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Questions tagged [topological-vector-spaces]

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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Continuity of representation of topological group

First, We set notations as follows. $G$ : topological group , $k$ : field , $V$ : linear topological space over $k$ , $\mathrm{Map}(V,V)$ : Set of all continuous maps from $V$ to $V$ $\mathrm{Aut}_k (...
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A linear topological space over real number field is locally compact ???

Let $V$ be a topological $\mathbb{R}$-vector space with ${\rm dim}(V) < \infty$ Then, $V$ is locally compact $??$
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1answer
50 views

About the locally convex topology

I know that if a locally convex space Hausdorff $(X,S)$ is first numerable then for the $\hat{0}\in X$ exists a countable local base $\{V_n, n \in \mathbb{N}\}$ and to each $V_n$ corresponds a ...
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Properties of topological vector spaces

I'd like to understand better the significance of certain properties of topological vector spaces. It would be great if someone could give me an intuitive picture for what makes them "special", and/or ...
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Cover set of open ball topology

Let $Q_k(x_k, r_k) $ be a cube of center $x_k$ and side length $\frac{r_k} {2} $ suppose that $1<a<a*$ such that $Q_k^{*} =a^{*}Q_k$ so $Q_k^{*}=Q_k(x_k, a^{*}r_k) $ Show that if $$O=\...
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1answer
24 views

What is known on the topology of everywhere converging Taylor series? [on hold]

Let $M$ be the subset of $\mathbb{R}^\infty$ with the property that $(a_0,a_1\cdots) \in M$ iff the Taylor series $$\sum a_ix^i$$ converges everywhere (for $x \in \mathbb{R}$). What is known about ...
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1answer
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Density of test functions in the space of distributions — a clarification

Let $U \subseteq \mathbb{R}^n$ be open and denote by $\mathcal{D}(U)$ the space of all compactly supported smooth functions $U \to \mathbb{R}$. Let $\mathcal{D}^\prime(U)$ be the space of all ...
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1answer
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A subspace of a dual Banach space $X^*$ is norming if and only if its weak$^*$ closure contains a multiple of the unit ball of $X^*$

Definition A subspace $Z$ of a dual Banach space $X^*$ is said to be norming if there exists $c>0$ such that $$\sup_{f\in B_Z}|f(x)|\geq c||x||$$ for every $x\in X$ (where $B_Z$ is $B_{X^*}\cap ...
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1answer
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Weak-* continuous functionals with bounded level sets

Let $X$ be an infinite-dimensional Banach space, $X^*$ its topological dual and $f:X^*\to\mathbb{R}$ some weak-* continuous functional (not necessarily linear). Is it possible for $f^{-1}(a)$ to ...
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1answer
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Rudin functional analysis theorem 4.13, (a) and (b)

Let $U$ and $V$ be the open unit balls in the Banach spaces $X$ and $Y$, respectively. If $T \in \mathcal{B}(X,Y)$ and $\delta > 0$, the the implications $$ (a)\to(b)\to(c)\to(d) $$ hold among ...
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Rudin Functional analysis, theorem 4.12 corollary (b)

Suppose $X$ and $Y$ are Banach spaces, and $T \in \mathcal{B}(X,Y)$ Then $$ \mathcal{N}(T^*) = \mathcal{R}(T)^{\perp} \;\;\text{and}\;\; \mathcal{N}(T) = ^{\perp}\mathcal{R}(T^*) $$ The corollary ...
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2answers
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How is Schwartz space different from Hilbert space?

I know that Schwartz space can be considered a dense subset of the Hilbert space isomorphic to $\ell^2$. What I wish to understand is, how really different Schwartz space is from the Hilbert space. ...
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1answer
28 views

Rudin functional analysis, theorem 4.9 part (b)

Let $M$ a closed subspace of a Banach space $X$. b) Let $\pi:X \to X/M$ be the quotient map. Put $Y = X/M$, For each $y^* \in Y^*$, define $$ \tau y^* = y^* \pi $$ Then $\tau$ is an isometric ...
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$+:X\times X\to X,(x,y)\mapsto +(x,y)=x+y$ and $\cdot:\Bbb{R}\times X\to X,(\lambda,x)\mapsto \cdot(\lambda,y)=\lambda\cdot x$ are weakly continuous

$$+:X\times X\to X,\\(x,y)\mapsto +(x,y)=x+y$$ and $$\cdot:\Bbb{R}\times X\to X,\\(x,y)\mapsto \cdot(\lambda,y)=\lambda\cdot x$$ are weakly continuous, where $X$ is an infinite dimensional normed ...
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1answer
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This convex hull is balanced?

