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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6
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2answers
76 views

Alternate definition for boundedness in a TVS

Let $X$ be a topological vector space over $\mathbb R$ or $\mathbb C$. A subset $B\subset X$ is defined to be bounded if for any open neighborhood $N$ of $0$ there is a number $\lambda>0$ ...
1
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2answers
375 views

Continuity in Frechet spaces

These are undoubtably simple questions, but I have no background in functional analysis and am wondering about them. The first is an exercise from Folland, the second is not, but both are claims I've ...
2
votes
3answers
264 views

Balanced but not convex?

In a topological vector space $X$, a subset $S$ is convex if \begin{equation}tS+(1-t)S\subset S\end{equation} for all $t\in (0,1)$. $S$ is balanced if \begin{equation}\alpha S\subset S\end{equation} ...
1
vote
1answer
60 views

Nice example where $D^{\alpha}\Lambda_{f}\neq\Lambda_{D^{\alpha}f}$?

Let $\Omega\subset\mathbb{R}^n$ be open and $f$ be a locally integrable function. The distribution associated with $f$, $\Lambda_{f}\in D'(\Omega)$, is defined via \begin{equation} ...
12
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1answer
2k views

If $A$ and $B$ are compact, then so is $A+B$.

This is an exercise in Chapter 1 from Rudin's Functional Analysis. Prove the following: Let $X$ be a topological vector space. If $A$ and $B$ are compact subsets of $X$, so is $A+B$. My guess: ...
3
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0answers
437 views

Translation invariant metric

Under what conditions can a metric vector space be given an equivalent metric that is translation invariant? I was wondering if the probability measures on real line can be embedded in vector space ...
2
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2answers
173 views

Why Strongly Continuous Representations?

When working with not-necessarily-finite-dimensional representations, the topology on $GL(V)$ makes a difference. My experience has been that usually people require that the representation $\pi ...
6
votes
3answers
238 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 ...
2
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0answers
51 views

Topology of $(\mathcal{A},*)$ determined by $\mathcal{A}_{sa}$?

Let $(\mathcal{A},*)$ be a $*$-algebra, we have the following observation: Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\mathcal{A}$ such that the involution is an isometry with respect to ...
3
votes
1answer
756 views

Proof that every normed vector space is a topological vector space

The topology induced by the norm of a normed vector space is such that the space is a topological vector space. Can you tell me if my proof is correct? Of course we have to show that addition and ...
-1
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1answer
116 views

Practical implications of a vector space being a topological vector space

I have a space $V$ and I lately discovered that it's a topological vector space. What are the practical implications of that?
3
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0answers
263 views

Definition of a topological module

A topological universal algebra of type $\Omega$ is a universal algebra $A$ of type $\Omega$ that is also a topological space, such that for any $n\!\in\!\mathbb{N}$ and any operation ...
4
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4answers
142 views

Question on Topological vector space 1

I have numbered this question as (1) because I will be posting series of questions where I don't understand. I hope its allowed. I want to prove the following : If $X$ is a topological vector ...
4
votes
1answer
442 views

Finding the topological complement of a finite dimensional subspace

I know that for any finite dimensional subspace $F$ of a banach space $X$, there is always a closed subspace $W$ such that $X=W\oplus F$, that is, any finite dimensional subspace of a banach space is ...
2
votes
0answers
190 views

Hairy ball theorem: references to applications

I'm looking for references to applications of the Hairy ball theorem. I already visited wikipedia and cited references, but I need a little more explanation in both meteorology and applications in ...
3
votes
1answer
304 views

Extreme boundary of a compact, convex, metrizable set is $G_\delta$

Let $X$ be a topological vector space (no assumptions about local convexity are made in the question, though I am worried they might be required). Suppose $K\subset X$ is a compact, convex, metrizable ...
1
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0answers
67 views

When is the orbit of a vector a minimal sequence? When does an operator have a minimal orbit vector?

Let $X$ be a banach space. We say a sequence $(x_n)$ is minimal if for each $k$, $x_k\notin [x_n]_{n\neq k}$, where $[x,y,z,\cdots]$ is the closed linear span of the vectors. For ...
6
votes
2answers
294 views

Local base of a topological vector space

I would like to prove that if $B$ is local base for a topological vector space $X$, then every member of $B$ contains the closure of some member of $B$. I would appreciate if somebody can guide me ...
2
votes
1answer
124 views

Limit on a topological vector space

in the Wikipedia article on Gâteaux derivative , the limit of a function between two topological vector spaces is taken. How is the limit defined on a topological space for a function ? I find ...
2
votes
2answers
299 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 ...
1
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1answer
71 views

Are seminorms convex and a question on local base?

How can I prove that seminorms are convex? Another question is that, when we talk about topological vector spaces, why do we emphasise neighborhoods about $0$? As far as I know, if we know the ...
0
votes
1answer
225 views

$X^*$ with its weak*-topology is of the first category in itself

Let $X$ be an infinite-dimensional Fréchet space. Prove that $X^*$,with its weak*-topology is of the first category in itself.
0
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0answers
69 views

Finding point distribution by eigen vectors

First of all I want to tell that my mathematics is poor, so I can’t use correct terms. Sorry for that. I have a point data set. This data represents some cylindrical objects surfaces (not exactly ...
4
votes
1answer
132 views

Connected components that are relatively open in $\sigma(T)$

Let $T$ be an bounded linear operator on a Banach space $X$. Suppose the spectrum of $T$, $\sigma(T)$ has infinitely many connected components, then $\sigma(T)$ must contain infinitely many ...
0
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2answers
234 views

Vector Analysis & Linear Algebra

I'm given a positive number, a unit vector $u \in \mathbb{R} ^n $ and a sequence of vectors $ \{ b_k \} _{ k \geq 1} $ such that $|b_k - ku| \leq d $ for every $ k=1,2,...$. This obviously implies $ ...
1
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2answers
177 views

Closure of a nontrivial normed vector subspace that is equal to the whole space

Can you show me an example of a normed vector subspace $S$ strictly included in a normed vector space $V$ whose closure is equal to the whole $V$?
0
votes
1answer
493 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 $\beta$ ...
8
votes
1answer
254 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 ...
19
votes
3answers
1k 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
vote
1answer
95 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 ...
7
votes
2answers
3k 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 $V$. ...
1
vote
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
94 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
votes
1answer
623 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
votes
1answer
146 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
172 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
votes
0answers
49 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
163 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
155 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 ...
1
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2answers
260 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 ...
2
votes
3answers
210 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
669 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
votes
3answers
472 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
votes
2answers
221 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
499 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 ...
1
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1answer
621 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 ...
12
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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 ...
0
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0answers
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, ...
2
votes
3answers
414 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
votes
1answer
395 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 ...