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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3
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1answer
47 views

Adjoint of completely continuous operator is completely continuous

In the proof of the fact that the adjoint operator $A^\ast$ of a completely continuous linear operator $A:E\to E$ mapping a Banach space into itself is also completely continuous on $E^\ast$ endowed ...
-2
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0answers
7 views

balanced convex hull questions [closed]

Let E topological vector space localy convex. Show that if B is bounded set in E then the balanced convex hull is bounded
1
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1answer
72 views

$\{x^nf(x)\}_{n\in\mathbb{N}}\subset L_2(a,b)$ as a complete system

I read in Kolmogorov-Fomin's (p. 430 here) the statement, sadly left without a proof, that if function $f:(a,b)\to\mathbb{C}$, measurable almost everywhere on $(a,b)$, where $-\infty\leq ...
0
votes
0answers
22 views

Condition for uniform convergence of Fourier series

Let $f$ be a Lebesgue summable periodic function on $[-T/2,T/2]$. I read in Kolmogorov-Fomin's (p.414 here) that if $f$ is bounded on a set $E\subset[-T/2,T/2]$ and for any $\varepsilon>0$ there is ...
2
votes
0answers
38 views

Topological vector space question

$C[0,1]=$ space of all continuous complex valued function over $[0,1]$. Define metric, $d(f,g)={\int_{0}^{1} \frac {|f(x)-g(x)|}{1+|f(x)-g(x)|}}$, for all $f,g\in C[0,1] .$ Let $(C[0,1],\sigma)$ ...
4
votes
1answer
55 views

A question about local convexity of the weak operator topology

By definition, I know a locally convex space is a topological vector space whose topology is defined by a family of seminorms $\cal P$ such that $$\bigcap_{p\in{\cal P}}\{x\colon p(x)=0\}=\{0\}.$$ ...
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0answers
23 views

Continuous functions dense in $L_p(X,\mu)$ if $X$ has a special property

Let $X$ be a metric space endowed with a measure $\mu$ satisfying the following condition: all the open and closed sets of $X$ are measurable and for any measurable set $M\subset X$ and any ...
0
votes
1answer
34 views

$L_2$ as a Hilbert space and $\ell_2$

I know that, if measure $\mu$, with which measure space $X$ is endowed, has a countable base, i.e. if for any measurable $M\subset X$ there exists a measurable set $A_k\in\mathscr{A}$, where ...
2
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0answers
27 views

Metric space with measure and a special property

Let $R$ be a metric space endowed with a (complete) measure $\mu$ satisfying the following condition: all the open and closed sets of $R$ are measurable and for any measurable set $M\subset R$ and any ...
0
votes
0answers
28 views

Co-ordinate chart and components of a vector field.

Q) Using a coordinate chart, give a formula for the components of the vector field $[v,w]$ in terms of the components of $v$ and $w$. Where $[v,w]: f \mapsto v(wf) - w(vf)$ I don't know what the ...
5
votes
2answers
92 views

A question about convex open set in a topological vector space.

Supose $E$ is a topological vector space(may not be Hausdorff). $U\subset E$ is an open set such that $U+U=2U$. How to show $U$ is convex? I can see if $E$ is $T_1$,then $E$ should be Hausdorff. ...
3
votes
1answer
61 views

A base of topology

Consider a space of smooth functions $C^{\infty}[a,b]$ and a set $$\tau=\left\{B(f,\varepsilon_0,\varepsilon_1...\varepsilon_r):f\in C^{\infty}[a,b],r\in\mathbb{N}\right\} $$ where $f$ is arbitrary ...
1
vote
2answers
26 views

Closure of linear subspace in Topological vector space

Let $X$ be a TVS, $x\in X$ and $M<X$ be a linear subspace. Does $x\in M+U$ for every open neighborhood $U$ of $0$ imply that $x$ is in the closure of $M$? EDIT: This argument is used here: ...
2
votes
1answer
33 views

Topological Vector Space not induced by Metric

Can anyone give me an example of a Topological Vector Space that is not metrizable? I know that the neighborhood base of $0$ needs to be incountable, and all I can construct then is no topological ...
1
vote
1answer
61 views

Existence of a Frechet topology on the dual of a barreled space

I have a Hausdorff separated locally convex barreled space $(X,\tau)$ with topological dual $X^*$. My questions are: $Q_1$ Is there a topology $\tau^*$ on $X^*$ that is finer than the weak-star ...
0
votes
1answer
17 views

A question about close line segment in TVS.