Let be (X,S) a locally convex space, and $B \subset X$ a nonempty sequentially closed bounded and convex set such that $\hat{0} \notin clB$,(the closure of B). Define the set T:=s-clco$\{B \cup-B \}$ ...
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1answer
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Prove that $\|f\|_{X^*} = \sup_{\|x\| \leq 1} |f(x)|.$

Let $X$ be a normed vector space. Let $f: X \rightarrow \mathbb{R}$ be a bounded linear functional. We define: $$\|f\|_{X^*} = \sup_{x \in X \\ x \neq 0} \dfrac{|f(x)|}{\|x\|_X}$$ Prove that: $$\|f\|...
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1answer
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Rudin's functional analysis, theorem 4.1.(completeness of $\mathcal{B}(X,Y)$)

Suppose $X$ and $Y$ are normed space. Associate to each $\Lambda \in \mathcal{B}(X,Y)$ the number $$\lVert \Lambda \rVert = \sup \left\{ \lVert \Lambda x \rVert : x \in X, \lVert x \rVert \leq 1 \...
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1answer
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Completeness condition for Fréchet space

In Wikipedia's definition of Fréchet space, it is stated that a Fréchet space is a topological vector space that satisfies the following: It is locally convex Its topology can be induced by a ...
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why a subspace is closed?

Let $E$ be $K-$vector space with a norm $\|\cdot\|$, and $F$ a subspace with dimension $n$. Show that $F$ is a closed set . I am trying to show that any convergent sequence of elements $(x_n)_{n \in ...
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1answer
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Rudin functional analysis theorem 3.28, $P$ is weak*-compact in $C(Q)^*$.

Follow up question to this one. It is later proved that defining $\phi : C(Q)^* \to X$ as $$ \phi(\mu) = \int_Q x d \mu (x) $$ if $P$ is the set of all regular probability measures on $Q$ we have $ ...
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1answer
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Rudin functional analysis theorem 3.28, application of Reisz representation theorem.

Suppose (a) $X$ is a topological vector space on which $X^*$ separates points, (b) $Q$ is a compact subset of $X$, and (c) the closed convex hull $\overline{H}$ of $Q$ is compact ...
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Is there a characterization of smooth $G$-equivariant functions $\mathbb R^n \to \mathbb R^m$?

If $G$ is a compact Lie group acting smoothly on $\mathbb R^n$ by orthogonal matrices, a result of G. Schwartz characteriezs the smooth $G$-invariant functions $f : \mathbb R^n \to \mathbb R$ as ...
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Rudin functional analysis, theorem 3.27

Few bits a bit unclear of a) $X$ is a topological vector space on which $X^*$ separates points, and b) $\mu$ is a Borel probability measure on a compact Hausdorff space $Q$. If $f : Q \to X$ ...
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Rudin functional analysis, Definition 3.26

Straight to the point Suppose $\mu$ is a measure on a measure space $Q$,$X$ is a topological vector space on which $X^*$ separates points, and $f$ is a function from $Q$ into $X$ such that the ...
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1answer
23 views

Closed sets with filters

I am studying topological vector spaces, and, being at the begin of the textbook (Sevres) I frequently encounter rephrasing and generalisations of known results about metric spaces in terms of filters....
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1answer
34 views

Rudin Functional Analysis Theorem 3.25

This is probably the first question regarding the theorem in the title... If $K$ is a compact set in a locally convex space $X$, and if $\bar{co}(K)$ is also compact, then every extreme point of $\...
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Hahn-Banach in dual space

Let $X$ be a Banach space. Let us take a subspace $V \subset X^*$ of the topological dual space and a linear functional $\varphi \colon V \to \mathbb R$. We assume that $\varphi$ is continuous w.r.t. ...
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If the dual of a topological vector space separates points, does it separate a point and a closed subspace?

The Hahn-Banach Theorem implies that if $X$ is a normed vector space, then the dual space $X^*$, consisting of continuous linear functionals on $X$, has the following two properties: $X^*$ separates ...
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Does the dual space of a topological vector space always separate points? [duplicate]

If $X$ is a normed vector space, then the dual space $X^*$, consisting of continuous linear functionals on $X$, separates points. What that means is that if $x_1,x_2\in X$, then there exists an $f\in ...
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1answer
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Is every completely regular topology induced by some topological vector space?

