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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15 views

Dense convex set in $*$-weak topology

Let $X$ be a Hausdorff topological vector space over $\mathbb{K}$. Suppose $W$ is a convex subset of its topological dual $X'$. How to prove that if for any $x\in X\setminus\{0\}$ set $\{f(x):f\in ...
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1answer
12 views

Bounded neighbourhood of zero in TVS

Is is true that in any topological vector space, which is $T_1$ there exists bounded neighbourhood of zero ? Is is still true if we omit $T_1$ axiom ?
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15 views

Space of polynomials as a continuous image of F-space

Let $X=\mathbb{R}[a,b]$. Is there any norm $\|\cdot\|$ on $X$ s.th. $X$ is a continuous image of some $F$-space. ($F$-space means that there exists complete metric s.th. $d(x+z,y+z)=d(x,y)$) ? My ...
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66 views

When is a diffeomorphism analytic?

I've read somewhere "a class $C^\infty$ diffeomorphism is said to be analytic" but I forgot to write down where I read this and now I can not find it, which makes me wonder if it's true? I'm working ...
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1answer
66 views

Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$

We can concluded that $\mathcal{D}(\Omega):=\bigcup_{K \in \mathcal{K}(\Omega)} \mathcal{D}_K(\Omega)$ (where $\mathcal{K}(\Omega)$ denotes the union of all compacts set content in a open subset ...
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48 views

Looking for an example of a bounded set.

Consider the local base over the space of complex continuous functions over $[0,1]$ (denoted by $\mathcal{C}[0,1]$) defined for each fixed $x\in [0,1]$ and $\epsilon>0$ by ...
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1answer
44 views

6.5 theorem in Functional Analysis by Rudin about topological vector space and $\mathscr{D(\Omega)}$

I am reading the proof of 6.5 Theorem in Rudin's book. For part (c), he wants to prove that a bounded subset $E \subset \Omega$ must be contained in a $\mathscr{D}_{k}$ for some compact subset $K ...
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1answer
35 views

Is a sequentially compact (non-metrizable) uniform space totally bounded?

First some topological definitions in terms of nets and sequences: A topological space $(X, \tau)$ is compact iff every net has a convergent subnet sequentially compact iff every sequence has a ...
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36 views

embeddings of TVS are continuous

How to show that the two embeddings of TVS are continuous: $C^{\infty}_c\subset S\subset C^{\infty}$? According to Wikipedia, the definition of "Continuously embedded" is that " one normed vector ...
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21 views

Check proof that the embedding of the unit ball $B\subset X$ into $X^{**}$ in weak-* dense

I have to prove the following theorem: Let $X$ be a (real) Banach space, and let $B$ denote its closed unit ball, and let $\tau (B)$ denote its canonical embedding into $B^{**}$, the closed unit ...
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1answer
34 views

Open sets in the unitary group $ U(\mathcal{H}) $ of a Hilbert space $ \mathcal{H} $.

Let $H$ be an infinite dimensional Hilbert space and let $(x_i)_1^\infty$ be an orthonormal basis for $H$. Consider $U(H)$ the unitary group of the continuous unitary operators on $H$. Equip $U(H)$ ...
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39 views

Existence of linear continuous functional on locally convex TVS

Let $c>1$ and $X$ be a locally convex TVS. Take $a,b\in X$ and a closed subspace $Y$ of $X$ such that $Y\cap \{(1-t)a+bt \ | \ t\in\mathbb{R}\}=\emptyset$. How to prove that there exist $f\in X'$ ...
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2answers
64 views

Is Topological Space a Metric Space?

What's the correct relationship between these two spaces? I think that topological space is a metric space, since the open is defined by a metric such that $d(x, a) < \epsilon$.
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1answer
37 views

The weak topology on $H$ is the weak* topology on $H^*$ pulled back via $\Phi$

I'm reading the following in Analysis Now by Pedersen: The map, $H$ a Hilbert space $$\Phi:H\to H^*: x\mapsto(\cdot\mid x)=[y\mapsto (y,x)]$$ is a conjugate linear isometry. Then define the weak ...
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1answer
20 views

Does $\frac{1}{n}x_0$ converge to the origin in any topological vector-space?

Let $X$ be a topological $\mathbf{R}$-vector-space (not necessarily Hausdorff) and $x_0 \neq 0$ a non-zero element of $X$. Then intuitively the sequence $(\frac{1}{n} x_0)_{n \in \mathbf{N}}$ ...
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1answer
42 views

Proof of a property regarding weak-star topology

I am reading a book in which a theorem is only proven analogously in the situation of weak topology. The proof does not seem to work in the case of weak$^{*}$ topology. Let $X$ be a normed linear ...
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2answers
14 views

If $V$ is completely normable, then is every norm complete?

Here is a theorem that motivated my question. Let $(V,||\cdot||_V)$ be a normed space over $\mathbb{K}$. Then, there exists a Banach space $(X,||\cdot||_X)$ such that $V$ is dense in $X$ and ...
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10 views

Is there a terminology to denote a topological vector space that admits an inner product?

