Y$. Now, $f$ is of course a bijection onto its image, but it's ...
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0answers
25 views

Generate a mesh from unsorted points (eight points)

I'm trying to generate a mesh from eight points. The challenge is that I don't know the order/label of the points, and I want it to work regardless of variations in the shape (see example below). The ...
13
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2answers
5k views

“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 ...
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2answers
26 views

If $E\subset X^{*}$ is bounded, then so is its weak* closure

If $X$ is a Banach space and $E\subset X^{*}$ is norm-bounded, I've shown that its weak* closure is also norm-bounded using Alaoglu's theorem. But perhaps using Alaoglu's theorem is not necessary? ...
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0answers
44 views

Weak* topology on Hilbert space

I am a little confused about the weak* topology on Hilbert space $H$. Beyond doubt, the weak* topology on $H^{**}$ is $\sigma(H^{**},H^*)$. Suppose $\tau$ is the natural embedding from $H$ onto ...
6
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1answer
65 views

Open neighbourhoods in topological vector spaces

It is well known that each open ball in a Banach space is homeomorphic to the whole space. Can we extend this to topological vector spaces? In other words, does every non-void open set in a ...
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0answers
17 views

A question involving normed spaces and strictly convex spaces

Let $(X, \| \cdot \|_X)$ be a normed space and let $\| \cdot \|$ be a norm on $X$ such that $(X, \| \cdot \|)$ is strictly convex. How can I find a strictly convex space $(Y, \| \cdot \|_Y)$ and a ...
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0answers
94 views

Is any closed and bounded subset of a reflexive Banach space compact in the weak topology?

It seems to me that Alaoglu's theorem implies that any closed and bounded subset of a reflexive Banach space is compact in the weak topology. Is convexity of the set also needed?
3
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1answer
53 views

Why doesn't Alaoglu's theorem imply that $X^{*}$ is locally compact in the weak* topology?

I must be missing something basic and simple: If $X$ is a normed vector space and the closed unit ball in $X^{*}$ is weak* compact, and translations and dilations are homeomorphisms, why isn't $X^{*}$ ...
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1answer
28 views

Trying to show that $(c_0, \| \cdot \|_s)$ is strictly convex, where $\| x \|_s = \underset{i = 1}{\overset{\infty}{\sum}} \frac{1}{2^i} | x_i |$

I'm trying to show that $ (c_0, \| \cdot \|_s) $ is a strictly convex space, where $$ \| x \|_s = \underset{i = 1}{\overset{\infty}{\sum}} \frac{1}{2^i} | x_i |,$$ $ x = (x_1, x_2, ..., x_i, ...) \in ...