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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58 views

In the Krein-Milman theorem, can the weak closure of the convex hull be replaced by norm-closure?

I have a question on the following formulation of the Krein-Milman theorem: Consider a vector space $X$ equipped with the weak topology induced by a separating space $X^*$ of functionals on $X$. ...
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2answers
54 views

from finite to $\sigma$-finite measure space [duplicate]

This might be rather elementary. I have put it at MSE for a while without getting any answers. Here is the question: In the proof of the following theorem, would anyone explain how the general case (...
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1answer
45 views

Vector space of all Lebesgue measurable functions

Let $L_0$ be the vector space of all Lebesgue measurable functions on $[0,1]$ with metric $d(f,g)$ = $\int_{0}^{1} |f(t)-g(t)|/( 1+ |f(t)-g(t)| ) dt $ . Show that $d(f_n,f) \to 0$ iff $f_n \to f$ in ...
3
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1answer
328 views

How can this theorem about weakly measurable functions on $\sigma$-finite measure spaces be deduced from the finite measure space case?

I am reading a theorem about measurability of vector-valued functions in a note on functional analysis: Theorem 3.6.1. If $X$ is a separable, metrizable locally convex space, $(\Omega, \Sigma, \mu)...
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1answer
214 views

Continuous inclusions in locally convex spaces

Let $(X, \left \| \cdot \right \|_X )$, $(X, \left \| \cdot \right \|_Y)$ two normed vector spaces with $X \subset Y$, by definition we have $X \hookrightarrow Y$ if $\left \| x \right \|_Y \leq C \...
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1answer
47 views

what is a cofinal sequence?

I understand that the subset $\Phi'$ of $\Phi$ is cofinal by looking at Wikipedia https://en.wikipedia.org/wiki/Cofinal_(mathematics) Would anybody explain what the cofinal sequence $(Y_n)$ means?
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1answer
34 views

The relation between Closed Operators (the graph is closed) and Closed Mappings (images of closed sets are closed)

Let $X$, and $Y$ be topological vector spaces and let $D$ be a dense vector subspace of $X$. An operator $T:D\to Y$ is called closed iff the graph of $T$, $\{(x,T(x))\in X\times Y|\,x\in D\}\subseteq ...
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1answer
35 views

Why is the dual space with the weak*-topology a topological vector space?

In my lecture-notes on functional analysis I've found the fact that the dual-space $X^*$ with the weak*-topology of a real vector-space $X$ is a topological vector-space. I've tried to prove it, but ...
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95 views

Are uncountable “Schauder-like” bases studied/used?

We could define the following notion of basis in a way analogous to unconditional Schauder basis: If $X$ is a topological vector space over $\mathbb R$ and $B=\{b_i; i\in I\}$ be a subset of $X$. ...
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35 views

Show that any convex subset of a Banach space X is closed with respect to the norm if and only if it is closed in the weak topology

How can it be proved? In a way which could be understood by a undergraduate math student. Specially if and "only if".
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13 views

If $Z$ is Hausdorff and if $Z=M\oplus N$, then $M$ and $N$ are closed.

Let $Z$ be a topological vector space over a field $K$. If $Z$ is Hausdorff and if $Z=M\oplus N$, then $M$ and $N$ are closed. Defn: $X$ is said to be topological vector space if $(i)$ $X$ ...
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0answers
76 views

Differential operators acting on the Schwartz space

I was asking me the following question and cannot find any answer to it. Any help/suggestion is most than welcome! Let $D$ be a linear differential operator with polynomial coefficients on $\mathbb{R}^...
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1answer
37 views

Kadets-Pelczynki criterion to prove Eberlein Smulian Theorem.

I am studying the proof of the Eberlein-Smulian Theorem via basic sequences in the book "Topics in Banach Space Theory" from F.Albiac and N. Kalton. Although, I found myself stuck in the following ...
2
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1answer
19 views

Let $H$ be a real Hilbert space (an euclidian vector space over the real numbers field).

Let $H$ be a real Hilbert space (an euclidian vector space over the real numbers field). If $\{x_n\}$ weakly converges to $x$ and $\|x_n\|\rightarrow \|x\|$, show that $\|x_n-x\|\rightarrow 0$. My ...
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1answer
68 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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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 ...
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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
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2answers
104 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
50 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, ...
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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
58 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 the ...
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1answer
24 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
48 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
94 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 ...
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68 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 $L/...
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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 $...
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0answers
16 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
83 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
38 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 ...
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2answers
95 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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55 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 $Y$....
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91 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
48 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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53 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 topology....
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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
48 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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28 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?
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1answer
85 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
87 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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36 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 $f:X\...
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1answer
54 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 $(c_n)$ is ...
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1answer
62 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 $C_{\infty}(...