Suppose $E$ a topological vector space,which need not be Hausdoff. $x,y\in E$ are different. How to prove the close line segment $\{\alpha x+(1-\alpha)y:\alpha\in[0,1]\}$ is closed. And should it be ...
1
vote
1answer
35 views

A question about linear functional on TVS

Let $E$ be topological vector space on field $\mathbb{R}$(or $\mathbb{C}$), which need not be Hausdoff. $f$ is a linear functional on $E$, and there are open set $U\subset E$ and $t\in \mathbb{R}$(or ...
1
vote
3answers
32 views

unreduced suspension

Is the definition $SX=\frac{(X\times [a,b])}{(X\times\{a\}\cup X\times \{b\})}$ of the unreduced suspension the standartdefininition? If I consider X=point, the suspension of X is a circle. But I saw ...
2
votes
1answer
67 views

Largest ideal of a local field on which a character is trivial

Let $K$ be a nondiscrete locally compact field. Then fixing a character $\chi$ on $K$, any character on $K$ can be written as $t \mapsto \chi(xt)$ for some $x \in K$. For $E \leq K$ a closed ...
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0answers
36 views

Initial topology coincides with the locally convex topology

Suppose that $\forall j\in J: X_j $ is a locally convex space, with defining family of seminorms $(q_{jk})_{k \in K_j}$. Also let $X$ be a vector space and $T_j: X \to X_j$ a linear map $\forall j\in ...
1
vote
1answer
72 views

From metric to topological vector space

Suppose that $E = C[0,1]$ and suppose we have a metric given by $$d(f,g) = \int_0^1 \min(|f(x)-g(x)|,1)dx$$ Why is it that the topology defined by this metric makes $E$ into a topological vector ...
0
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0answers
23 views

Computing time-dependent vector field

Let $H : \mathbb{R} \rightarrow \operatorname{Herm}(C^2) $ be a smooth function and $$ t \mapsto g(t) = e^{iH(t)} $$ be the associated curve of diffeomorphisms of $\mathbb{C}^2$. Compute the ...
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0answers
38 views

Closure of non-trivial ideal of algebra over $\mathbb{C}$ non-trivial?

Let $I$ be a non-trivial ideal of a topological algebra $X$ over $\mathbb{C}$, defined by the continuity of the multiplication $X\times X\to X$. I know that, if $X$ is a normed algebra, then the ...
0
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0answers
42 views

Continuity of multiplication in algebras over $\mathbb{C}$

friends! Let $X$ be a topological vector space equipped with the structure of an algebra over the field $\mathbb{C}$. Is the multiplication $X\times X\to X ,(x,y)\mapsto xy$ continuous with respect ...
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1answer
27 views

Is the closure of a linear variety a linear variety?

I know and have been able to prove that, if $L$ is a linear variety of a normed space $X$, i.e. a vector subspace $L$ of $X$ regarded as a vector space, then its closure $\bar{L}$ with respect to the ...
1
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1answer
19 views

Minkowski functional of a open set

Given $X$ is a topological vector space and $V$ is a convex, balanced neighbourhood of $0$ in X. Then for $x \in V$, the Minkowski functional $\mu_{V}(x)$ = inf $\{t>0: t^{-1}x\in V\}$ $<$ ...
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votes
1answer
30 views

An Open Mapping Problem [duplicate]

Consider two topological vector spaces $X$ and $Y$ where $Y$ is finite dimensional. Let $f:X \rightarrow Y$ be a surjective linear map. Prove that $f$ is an open mapping.
4
votes
1answer
101 views

Relative countable weak$^{\ast}$ compactness and sequences

I am finding serious difficulties in understanding some things about relative countable compactness and the use of sequences in proving it by my functional analysis text, Kolmogorov-Fomin's. For ...
1
vote
1answer
34 views

$A,B$ bounded $\Rightarrow$ $A+B$ bounded

I have read that if the subsets $A$ and $B$ of a topological vector space are bounded, i.e. for any neighbourhood $U$ of $0$ there is an $n>0$ such that, for all $|\lambda|\geq n$, $M\subset\lambda ...
0
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0answers
30 views

Kernel of linear operator closed if domain non-$T_2$?

I read on my functional analysis text that the kernel of a linear operator $A:V\to W$ between two topological linear spaces is closed. My book don't require topological linear spaces to be Hausdorff ...
0
votes
1answer
25 views

Extending Banach Space of Functions

The idea is that one could in principle consider the space of functions: $$\{f:\Omega\to V\}$$ with pointwise operations and uniform convergence: $$f_\lambda\to f:\iff\|f_\lambda-f\|_\infty\to 0$$ ...
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2answers
67 views

Reflexivity of $C[a,b]$

I find the statement that the normed, complex or real, linear space $C[a,b]$ is reflexive, i.e. the natural map of the space $C[a,b]$ into the bidual space $C[a,b]^{\ast\ast}$, defined by ...
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0answers
22 views

Separability=$T_1$ for TVS?

