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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3answers
42 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
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1answer
69 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
43 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 ...
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1answer
75 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 ...
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0answers
27 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
52 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 ...
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0answers
47 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
31 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 ...
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1answer
47 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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1answer
37 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
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1answer
120 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 ...
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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
53 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
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1answer
30 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
78 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
26 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
37 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
75 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 ...
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0answers
64 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
75 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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2answers
99 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
95 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
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0answers
38 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 ...
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1answer
37 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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1answer
85 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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3answers
316 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
80 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
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0answers
19 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
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1answer
46 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
45 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
73 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 ...
3
votes
1answer
74 views

Absorbing at Each point means Open?

In a topological vector space, it seems that open sets have the property that they are absorbing at each of their points, since $(\alpha,x)\to \alpha x$ is continuous. I am wondering if the converse ...
2
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1answer
41 views

How to construct a particular convex set (when defining inductive limits of Frechet spaces in Reed and Simon)

Let $X$ be a topological vector space (over $\mathbb{C}$) whose topology is defined by a family of separating seminorms $\{\rho_\alpha\}$. Let $X_1$ be a vector subspace of $X$ whose topology is the ...
1
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1answer
51 views

Almost Everywhere Function Space

Problem Let $\Omega$ be a measure space with measure $\mu$ and $V$ a topological vector space not necessarily Hausdorff as well as the function space $\mathcal{F}:=\{f:\Omega\to V\}$ topologized by ...
4
votes
1answer
77 views

Topology Book including specific aspects

I am looking for a basic book about Topology (maybe also a bit of Functional analysis but basically Topology) including the following points (in addition to the basic points): $\bullet$ Seminorms ...
1
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1answer
39 views

Topological Tensor Product is a Topological Ring Independent of the Choice of Basis

Let $A, B$ be commutative rings containing a field $k$, with $B$ a finite dimensional $k$-module, $w_1, ... , w_N$ a basis. If $w_iw_j = \sum\limits_{n=1}^N c_{ijn}w_n$, then we can define $C$ to be ...
1
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1answer
34 views

How to show $\mu_A(x)=\inf\{\alpha>0: \alpha^{-1} x\in A\}$ where $\mu_A(x)$ is the Minkowski functional of $A$?

Let $X$ be a $\mathbb K$-vector space ($\mathbb K=\mathbb R$ or $\mathbb C$) and suppose $A\subset X$ is convex (and absorbing). How to show $$\{x\in X: \mu_A(x)<1\}\subset A?$$ Above ...
3
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0answers
65 views

When a locally convex space is metrizable

Let $X$ be a locally convex Hausdorff topological space, whose topology if generated by the countable family of seminorms $\{p_i:\space i\in\mathbb{N}\}$. I'd like to prove that $X$ is metrizable. So, ...
0
votes
1answer
57 views

Balanced Core: $U\text{ open }\implies U^*\text{ open}$

I need one last lemma for the proof of finite dimensional subspaces are closed: Is it true that if a subset is open so is its balanced core??
5
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0answers
124 views

Connections and dependences between topological and algebraic basis in topological vector space

On my last functional analysis exam, one of the tasks was to show that if normed vector space $X$ have countable Hamel basis, then $X$ is separable space (over field $\mathbb{R}$). I am not sure if ...
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1answer
61 views

TVS: Uniform Structure

Disclaimer: This thread is meant informative and therefore written in Q&A style. Of course everybody is encouraged to give an answer as well! Prove that any topological vector space gives rise ...
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2answers
97 views

Topological Vector Space: $\dim V=n\implies V\cong\mathbb{K}^n$

Let $V$ be a finite dimensional Hausdorff topological vector space. Prove that it is is isomorphic to the Euclidean vector space of the same dimension: $$\dim V=n\implies V\cong\mathbb{K}^n$$
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1answer
25 views

Balanced Core: Explicit Expression?

Denote the collection of all balanced subsets by: $\mathcal{B}:=\{B\subseteq X: B\text{ balanced}\}$ Since the union of arbitrary balanced sets is balanced we can form the balanced core of arbitrary ...
2
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0answers
26 views

Criteria for a sequence in $C[0,1]$ to converge weakly [duplicate]

Is there a criterion for a sequence in $C[0,1]$ to converge weakly? Let $\{f_n\}$ be a sequence in $C[0,1]$, $f\in C[0,1]$. Suppose $f_n\to f$ (weakly). Then for each $x\in [0,1]$ $ev_x(f_n)\to ...
0
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1answer
32 views

Various convergences in the space of bounded operators

Could you please help me to find some classical (counter)examples in functional analysis? Let $X$ and $Y$ be some normed spaces over $\mathbb{C}$. By $\mathcal{B}(X,Y)$ we denote the space of bounded ...
2
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1answer
29 views

Closure of a von Neumann bounded set

Let $V$ be a topological vector space and $B \subseteq V$ bounded. Then the closure $\overline{B}$ is bounded. This appears on the Wikipedia page ...
4
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0answers
63 views

Infinite Dimensional Topological Vector Space

Let $V$ is a finite-dimensional vector space over $\mathbb{R}$ (or $\mathbb{C}$). To make $V$ a topological space, we may choose the sets $f^{-1}(U)$ as a sub-basis, where $f$ ranges over all linear ...
2
votes
2answers
91 views

Show that no topological vector space is bounded.

I am studying the concept of topological vector spaces in Grubb's Distributions and Operators. A vector space $X$ (over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$) is called a topological vector ...
2
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1answer
53 views

Please verify this definition of locally convex vector space

A locally convex vector space is a vector space $V$ equipped with a family $P$ of separating semi-norms. Is this a correct definiton? So to determine whether a norm induces a locally convex topology ...