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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86 views

Can 2 parallel lines be discriminated as 'away', 'beside' with respect to 3rd parallel line?

I have nearly parallel several 3D line segments. some line segments locate (blue line) beside to a spefic line segment (black line) and some other (red line) locate away from that line segment. i want ...
2
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0answers
48 views

When is a subset of {0,1} valued borel functions on a standard borel space (polish space) complete (see *) under the pointwise convergence topology?

*In other words, what restrictions on a family F of {0,1} valued borel functions will tell us that the pointwise limit of any net in F is borel. I feel like there must be lots known about this but I ...
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158 views

Are topological vector spaces completely regular?

Every uniformizable space is a completely regular topological space. topological vector spaces are uniform spaces. every Hausdorff topological vector space is completely regular. From ...
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2answers
148 views

Completability of a uniform space, metric space and topological vector space?

From Wikipedia In particular, topological vector spaces are uniform spaces and one can thus talk about completeness, uniform convergence and uniform continuity. (This implies that every ...
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2answers
253 views

“A linear operator $f$ is continuous if $f(V)$ is bounded for some neighborhood $V$ of $0$.”

From Wikipedia a linear operator $f$ between two topological vector spaces is continuous if $f(V)$ is bounded for some neighborhood $V$ of $0$. I wonder why it is true? If I understand ...
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3answers
201 views

Boundedness in a topological space?

I was wondering if there is a concept of boundedness for subsets of a topological space? If yes to 1, is it this one from Wiki Elements of a Bornology B on a set X are called bounded sets and the ...
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1answer
615 views

Definition of locally convex topological vector space

From Wiki A locally convex topological vector space is a topological vector space in which the origin has a local base of absolutely convex *absorbent* sets. Also from Wiki Locally ...
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3answers
430 views

Metric linear space and locally convex topological vector space

I was wondering which one is more general, metric linear spaces or locally convex topological vector spaces? Is a metric linear space a locally convex topological vector space? Vice versa? In terms ...
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2answers
213 views

“The two notions of boundedness coincide for locally convex spaces”

From Wiki The boundedness condition for linear operators on normed spaces can be restated. An operator is bounded if it takes every bounded set to a bounded set, and here is meant the more ...
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2answers
466 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 ...
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1answer
559 views

Continuous linear mapping and bounded subsets

Continuous linear mappings between topological vector spaces preserve boundedness. I was wondering if it means that the inverse image of a bounded subset under a continuous linear mapping is ...
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2answers
3k 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 ...
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39 views

A subset that can be scaled to be the whole space, or that can contain a scaled version of the whole space

The following is from Mariano's comments on my earlier question In a topological vector space, why is the following true: if a neighborhood U of zero contains a scaled copy of the whole space, ...
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3answers
378 views

Why is boundedness defined so differently in a topological vector space and in a metric space?

From Wikipedia A subset S of a metric space (M, d) is bounded if it is contained in a ball of finite radius Also from Wikipedia a set in a topological vector space is called bounded or von ...
2
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1answer
358 views

Are operator and mapping the same concept?

I was wondering what differences and relations are between a mapping and an operator generally? For topological vector spaces or functional analysis, it seems like an operator and a mapping are the ...
8
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1answer
714 views

The dual of a Fréchet space.

Let $\mathcal{F}$ be a Fréchet space (locally convex, Hausdorff, metrizable, with a family of seminorms ${\|~\|_n}$). I've read that the dual $\mathcal{F}^*$ is never a Fréchet space, unless ...
4
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1answer
290 views

Constructing a countable family of seminorms in a metrizable LCS.

Here's some context before my question. Let $\mathbb{V}$ be a topological vector space, which is Hausdorff and such that its topology is generated by some arbitrary family of seminorms ...
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1answer
283 views

How to get a projected 3d line segment, lie on another 3d line parallel to that line segment.

I have a 3D line segment and another 3D position which locate slightly away from the line segment. I want to get the projected line segment (3D) which lies on imaginary 3D line which passes through ...
3
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1answer
146 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
232 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
445 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
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1answer
702 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 ...
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1answer
221 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
98 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. ...
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3answers
584 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 ...
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102 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 ...
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1answer
247 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 ...
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1answer
470 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 ...
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382 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 ...
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1answer
203 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
342 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
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1answer
150 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 ...
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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 ...
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2answers
260 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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0answers
102 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 ...
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2answers
954 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
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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
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1answer
185 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 $, ...
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2answers
956 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
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1answer
256 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 ...
31
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0answers
722 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
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1answer
926 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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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
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2answers
250 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
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1answer
423 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
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1answer
316 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
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2answers
493 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 ...
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1answer
198 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
585 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 ...
10
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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!