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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263 views

Do the notions of weak and weak* convergence coincide for $\ell^1(\mathbb{N})$?

As my friends and I were studying for our real analysis final exam yesterday, we were playing with various examples and found ourselves asking this question: The space $\ell^1(\mathbb{N})$ is the ...
20
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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 ...
1
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1answer
103 views

Topological fields questions

From Wikipedia: "Let $K$ be a topological field, namely a field with a topology such that addition, multiplication, and division are continuous. In most applications $K$ will be either the field of ...
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2answers
4k 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 $V$. ...
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1answer
1k views

Must a complete space be locally compact?

There are two versions of second category space: one is complete metric space, the other is locally compact space. As we know, an open interval is locally compact but not complete. But how about the ...
3
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0answers
97 views

Consequence of metrizability proof - disregard, the question is an error

In Marian Fabian et al's Functional Analysis and Infinite-Dimensional Geometry, Proposition 3.22 states/proves that if $X$ is a separable Banach space, then the (closed) unit ball, $B_{X^{*}}$ of ...
6
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1answer
710 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 ...
0
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1answer
148 views

How is the topology on a vector space induced from its base field $\mathbb{R}$?

I don't remember where on this site, but I vaguely remember seeing that for a vector space with its base field being $\mathbb{R}$ (or more generally, a topological field?), there can be a natural ...
0
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2answers
191 views

Does “two topological vector spaces in duality” means they are each other's continuous dual?

When saying two topological vector spaces $E$ and $F$ are in duality, does it mean that they are each other's continuous dual, i.e. $E = F^*$ and $F=E^*$, or just that one is the other's continuous ...
2
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0answers
52 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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0answers
184 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
163 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
283 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 ...
5
votes
3answers
259 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 ...
3
votes
1answer
781 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 ...
3
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3answers
547 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 ...
4
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2answers
250 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 ...
3
votes
2answers
574 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
743 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 ...
13
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2answers
4k 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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0answers
45 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, ...
2
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3answers
447 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
462 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 ...
9
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1answer
873 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
votes
1answer
406 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
421 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 ...
4
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1answer
163 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 ...
1
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1answer
279 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 ...
2
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1answer
599 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
818 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
225 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 ...
2
votes
1answer
104 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. ...
9
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3answers
693 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
106 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
259 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 ...
8
votes
1answer
569 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
446 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 ...
4
votes
1answer
215 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 ...
4
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2answers
391 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
154 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
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1answer
102 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 ...
5
votes
2answers
291 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
104 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 ...
4
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2answers
1k 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
139 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
207 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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3answers
1k 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
264 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 ...
60
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2answers
1k 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
1k 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 ...