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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140 views

Question about using continuity of a seminorm

I almost have a homework problem solved but I've used a claim that might be dubious. The setting is this: Let $(V,Q)$ be a locally convex space ($Q$ is the family of seminorms inducing the topology ...
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1answer
219 views

Relationship between Convergence and Open sets

If you show that convergence of nets in a topological vector space $V$ with topology $\tau$ is equivalent to convergence of nets in a topological vector space $V$ With topology $\sigma$, does it ...
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1answer
409 views

Continuity of Minkowski functional

I'm working through a proof and the setting is that I have a convex, balanced, open subset $U$ of a topological vector space $V$. The claim that I can't verify is made briefly: the Minkowski ...
9
votes
1answer
671 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 ...
10
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1answer
220 views

Is the standard structure of a topological vector space on reals unique?

The standard stucture of a topological vector space on reals is this given by the metric d(x,y)=|x-y| on the vector space $\mathbb{R},$ with the field of scalars $\mathbb R$ with standard topology. I ...
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1answer
97 views

How can I verify the topology of locally convex vector spaces induced by seminorms is closed under unions?

The definitions I am using are as follows: A vector space $V$ equipped with a family $P$ of semi-norms such that $\cap_{p\in P}\{x\in V : p(x) = 0\} = \{0\}$ is called a locally convex vector space. ...
8
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3answers
571 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 ...
3
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0answers
100 views

Self-absorbing subsets in a vector space

From planetmath Let $V$ be a vector space over a field $F$ equipped with a non-discrete valuation $|\cdot|:F\to \mathbb{R}$ . Let $A$ and $B$ be two subsets of $V$. Then $A$ is said to absorb ...
2
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1answer
239 views

Definition of boundedness in topological vector spaces

From Wikipedia: Given a topological vector space $(X,τ)$ over a field $F$, $S$ is called bounded if for every neighborhood $N$ of the zero vector there exists a scalar $α$ so that $$ S ...
5
votes
1answer
452 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 ...
3
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0answers
356 views

Understanding examples - metric spaces, Minkowski functionals and topologies

I'm teaching myself a course on functional analysis but having trouble understanding the notes I've been using. I was hoping I could write out a section of the content and you might be able to help me ...
3
votes
1answer
198 views

Dual space $E^*$ metrizable iff E has a countable basis

I have trouble proving the following theorem: If $E$ is a locally convex, Hausdorff topological vector space, then $E^*$ is metrizable if and only if $E$ has an (at most) countable basis. I've ...
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2answers
332 views

Example of a topological vector space

I have the following question: give an example of a topological vector space $E$ with subspace $M$ and $N$, such that $E = M \oplus N$ algebraically, but not topologically (so $E \ncong M \sqcup N$). ...
4
votes
1answer
149 views

Why does the continuity of $(x,y) \rightarrow x-y$ mean the commutativity of a topological group?

The following is a part from P.13 in Topological Vector Spaces (Third Printing Corrected 1971) by H.H.Schaefer Given a vector space $L$ over a (not necessarily commutative) non-discrete ...
2
votes
1answer
100 views

Is it always true that |1+1|>1 in an Archimedean valuated field?

The following is a sentence from the proof of the theorem 1.2 (P.14-15) in Topological Vector Spaces (Third Printing Corrected 1971) by H.H.Schaefer Finally, if $K$ is an Archimedean valuated ...
4
votes
2answers
257 views

If you know the convergent sequences, how do you know the open sets?

I have a homework problem which I feel should be simple but is actually surprisingly tricky. This is why I love math sometimes.... Let $X$ be a normed linear space. Suppose $\|\cdot\|_1$ and ...
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votes
0answers
100 views

Is the dual cone of the dual cone equal to the original cone? [duplicate]

Possible Duplicate: Dual of a dual cone I try to prove the following statement: Let $V$ be a finite-dimensional ordered topological vector space ($V^{**} \cong V$) with a closed positive ...
2
votes
2answers
916 views

Original proof of Uniform Boundedness Principle (Banach Steinhaus) and related questions

