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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Weak operator topology is the smallest topology on $B(H)$

Show that weak operator topology is the weakest locally convex topology on $B(H)$ such that every $\phi\in F(H)$ is continuous. (F(H) means finite rank operators on $H$). To show it , let $\tau$ ...
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37 views

Closed unit ball of $B(H)$ with wot topology is compact

The following is a Theorem of Conway's operator theory: I can not understand how he proves it. I think $\phi(\text{ ball B(H)})$ is compact if $\phi(\text{ ball B(H)})$ is closed subset of compact ...
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42 views

Does $B(H)$ satisfy in Heine-Borel property?

Based on here, I know that every bounded and closed subset of a space is not compact. I really want to know that $B(H)$, the space of bounded linear operators, satisfies in Heine - Borel property. ...
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97 views

On separable Hilbert space $H$, weak operator topology is metrizable on bounded parts of $B(H)$

The following is a theorem of Takesaki's operator theory: In this proof, weak topology means weak operator topology. I'm wonder why the theorem holds just for bounded parts of $B(H)$ and also ...
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1answer
23 views

Proving linearity implies (or can imply under opportune conditions) lower semicontinuity

A function $f:X\to\mathbb{R}$, with $X$ being a topological space, is termed as lower semicontinuous (lsc) at $x_0\in X$ if: $$\forall\epsilon>0\,\,\exists V\text{ an open neighborhood of }x_0:x\in ...
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1answer
50 views

Proof that difference of compact and closed sets is also closed [duplicate]

I am trying to prove that for any $A$ compact, $B$ closed sets $\Rightarrow A-B = \{a-b | a\in A, b\in B\}$ is also closed, where A and B are subsets of a topological vector space $X$.
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1answer
40 views

Is this extension a Hilbert space?

Let $V$ be an inner product space over $\mathbb{F}$. Let $H$ be a complete subspace of of $V$ and $x\in V\setminus H$ Define $K= span(H\cup \{x\})$. Is $K$ a Hilbert space? How do I prove it?
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82 views

Metrizability of the unit ball $B_{X^*}$.

I am trying to prove the assertion: If $X$ is a separable normed space, then the unit ball in $X^*$ with the weak* topology, $(B_{X^*},\sigma(X^*,X))$, is metrizable. Firstly, I took ...
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1answer
30 views

Regarding direct sums in topological vector spaces

If $E=E_0\oplus E_1$ is a topological vector space and $A\subseteq E_0$ open in the induced topology on $E_0$, can I conclude that $A+E_1$ is open in $E$? Possibly if I assume $E$ to be locally ...
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1answer
21 views

Regarding embeddings of locally convex spaces

If $f:E\rightarrow E'$ is a linear embedding of locally convex topological vector spaces, and $A\subseteq E$ open and convex, can we always find $A'\subseteq E'$ open and convex sucht that ...
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78 views

A typical example of Homeomorphism

The set $\mathbb{R}^2-\{(0,0)\}$ with the usual topology is: (A) Homeomorphic to the open unit disc in $\mathbb{R}^2$ (B) the cylinder $\{(x,y,z)\in \mathbb{R}^3/ x^2+y^2=1 \}$ (C) the ...
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1answer
64 views

Topological supremum of family of linear topologies

In Hans Jarchow - Locally Convex Spaces 2.4.4 (c) it says: The topological supremum of any family of linear topologies on a fixed vector space is linear. I couldn't find a proof in the book and ...
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75 views

Continuity of implicit function

I find a proof of the following theorem in A.N. Kolmogorov and S.V. Fomin's Элементы теории функций и функционального анализа (pp. 492-493 here): Let $X,Y,Z$ be Banach spaces, $U$ a neighbourhood ...
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1answer
31 views

Operator topologies and examples

In class we covered several operator topologies: the weak topology, the weak* topology, the weak operator topology, and the strong operator topology. The first two are defined on a normed vector ...
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57 views

Closedness of convex sets in a locally convex space

Let $C$ be a convex subset of a locally convex topological vector space. Consider the properties: a) $C$ is closed. b) $C$ is weakly closed. c) $C$ is weakly sequentially closed. d) $C$ is ...
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2answers
153 views

Topology on the space of test functions

I try to read into the theory of distributions and there is one thing which bothers me. I read that a distribution is a linear, continuous functional from the space of test functions, which, depending ...
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61 views

Limit topology of a sequence of topological vector spaces

Under which circumstances is the limit topology of an increasing sequence $E_0\subseteq E_1\subseteq E_2\subseteq\cdots$ of topological vector spaces, where the inclusion maps are linear and ...
3
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1answer
59 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 ...
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1answer
86 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 ...
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0answers
31 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 ...
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47 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 $(C[0,1],\sigma)$ ...
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1answer
66 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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28 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 ...
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1answer
44 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 ...
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33 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 ...
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40 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 ...
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2answers
99 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
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1answer
69 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 ...
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36 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: ...
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1answer
38 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 ...
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1answer
68 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 ...
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1answer
19 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 ...
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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 ...
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3answers
39 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 ...
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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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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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26 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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51 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
45 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
45 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
35 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.
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1answer
118 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 ...
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0answers
51 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 ...
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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
77 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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25 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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36 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 ?