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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60 views

How are the assumptions used in the proof of Bourbaki-Alaoglu Theorem?

This is a follow up question to a previous one. In the proof of the following theorem, where are the assumptions "Hausdorff" and "locally convex" used?
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2answers
25 views

Let $E$ be a topological linear space, if $U$ is a neighbourhood of $0$ why is $U+x$ a neighbourhood of $x$?

Let $E$ be a topological linear space, if $U$ is a neighborhood of $0$ why is $U+x$ a neighborhood of $x$? With linear space I mean a vector space over the real or complex numbers. I know the ...
2
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0answers
30 views

Linear finite-dimensional topological vector space is closed.

Suppose $X$ is a topological vector space and let $Y \subset X$ be its subspace with $\dim Y < \infty$. The goal is to prove that $Y$ is closed in $X$. I know how to prove this fact when $X$ is ...
2
votes
2answers
101 views

Closedness in the proof of the Alaoglu Theorem

I'm reading a proof the Bourbaki-Alaoglu Theorem: Could someone explain how the closedness of $\Phi(V^\circ)$ (namely, $\Phi(V^\circ)$ contains all of its limit points) is done in the proof? I ...
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1answer
25 views

Uniqueness of separated (Hausdorff) topology on a $n$-vector space

I recently asked a question on a french forum about the proof that any norms on a finite dimensional (real or complex) vector space are equivalent. Someone showed me the proof of a more powerful ...
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1answer
47 views

neighborhood base for the Mackey topology

I'm reading the proof of a theorem due to Mackey in a note of functional analysis: I don't see why the first sentence is clear. By the definition of neighborhood base and locally convex topology, ...
2
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0answers
67 views

Show that $df_x: TM_x\rightarrow TN_{f(x)}$ is well-defined

Let $f\colon M\rightarrow N$ be a $C^\infty$ function between $C^\infty$ manifolds. Show that $$df_x: TM_x\rightarrow TN_{f(x)}$$ defined by $$df_x([c_0]) := [f \circ c_0]$$ is a well-defined map. ...
2
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1answer
56 views

Natural isomorphism between linear space to bilinear space

Let $V$ and $W$ be (not necessarily finite-dimensional) vector spaces. Show that there is a natural isomorphism (meaning an isomorphism that can be described without reference to a basis) between ...
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1answer
23 views

Prove that continuity $\Leftrightarrow$ component continuity

Theorem: Let $\mathbf{f}: S \to \mathbb{R}^n$, where $S \subseteq \mathbb{R}^n$. $\mathbf{f}$ is continuous if and only if its components $\mathit{f}_k, k = 0, \dots, n-1$ are continuous. ...
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1answer
27 views

linear independence in a dual pair and its consequence

This is a follow-up question to a previous one: linear independence in a dual pair. The following is from the Topological Vector Spaces by Schaefer: The corollary has been proven independently ...
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0answers
15 views

$P(D)f=g$ has a solution $f\in C^\infty(\Omega, E)$ for every $g\in C^\infty(\Omega, E)$?

I read in a paper the following statement: Let $P(D)$ be an elliptic linear partial differential operator with constant coefficients, $\Omega\subset \mathbb R^n$ an open subset and $E$ a Fréchet ...
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1answer
46 views

linear independence in a dual pair

The following is an excerpt from the Topological Vector Spaces by Schaefer: I don't see how the underscored sentence work. Suppose $$ \langle x,y_n\rangle=0 $$ for all $x\in F_n$. Why this implies ...
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1answer
33 views

weak topology and dual pairs

I'm reading a note on functional analysis and the following statement is given without a proof: Let $(X,Y,\langle,\rangle)$ be a dual pair and $\tau$ a topology on $X$. Let $\sigma(X,Y)$ be the ...
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1answer
72 views

Do I need topology to study stochastic process?

So far I dealt with probability from a very intuitive point of view, like guessing frequencies etc. But while studying stochastic process (particularly with application to finance), I came across ...
4
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1answer
66 views

Advantange of having a complete topology on test functions

Let's consider $\mathscr D(\Omega)$, the space of test functions on $\Omega \neq \emptyset \subseteq \mathbb R^n$ as usually defined. For the sake of clareness, $$\mathscr D(\Omega) = \cup_K ...
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1answer
32 views

In a proof of the representation of linear functionals of topological vector space

I'm reading the proof of the following theorem in a note on functional analysis: Here $p_F$ is defined as $p_F(x)=\max_{y\in F}|\langle x,y\rangle|$. Could anyone show me why the underscored ...
2
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1answer
23 views

Quotient topology of a topological vector space is translation-invariant

Let $(L,\tau)$ be a topological vector space over $\Bbb{C}$ and $M$ be a subspace of $L$ and let $$f:L\to L/M$$ be the canonical map of $L$ onto $L/M$.Let $ \hat \tau$ be the quotient topology on ...
1
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1answer
29 views