Let a topological linear space be defined by the continuity of the linear operations only. I read on an Italian language functional analysis book, which doesn't show the proof, that any locally ...
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0answers
30 views

Motivation for the notion of locally convex topological vector space

Is the only motivation for the notion of locally convex topological vector space that the local bases have some nice property i.e. convex, balanced, absorbing ?
0
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1answer
71 views

Continuous functional such that $f(x_0)\ne 0$

I read that in any locally convex topological space $X$, not necessarily a Hausdorff space but with linear operations continuous, for any $x_0\ne 0$ we can define a continuous linear functional ...
0
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0answers
27 views

Comparision between Hamming distance and cosine similarities?

I want to check the similarities between binary vectors of different length and I am using cosine similarities and hamming distances for calculations. These are of length 1000 elements(0 and 1). ...
3
votes
1answer
60 views

Sum of Neighborhoods of Zero

When do two neighborhoods of zero over a topological vector space add up as: $$aN+bN=(a+b)N\quad a,b\geq 0$$ I could imagine something like balanced might suffice... The problem is that I'd like to ...
0
votes
1answer
43 views

Locally convex topological vector space using semi norms

Given a vector space and a family of semi-norms defined on it, I have to prove that it becomes a locally convex topological vector space. To prove that it becomes a locally convex space I have to ...
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vote
2answers
90 views

Continuity of Function Related to $F$-norms

Let $X$ be a locally bounded $F$-space and $\left\|\cdot\right\|$ be an $F$-norm on $X$. Suppose that $\left\|\cdot\right\|$ is concave: for all $x\in X$ fixed, the function ...
2
votes
2answers
61 views

Definition of bounded set in a topological vector space

What is the motivation behind the definition of bounded set in a topological vector space? The definition is different from the boundedness definition in metric space. Why is it not simply defined as ...
2
votes
0answers
30 views

Extension of function with values in a Banach space

I want to prove the following Let $E,X$ be Banach spaces, and $Y\subset E$ a closed subspace with codimension $1$. Let $T:Y \to X$ be a continuous linear function. Then there exists a continuous ...
0
votes
0answers
46 views

Completation of an n.v.s. and dimensions of subspaces.

I don't know if the following statement is true: Let $X$ be an n.v.s. with $\text{dim}(X)=\infty$ and not Banach; and $\bar X $ its completation in the bidual space. Let $Y$ be a closed subspace ...
1
vote
1answer
32 views

Existence of right inverse.

We know that a surjective continuous linear map $ T : X \to Y$ has a right inverse iff $ \ker(T)$ is complemented. Here $X$ and $Y$ are Banach spaces. Is this result true for locally convex ...
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vote
1answer
64 views

Quotients and Topological Vector Spaces

Suppose $X$ is a topological vector space and $M$ is a closed linear subspace of $X$. Give $X/M$ the quotient topology induced by the mapping $p:X \to X/M$ defined by $p(x)= x + M$. The show that ...
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vote
3answers
273 views

Counterexample for Mazur–Ulam theorem

We know, Mazur–Ulam theorem states that if $V$ and $W$ are normed spaces over $\mathbb{R}$ and the mapping $f\colon V\to W$ is a surjective isometry, then $f$ is affine. Can somebody say ...
2
votes
1answer
59 views

For an inductive limit $X = \bigcup X_n$ of vector spaces, show that $X$ is complete if $X_n$ is complete for all $n$

Let $X$ be a vector space. Suppose that $\{X_n\}_{n=1}^\infty$ is a sequence of vector subspaces such that $X_n \subseteq X_{n+1}$ for all $n$, Each $X_n$ is a locally convex topological vector ...
2
votes
0answers
17 views

T1 axiom in dual space

My books talks about the conjugated space, but does it mean dual space? Are not the same thing? I don't understand why in the dual space $E^\ast$ of $E$, the separation axiom $T_1$ is satisfied and ...
0
votes
1answer
36 views

Linear bijection non-preserving Hausdorff propery

My question is: If $f: X \to Y$ is a continuous and linear bijection between topological vector spaces, is it possible that $X$ is Hausdorff and $Y$ is non-Hausdorff? (TVSs are considered in the more ...
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1answer
36 views

Question about the basis for the topology on an inductive limit of Frechet spaces

Let $X$ be a complex vector space, and let $\{X_n\}_{n=1}^\infty$ be a sequence of vector subspaces such that $X_n \subseteq X_{n+1}$ for all $n$ and $X = \bigcup_n X_n$. Furthermore, suppose that: ...
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0answers
69 views

Can complex-valued affine function be approximated?

If $K$ is a compact convex set of a locally convex Hausdorff space $V$ over $\mathbb{R}$ and $A(K)$ is the set of all continuous affine real-valued function on $K$, then the set of all restrictions to ...