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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16
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2answers
7k 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 ...
8
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2answers
6k 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 $V$. ...
5
votes
2answers
605 views

How to endow topology on a finite dimensional topological vector space?

This post may be coincide with some of the contents here. From Conway, A course in functional analysis, page 104. If $H$ is a finite dimensional vector space and $F_{1},F_{2}$ are two topologies ...
2
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1answer
684 views

Weak *-topology of $X^*$ is metrizable if and only if …

Let $X$ be a topological vector space on which $X^*$ separates points. Prove that "the weak *-topology of $X^*$ is metrizable if and only if $X$ has a finite or countable Hamel basis"? (A set $\beta$ ...
12
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2answers
2k views

Dual of a dual cone

Any hint on how to prove the following please: Let $K$ be a convex cone, and $K^*$ its dual cone. Prove that $K^{**}$ is the closure of $K$. Thanks!
10
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3answers
806 views

Do continuous linear functions between Banach spaces extend?

Just wondering... Let $E$, $G$ be Banach spaces, let $U\subset E$ be a subset of $E$, and let $f:U\rightarrow G$ be a continuous linear function. Can $f$ be extended to a continuous linear function on ...
5
votes
1answer
991 views

Is any Banach space a dual space?

Let $X$ be a Banach space. Is there always a normed vector space $Y$ such that $X$ and $Y^*$ are isometric or isomorphic as topological vector spaces (that is, there exists a linear homeomorphism ...
13
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1answer
987 views

If weak topology and weak* topology on $X^*$ agree, must $X$ be reflexive?

Let $X$ be a Banach space and suppose that the weak topology on $X^*$ agrees with the weak* topology on $X^*$. Must $X$ be reflexive? To prove the contrapositive, it will suffice to assume that $X$ ...
5
votes
1answer
120 views

Part (a) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (a) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
4
votes
1answer
610 views

$\omega$ - space of all sequences with Fréchet metric

I'm working on to prove the following: Show that the convergence in the space $\omega$ (space of all sequences with respect to the Fréchet metric) is the coordinate convergence. Any hint is ...
18
votes
2answers
1k views

When is a notion of convergence induced by a topology?

I'm interested in sufficient conditions for a notion of sequential convergence to be induced by a topology. More precisely: Let $V$ be a vector space over $\mathbb{C}$ endowed with a notion $\tau$ of ...
5
votes
1answer
638 views

What is the topological dual of a dual space with the weak* topology?

I'm trying to understand a claim I heard in class. To be concrete, suppose $X$ is a compact, hausdorff topological space, and let $C(X)$ be the space of continuous functions on $X$ with the supremum ...
4
votes
2answers
637 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 ...
8
votes
2answers
2k views

Sum of closed and compact set in a TVS

I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin's Functional Analysis), where $A$ and $B$ are subsets of a topological vector space ...
3
votes
1answer
324 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, ...
2
votes
2answers
345 views

First theorem in Topological vector spaces.

I came across this theorem and I am disappointed not being able to understand or to have intuition to understand it . I would be glad to get help . Theorem : If $K$ and $C$ are subset of topological ...
0
votes
1answer
1k views

Topology: Proof that a finitely generated cone is closed

Looking for the proof of the lemma asserting that the conical surface (envelope) is a closed space. Thank you.
5
votes
1answer
174 views

The weak topology of a product

Let $E$ and $F$ be normed vector spaces and let $E^{\sigma}$, resp. $F^{\sigma}$ be $E$, resp. $F$ with the weak topology associated with the elements of the duals $E^*$, resp. $F^*$. Then, for ...
2
votes
0answers
94 views

Part (b) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (b) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
2
votes
2answers
305 views

Openness of linear mapping

Let $X$ be a topological vector space over the field $K$, where $K=\mathbb R$ or $K= \mathbb C$, and let $f:X\rightarrow K^n$ ($n \in \mathbb N$) be a linear and surjective functional. How to prove ...
0
votes
1answer
51 views

How can I prove this is a metric?

While proving the Banach-Alaoglu theorem, one needs to prove the topology induced by a countable family of seminorms $\rho_n$ on a vector space $X$ is metrizable if $X$ is Hausdorff with that ...
22
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3answers
2k views

When do weak and original topology coincide?

Let $X$ be a topological vector space with topology $T$. When is the weak topology on $X$ the same as $T$? Of course we always have $T_{weak} \subset T$ by definition but when is $T \subset ...
13
votes
3answers
2k views

Does every $\mathbb{R},\mathbb{C}$ vector space have a norm?

Is there a canonical way to define on any vector space over $\mathbb{K}=\mathbb{R},\mathbb{C}$ a norm ? (Or, if there isn't, can someone give me an example of a vector space over $\mathbb{K}$ that is ...
6
votes
1answer
109 views

local convexity of $L_p$ spaces

wiki says The spaces $L_p([0, 1])$ for $0 < p < 1$ are equipped with the F-norm they are not locally convex, since the only convex neighborhood of zero is the whole space Why is this so? ...
6
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1answer
894 views

Generic topology on a vector space?

For a (possibly infinite-dimensional) vector space $V$, I thought about the following topology $\tau$: Let $O \in \tau$ if every $x \in O$ has the property that for every $v \in V$, there is an ...
5
votes
1answer
525 views

An explicit construction for a “doubly weak” topology

Let $(X,s)$ be a topological vector space over $\mathbb{F}$ with linear topology $s$, which we will henceforth refer to as the strong topology. Then, as usual we can construct the continuous dual ...
3
votes
2answers
699 views

Meaning of “a mapping preserves structures/properties”

Sometimes I see something like "a mapping preserves the structures of its domain and of its codomain". From Wiki about morphisms in category theory: a morphism is an abstraction derived from ...
8
votes
3answers
299 views

Can continuity of inverse be omitted from the definition of topological group?