Let $(V,\tau)$ be a topological vector space over $\mathbb{K}$. If there is a norm $||\cdot||$ on $V$ such that the metric topology induced by $||\cdot||$ is $\tau$, then we call $V$ is normable. ...
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1answer
45 views

Complete as a semimetric space but not as a topological group

I shall begin with some definitions. 1) Suppose that $X$ is a topological (additive) group and $(x_{s})\subseteq X$ is a net, we said that $(x_{s})$ is Cauchy whenever $U$ is a neighbourhood of $0$, ...
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1answer
50 views

Example of a subnet that have no subsequence.

I have an elementary question on nets because I'm not familiar with this concept. Here are two basic facts: Every subsequence of a sequence is a subnet; Not every subnet of a sequence is a ...
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1answer
37 views

open and closed unit balls in topological vector space

It is a famous fact that in any seminormed space $X$, the open unit ball $B$ and closed unit ball $C$, which are defined by $B=\{x\in X: \|x\|_{X}<1\}$ and $C=\{x\in X: \|x\|_{X}\leq 1\}$ ...
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19 views

Core points of a convex set

In the book of Gamelin "Unifrom Algebras" I found the following definition: Let $K \subset V$ be a convex set in a vector space. An element $x \in K$ is called core point of $K$ if whenever $z ...
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2answers
38 views

Example of a topological vector space which is not locally convex

I'm currently studying Functional Analysis and the professor gave an example for a TVS (which we have defined to be a vector-space $X$ in which addition $X \times X \rightarrow X, (x, y) \mapsto x + ...
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1answer
29 views

Quotient map $\pi : X \rightarrow X / \mathrm{ker}(A)$ is open for a bounded linear operator $A$

I'd like to show: if $A : X \rightarrow Y$ is a bounded linear operator between Banach-spaces, then $\pi : X \rightarrow X / \mathrm{ker}(A)$ is a open map. I found a proof, which I do not really ...
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1answer
9 views

Semi-norm generating the standard locally convex topology on the space of locally finite Borel measures

In several articles available over the internet, it is written that: .. $M_{loc}(\Omega)$ is the space of locally finite Borel Measures on $\Omega$ with the standard locally convex topology ...
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2answers
49 views

A $T_0$ topological vector space is Hausdorff

I know a $T_0$ topological group is $T_1$, and if we have a topological vector space, there should be a way of using scalar multiplication to get disjoint neighborhoods. Right?
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1answer
31 views

Dual topology and Mackey–Arens theorem

I read only by wikipedia the Mackey–Arens theorem, that is: Given dual pair $(X, X')$ with $X$ a locally convex space and $X'$ its continuous dual, then $\mathcal{T}$ is a dual topology on $X$ if ...
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1answer
65 views

Equivalent definitions of locally convex topological vector space

This Wikipedia article gives two equivalent definitions of locally convex space (l.c.s). I don't see clearly the equivalence and I'd like to make it crystal clear. Definition 1 Let $(V,\tau)$ be a ...
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2answers
59 views

Continuity of seminorms

The following is from Wikipedia: A locally convex space is defined to be a vector space $V$ along with a family of seminorms $\{p_α\}_{α ∈ A}$ on $V$. A locally convex space carries a natural ...
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1answer
38 views

Different definitions of absorbing sets from the Wikepedia

Consider a vector space $X$ over the field $\mathbb{F}$ of real or complex numbers and a set $S\subset X$. In this Wikipedia article about absorbing sets, $S$ is called absorbing if for all ...
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1answer
81 views

Discontinuous surjective linear map which is not open

The following statement is true: Assume that $X$ and $Y$ are topological vector spaces where $Y$ is finite-dimensional Hausdorff, if $A:X\rightarrow Y$ is a continuous surjective linear map then $A$ ...
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2answers
139 views

Proving that scalar multiplication is continuous

Let $\mathbb{K} \in \{ \mathbb{R} , \mathbb{C} \}$ and $s= \mathbb{K}^{\omega}$ be the usual sequence set with entries on $\mathbb{K}$. I proved that $\mathbb{K}$ induces a $\mathbb{K}$-vector space ...
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2answers
27 views

Why is $\{x-y: x,y\in K,\ d(x,y)\geq 1/n\}$ a compact set in a metrizable topological space with compact $K$?