Basic neighborhoods in weak topology

I am trying to visualize the basic neighborhoods of the form $V(x_0;\varepsilon,f_1,...,f_n) = \bigcap_{j=1}^n \{ x \in E : |f_j(x-x_0)|<\varepsilon \}$ where $x_0 \in E$, $\varepsilon>0$ and ...
0
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0answers
15 views

cosine similarity among multiple sets

my background is not mathematics, so I will try to do my best to describe. I have multiple sets of elements A0 = {... }, A1 ={..}, .., An ={} I want to: 1. compute a similarity (here cosine for ...
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1answer
18 views

Cardinal functions on TVS

Its a well know fact that, in the class of metric spaces, the cardinal invariants density and weight agree. I would like to know if there is a example of a topological vector space $X$ for whose ...
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3answers
76 views

The existence of a limit point of a closed set

Walter Rudin define a closed set as: 2.18 (d) A subset E of a metric space is closed if every limit point of E is a point of E. I don't see in this definition nothing about the existence of a ...
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1answer
37 views

equivalent definitions of weak topology on a topological vector space

Let $(X,\tau)$ be a topological vector space over $\Bbb{R}$ and $X^*$ be the topological dual of $X$. There are two ways to define the weak topology $\tau_w$ on $X$: $\tau_w$ is the initial topology ...
5
votes
2answers
92 views

Is every convergent sequence in a topological vector space bounded?

It is known that in a metric space, every convergent sequence is bounded. In a topological vector space (TVS), one also has the notion of bounded sets. Here is my question: Is every convergent ...
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0answers
46 views

Counterexamples of topological vector space

How can I prove that an vector space with infinitely many elements endowed with the cofinite topology is not topological vector space? I have an indirect proof: Such an infinite space with the ...
2
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1answer
35 views

continuity of linear operators between topological vector spaces

Let $X$ and $Y$ be topological vector spaces and $T:X\to Y$ a linear operator. If $T$ is continuous at $0\in X$ then $T$ is continuous. Suppose $x_0\in X$ and let $U$ be a nbhd of $T(x_0)$ in ...
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0answers
79 views

Are Lusin and Souslin spaces sequential or even Fréchet-Urysohn?

First some definitions: A Polish space is a separable and completely metrizable topological space. A Hausdorff space is Lusin if it is the image of a Polish space under a bijective continuous map. A ...
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0answers
47 views

If $A$ is a balanced and convex set, then $\lambda A + \mu A = (|\lambda| + |\mu|)A$?

I have proven that $\lambda A + \mu A \subset (|\lambda| + |\mu|)A$. How do I prove the reverse inclusion?
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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 ...
0
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1answer
38 views

How do I prove that if $A$ is a balanced subset of a vector space $E$ then $\lambda A = |\lambda| A$?

I have to somehow use $\left|\frac{\lambda}{|\lambda|}\right|=1=\left|\frac{|\lambda|}{\lambda}\right|$, I don't know how, however.
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3answers
47 views

Is every function from a subset of $[0,\infty)$, which is closed under addition and contains $0$, into a Polish space continuous?

Let $I\subseteq[0,\infty)$ be closed under addition with $0\in I$ and $(E,d)$ be a Polish space. Is every function $f:I\to E$ continuous? The answer seems to be "yes", if $I\subseteq\mathbb N_0$, ...
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0answers
27 views

Is it possible to compute the volume of a cone on a inner product space?

This is a matter of curiosity for me. Volumes are often compute using triple integration. But is it possible to compute volumes on a vector space with an inner product defined on that vector space?
4
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1answer
79 views

characterization of topological vector spaces

It is known that if $X$ is a topological vector space (TVS), then all the translations and nontrivial scalar multiplications are homeomorphisms. I'm curious about the following question about which ...
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1answer
82 views

Linear span of general set in topological linear spaces

I'm studying Functional Analysis and I'm in doubt with the definition of linear span. The book states that: Let $\mathscr{L}$ be a topological linear space and let $\mathscr{M}$ be a linear ...
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0answers
35 views

“Algebraic isomorphism $\implies$ Homeomorphism” in the topological vector space context

This question just pop up when I was trying to solve another problem. Let $X,Y$ be vector topological spaces over the field $\mathbb{K}$ ( with $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ ) and let ...
0
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1answer
44 views

How does the definition of compactness imply that all continuous operators are compact in finite dimensional spaces?

Let $S \subset X, Y$ be normed spaces over $K$. An operator $A:S \to Y$ is called compact if: $A$ is continuous $A$ transforms bounded set into relatively compact sets i.e. if $(c_n)$ ...
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1answer
54 views

Is $C^{\infty}(\mathbb{T})$ dense in $C(\mathbb{T})$?