Can someone please provide me with any of the things listed below : a list of different proofs of (some version of) the Uniform Boundedness Principle (also known as the Banach Steinhaus theorem), I ...
3
votes
1answer
137 views

The topology of $L_\mathrm{loc}^2 (\mathbb{R})$

In fact I am reading the book of Ohsawa, Analysis of Several Complex Variables, and I came across this line on page 13, ... $L^{2}_\mathrm{loc}(\Omega)$ with respect to the topology induced by the ...
4
votes
1answer
179 views

Existence of balanced neighborhoods in a topological vector space

I'm wondering about the following: Let$\ X $ be a topological vector space. Then one could pick balanced neighborhoods$\ W $ and$\ U $ of$\ 0 $ such that $\ \overline{U} + \overline{U} \subset W $, ...
11
votes
2answers
857 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 ...
8
votes
1answer
253 views

Strong topology vs Natural topology

Let $X$ be a locally convex space and $\left< X, X^{\prime} \right>$ stands for the dual pair. The bidual of $X$ is denoted by $X^{\prime \prime}$ and this is a dual of $X^{\prime}$ with a ...
27
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0answers
614 views

Differential forms on fuzzy manifolds

This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me. Question In the usual setting of open subsets of ...
5
votes
1answer
886 views

Convex functions and families of affine functions

I know that the supremum of a family of affine functions is convex. Just wondering if it is true (and if so how one proves) that the converse -- any $C^1$ convex function is the supremum of some ...
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0answers
95 views

Are nuclear Montel spaces projective?

The well known fact about $\ell^1$ says that the Schauder basis of $\ell^1(I)$ behaves more-less like a Hamel basis, namely if $X$ is any Banach space and $\mathcal{E}=(e_i)_{i\in I}$ is a basis for ...
2
votes
2answers
239 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 ...
10
votes
1answer
409 views

Isomorphisms of Fréchet Spaces

What is the proper notion of an isomorphism between Fréchet spaces? Obviously it should be a linear map. I'm just worried about the analytic structure. Should one be able to order the seminorms on ...
6
votes
1answer
308 views

Uniqueness of the derivative in locally convex topological vector space

I need a hint of proof of uniqueness of the derivative in locally convex topological vector space (it's asserted in Lang's "Introduction to differentiable manifolds"). Define derivative of a function ...
3
votes
2answers
469 views

projection operators on topological vector spaces

Suppose $A \in \mathbb{R}^{m\times n}$. Then there exists a projection matrix $P$ onto the range of $A$. In other words, there exists a matrix $P \in \mathbb{R}^{m\times m}$ such that $P^2=P$, and ...
1
vote
1answer
192 views

Understanding Hom-bundle on vector bundles (Atiyah)

I am trying to make sense of the following statment (from Atiyah's K-theory book) Suppose $V,W$ are vector spaces, and that $E=X \times V, F=X \times W$ are the corresponding product bundles. Then ...
11
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3answers
570 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 ...
8
votes
2answers
1k 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!
4
votes
1answer
205 views

Why This Map is Closed?

Consider the following definition of closed maps, defined in the book Nonlinear Programming by Bazaraa et al.: Let $X$ and $Y$ be nonempty closed sets in $\mathbb{R}^p$ and $\mathbb{R}^q$, ...
3
votes
1answer
379 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 ...
16
votes
2answers
784 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
294 views

How to define the derivative of Radon measures

Let $M$ be the positive borel measures on a hausdorff topological space $X$, which are finite on compacts sets $--$ i.e. the real cone of radon measures. I am given a definition of a derivative of ...
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2answers
646 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.
2
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1answer
140 views

Simple Continuity Question concerning Vector Bundles

I'm a bit confused about the following part in Sir Michael Atiyah's "K-Theory." Let $E = X \times V$ and $F = X\times W$, and let $\phi: E\to F$ be a vector bundle homomorphism. Why is the induced ...
4
votes
1answer
155 views

Finest topology on a space of banach space operators

let $X$ be some Banach space. Let $L(X)$ be the set of continuous operators on $X$ into $X$. Let $(\tau_i)_i$ be a set of topologies on $L(X)$ s.t. $L(X)$ is topological vector space (i.e. addition ...