Assume that $K$ is a compact convex set in a Hausdorff locally convex space, and $K$ is metrizable with the induced topology. Let $d$ be a metric defining the induced topology. Show that the set ...
3
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1answer
37 views

Locally compact Hausdorff topological vector spaces are finite-dimensional

It's an important fact that locally compact Hausdorff topological vector spaces are finite-dimensional, a proof can be found here. I'm somewhat stuck with the proof. If ${U}$ is a neighbourhood of ...
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1answer
15 views

Second countable normed vector space

I know, that if $X$ and $Y$ is second countable, then $X_{/Y}$ is second countable. Is it true, that if $X$ is normed vector space, $Y$ is closed subspace of $X$, and $Y$ is second countable, and ...
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1answer
12 views

Show that $ F_n=\{x\in K:\textrm{there exist }\ y,z\in K\textrm{ such that }x=(y+z)/2\textrm{ and }d(x,y)\geq 1/n\}$ is a closed set

This is a follow-up question to a previous one. Assume that $K$ is a compact convex set in a Hausdorff locally convex space, and $K$ is metrizable with the induced topology. Then the set ...
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1answer
27 views

properties of non-extreme points

I'm reading a proof of the following lemma Assume that $K$ is a compact convex set in a Hausdorff locally convex space, and $K$ is metrizable with the induced topology. Then the set ...
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2answers
23 views

Can a metrizable TVS be induced by a non-translation invariant metric?

Is it possible to have a topological vector space $(X, \tau)$ with its topology induced by a metric $d$ which is not translation invariant? I'm asking this because in Rudin's 'Functional Analysis' ...
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26 views

$W^{1,p}$ is separable for $1\leq p<\infty$

I've been asked to prove that the Sobolev spaces $W^{1,p}(\Omega)$, $\Omega$ open in $\mathbb R^n$, are separable for $1\leq p <\infty$ using the map $$i\colon W^{1,p}(\Omega)\to L^p(\Omega)\times ...
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0answers
72 views

Local boundedness of monotone operators in general spaces

A classical result states that: If $X$ is a Banach space then every multi-valued monotone operator $T:X\to 2^{X^*}$ is locally bounded on $\operatorname*{int}D(T)$ (the interior of its domain). I ...
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1answer
28 views

Why would $\{f_c:x\mapsto c\|c\in [-1,1]\}\subset (C_\mathbb{R}[0,1],||\cdot||)$ be closed in the wk* topology?

When proving that $(C_\mathbb{R}[0,1],||\cdot||_\infty)$ is not a dual space of any Banach space using Alaoglu's theorem and Krein-Milman's theorem I'm stuck with the conclusion that $S=\{f_c:x\mapsto ...
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1answer
37 views

properties of a separable metrizable locally convex space

Let $X$ be a separable, metrizable locally convex space. Suppose $V$ is a neighborhood of $0$ and a barrel (closed, absolutely convex, and absorbing). Show that there exist points $y_n\in ...
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1answer
46 views

In the Krein-Milman theorem, can the weak closure of the convex hull be replaced by norm-closure?

I have a question on the following formulation of the Krein-Milman theorem: Consider a vector space $X$ equipped with the weak topology induced by a separating space $X^*$ of functionals on $X$. ...
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2answers
43 views

from finite to $\sigma$-finite measure space [duplicate]

This might be rather elementary. I have put it at MSE for a while without getting any answers. Here is the question: In the proof of the following theorem, would anyone explain how the general case ...
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1answer
38 views

Vector space of all Lebesgue measurable functions

Let $L_0$ be the vector space of all Lebesgue measurable functions on $[0,1]$ with metric $d(f,g)$ = $\int_{0}^{1} |f(t)-g(t)|/( 1+ |f(t)-g(t)| ) dt $ . Show that $d(f_n,f) \to 0$ iff $f_n \to f$ in ...
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1answer
310 views

How can this theorem about weakly measurable functions on $\sigma$-finite measure spaces be deduced from the finite measure space case?

I am reading a theorem about measurability of vector-valued functions in a note on functional analysis: Theorem 3.6.1. If $X$ is a separable, metrizable locally convex space, $(\Omega, \Sigma, ...
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164 views

Inclusions continuous space of test functions

Let $(X, \left \| \cdot \right \|_X )$, $(X, \left \| \cdot \right \|_Y)$ two normed vector spaces with $X \subset Y$, by definition we have $X \hookrightarrow Y$ if $\left \| x \right \|_Y \leq C ...
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1answer
31 views

what is a cofinal sequence?

I understand that the subset $\Phi'$ of $\Phi$ is cofinal by looking at Wikipedia https://en.wikipedia.org/wiki/Cofinal_(mathematics) Would anybody explain what the cofinal sequence $(Y_n)$ means? ...
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1answer
28 views

The relation between Closed Operators (the graph is closed) and Closed Mappings (images of closed sets are closed)

Let $X$, and $Y$ be topological vector spaces and let $D$ be a dense vector subspace of $X$. An operator $T:D\to Y$ is called closed iff the graph of $T$, $\{(x,T(x))\in X\times Y|\,x\in D\}\subseteq ...
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1answer
32 views

Why is the dual space with the weak*-topology a topological vector space?

In my lecture-notes on functional analysis I've found the fact that the dual-space $X^*$ with the weak*-topology of a real vector-space $X$ is a topological vector-space. I've tried to prove it, but ...