For a topological space $X$, the space of smooth functions with compact support (denoted by $C^{\infty}_0(X)$) is dense in the space of continuous functions vanishing at infinity (denoted ...
2
votes
1answer
54 views

How strong is the operator norm topology?

Let $(V,\tau_V), (W,\tau_W)$ be normable topological vector spaces. Let $||\cdot||_V, ||\cdot||_W$ be norms on $V,W$ inducing $\tau_V, \tau_W$ respectively. Let $||\cdot||_{op}$ be the operator norm ...
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0answers
34 views

Norms on $L(V,W)$

Let $V,W$ be normable topological vector spaces over $\mathbb{F}$. Let $C(V,W)$ be the set of continuous linear transformations $T:V\rightarrow W$. Let $||\cdot||_V, ||\cdot||_W$ be norms on $V,W$ ...
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1answer
27 views

Does every Hamel basis for an infinite dimensional topological vector space have a maximal countable spanning subset?

‎‎‎‎Let ‎$ X $ ‎be an infinite dimensional topological vector space and ‎$\{ e‎_{i}: i ‎\in I ‎\}$‎ be a Hamel basis for $X$. Does there exist a maximal countable subset ‎$J‎\subset ‎I$ ‎such ...
5
votes
1answer
68 views

Sequence of topological spaces

A friend of mine did an exercise where a part of the text was: In $\mathbb{R}^3$, with euclidian topology, we consider $X=\mathbb{S}^2 \setminus \{ N \}$, where $N= (0,0,1)$ and $E=\{(x,y,z) \in ...
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0answers
33 views

The restriction of a discontinuous linear functional to any open set is surjective.

Problem. Let $X$ be a topological vector space and $f:X\to\mathbb{K}$ a linear mapping. Prove that if $f$ is discontinuous, then $f(A)=\mathbb{K}$ for all nonempty open set $A\subset X$. I'd like ...
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2answers
68 views

Question about vector spaces with the discrete topology

Is it true that every vector space with the discrete topology is a topological vector space? (That is, a topological space with continuous addition and scalar multiplication whose singletons are ...
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2answers
23 views

Is multiplication by scalars a homeomorphism?

Let V be a complex topological vector space. For any nonzero v in V, consider the map x -> x * v from C to V. This map is continuous, but is it a homeomorphism onto its image? Thank you.
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1answer
113 views

Example of Topological Vector Space

Is there a topological vector space such that, for every $x\in X$, there is a proper neighbourhood $V$ of $x$ in $X$ which is convex, but the whole space is not locally convex (i.e. $X$ has a local ...
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0answers
143 views

Hilbert Cube and Metric Space

Given that $d(x,y)=\sum_{n=1}^{\infty}2^{-n}|x_{n}-y_{n}|$ defines a metric on $H^{\infty}$ where $H^{\infty}$ is the Hilbert Cube, a collection of all real sequence $x=(x_{n})$ with $|x_{n}|\leq 1$ ...
2
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1answer
57 views

Topology question with closed sets.

Let $ K\subseteq \mathbb{R}^n$ be a compact set and let $E\subseteq \mathbb{R}^n$ be a closed set. ***Its also given that $ \inf \{d(x,y)|x\in K, y\in E\}=0$. $ d(x,y)=\sqrt{\sum_j (x_j-y_j)^2}$ ...
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0answers
59 views

A consequence of “every norm is equivalent to the sup-norm” in a finite dimensional normed vector space

So, I am reading the following proposition in Neukirch's Algebraic Number Theory: Proposition. Let $K$ be a complete field with respect to the valuation $|\:\: |$ and let V be an $n$-dimensional ...
2
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1answer
33 views

Is the product of two complete TVS a complete TVS?

I'm reading about Topological Vector Spaces (using Treves's book) and I failed to answer the question on title. I'll be more precise: 1) We say that a TVS $E$ is complete if every Cauchy filter in ...
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0answers
23 views

Topological vector space with separating dual space

Let ‎$ X $ ‎‎‎be a topological vector space with dual ‎$ X‎^{‎\ast‎}‎ $. Has ‎$ X $ Hahn-Banach Extension Property, ‎i‎f ‎$ X‎^{‎\ast‎} $‎ ‎‎ separates points of ‎$ X $ ? (Hahn-Banach Extension ...
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1answer
42 views

Locally bounded topological vector spaces

Let ‎$ X $ ‎‎‎be a topological vector space ‎such that every neighborhood of zero contains an infinite dimensional subspace. Then ‎$ X $ is not locally bounded. I do not know, why? We know ‎$ X $ is ...