According to Wikipedia, a topological group $G$ is a group and a topological space such that $$ (x,y) \mapsto xy$$ and $$ x \mapsto x^{-1}$$ are continuous. The second requirement follows from the ...
12
votes
3answers
740 views

Topology on the general linear group of a topological vector space

Let $K$ be a topological field. Let $V$ be a topological vector space over $K$ (if it makes things convenient, you may assume it is finite dimensional). Naive Question: Is there a canonical way of ...
4
votes
1answer
2k views

Question about proof that finite-dimensional subspaces of normed vector spaces are direct summands

I am reading a proof that finite-dimensional subspaces of normed vector spaces have closed direct sum complements. This is the proof: Let $\{e_1, ..., e_n\}$ be a basis for $\mathcal M$. Every $x ...
3
votes
0answers
141 views

What do we call a Schauder-like basis that is uncountable?

In a topological vector space, every Schauder basis is assumed countable, by definition. Supposing we drop the countability condition, we call this a [what goes here?] basis?
1
vote
2answers
85 views

$C_c(X)$ is dense in $C_0(X)$

Let $X$ be a topological space, and let $C_0(X)$ be the $\mathbb{C}$-vector space of continuous functions $g:X \rightarrow \mathbb{C}$ with the property that for any $\epsilon > 0$, there exists a ...
7
votes
2answers
277 views

Topological vector space with discrete topology is the zero space

Hello i have a question about topological vector spaces. To remind the definition of such a space: A topological vector space is a pair $(X,\tau)$ with $X$ a vector space and $\tau$ a topology on ...
7
votes
1answer
186 views

Rotation of $\mathbb{R}^3$ by using quaternion

Express the rotation of $\mathbb{R}^3$ by $\frac{\pi}{3}$ about the $x=y=z$ axis by using quaternions and identifying $\mathbb{R}^3$ with $(i,j,k)$-space. Thoughts: From my point of view, every ...
2
votes
1answer
88 views

Part (c) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (c) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
2
votes
1answer
48 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 ...
2
votes
1answer
197 views

Why are the Differential- and multiplication mapping on $C^{\infty}(\Omega)$ continuous?

Let $\Omega\subset\mathbb{R}^n$ be open and $\Omega\neq\varnothing$ and suppose we have the Fréchet topology on $C^{\infty}(\Omega)$ (this can be obtained by the topology construction from out ...
1
vote
1answer
81 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 ...
1
vote
1answer
85 views

About a weak topology on TVS

Let $(X,\tau)$ be a topological vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each ...
0
votes
0answers
51 views

Doubt Concerning the Definition of a Locally Convex Space Structure through Seminorms?

In the book Introduction to Functional Analysis written by A. E. Taylor there are the following theorems: Theorem 1. Suppose that $X$ is a linear space and that $\mathscr{U}$ is a nonempty family ...
16
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1answer
2k views

Semi-Norms and the Definition of the Weak Topology

When I was searching for the definition of the weak topology, I found two different definitions. One defines the weak topology in terms of a family of semi-norms, while the other defines it in terms ...
12
votes
1answer
955 views

Hahn-Banach theorem: 2 versions

I have a question regarding the Hahn-Banach Theorem. Let the analytical version be defined as: Let $E$ be a vector space, $p: E \rightarrow \mathbb{R}$ be a sublinear function and $F$ be a subspace of ...
9
votes
1answer
225 views

Why do we give $C_c^\infty(\mathbb{R}^d)$ the topology induced by all good seminorms?

Briefly, my question boils down to the following: What benefits do we gain from considering the space of test functions in the topology induced by all "good" seminorms, as opposed to other topologies ...
8
votes
1answer
665 views

Sequential and topological duals of test function spaces

Given a test function space, in particular $\mathcal{S}=\mathcal{S}(\mathbb{R}^n)$ (the Schwartz space) or $\mathcal{D}=\mathcal{D}(\mathbb{R}^n)$ (the space of compactly supported smooth test ...
8
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1answer
234 views

Is there finest topology which makes given vector space into a topological vector space?

I we are given a vector space $(V,+,\cdot)$ over a field $\mathbb K$ (where $\mathbb K=\mathbb R$ or $\mathbb K=\mathbb C$), is there the finest topology $\mathcal T$, such that $(V,\mathcal T)$ is ...
5
votes
1answer
147 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 ...
4
votes
1answer
773 views

proving that $SO(n)$ is path connected

Our professor gave us exercise to show that $G=SO(n,\mathbb R)$ is path connected. He gave some hints, using them I have come upto this far: I have shown that $SO(n)$ acts on $S^{n-1}$ transitively ...
3
votes
2answers
299 views

Left topological zero-divisors in Banach algebras.

Let $ A $ be a unital Banach algebra. Define $ \zeta: A \longrightarrow [0,\infty) $ by $$ \forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|, $$ where $ ...
2
votes
1answer
156 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 ...
6
votes
1answer
2k views

Interior of a convex set is convex [duplicate]

A set $S$ in $\mathbb{R}^n$ is convex if for every pair of points $x,y$ in $S$ and every real $\theta$ where $0 < \theta < 1$, we have $\theta x + (1- \theta) y \in S$. I'm trying to show that ...