Every topological vector space is completely regular. My question is, is the converse true? That is, is every completely regular topology induced by some topological vector space? If not, does ...
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In TVS, is it true that every neighbourhood of zero is sum of some two other neighbourhoods of zero

Let $V$ be a neighbourhood of zero. Is it true that there always exists neighbourhoods of zero A and B such that $V=A+B$? If this is true, then could it be generalized to neighbourhood of any point, ...
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Sufficient conditions for minimum to exist in non-compact subset of normed vector space

Let $X$ be a non-compact subset of a normed vector space. Let $f : X \to \mathbb{R}$ be a differentiable convex function that is bounded from below. Given that $X$ is not compact: 1) Is there a ...
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1answer
38 views

Is the addition and scalar multiplication continuous with respect to this particular topology?

Suppose $X$ is a normed linear space. If $\mathcal {B}$ consists of $\phi$, $X$, all open balls centered at origin, and all open annulus centered at origin, then it is clear that $\mathcal {B}$ is a ...
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1answer
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Rudin functional analysis, theorem 3.23. Use of separation property.

Still following the theorem 3.23 (Krein-Milman), follow up to this question. The proof again continues It now follows from (b) that every $\Lambda \in X^*$ is constant on $M$, since $X^*$ ...
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1answer
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Rudin functional analysis, theorem 3.23. Finite intersection property

This is a follow up question to this one. Continuing through the proof there's a subtlety I don't get, but I think it would be educational to understand to improve my background in general topology. ...
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1answer
43 views

Rudin's functional analysis, Theorem 3.23 point (b)

Reading through the proof of the Krein-Milman theorem Suppose $X$ is a topological vector space on which $X^*$ separates points. If $K$ is a nonempty compact convex set in $X$, then $K$ is the ...
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1answer
28 views

Two notions of boundedness in metrizable topological vector space.

In a metrizable topological vector space $X $ with the metric $d $, a subset A is said to be bounded if it can be absorbed by any neighbourhood of $0$ and a subset A is said to be d-bounded if its ...
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1answer
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Relation Between Nbhd Base at $e$ and the Uniform Structure on a Topological Group

We have the following theorem (from Husain's Introduction to Topologcal Groups), slightly rephrased: The following are true (apologies for the sloppy formatting): (1) in any topological group $G$, ...
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0answers
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How to define convergence in probability in topological spaces?

I want to expand on the question already posed here: Does convergence in probability w.r.t. a topology make sense? . Suppose we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a ...
3
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1answer
43 views

Example of a metric.

I am looking for an example of a metric $d(x,y) $ on a vector space $X $ such that is neither a discrete metric nor induced by a norm and satisfies: $$d(0,ax+(1-a)y)\leq {ad(0,x)}+(1-a)d(0,y),\ \...
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1answer
25 views

Rudin's functional analysis theorem 3.21.

Small proof of Suppose $X$ is a topological vector space on which $X^*$ separates points. Suppose $A$ and $B$ are disjoint, nonempty, compact, convex sets in $X$. Then there exits $\Lambda \in X^*$ ...
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1answer
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Theorem 3.20 rudin's functional analysis, compactness of $K = f(S \times A)$

Reading through theorem 3.20, Rudin's functional analysis (point (a)). If $A_1,\ldots, A_n$ are compact convex sets in a topological vector space $X$ then $co(A_1 \cup \ldots \cup A_n)$ is compact. ...
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1answer
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Every weak neighborhood of $0$ contains a neighborhood of the form $V = \left\{ x : |\Lambda_i x | < r_i \;\text{for} \; 1 \leq i \leq n\right\}$

As usual I struggle with the weak topologies... There's a probably silly detail I'm missing in "Rudin's Functional analysis" section 3.11. "The weak topology of a topological vector space". At some ...
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1answer
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Is this estimate true for functionals on Frechet spaces?

In class the other day my professor made the following claim about the Schwartz class $\mathcal S$: Let $u: \mathcal S \to \mathbb C$ be linear. $u$ is continuous iff there exist $C,N>0$ such ...
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1answer
29 views

$d(0,\alpha x+\beta y) \leq |\alpha|d(0,x)+|\beta|d(0,y);\ |\alpha|+|\beta|\leq 1$

Let $X$ be a metrizable topological vector space with the metric $d$ and $\alpha, \beta$ be scalars (complex or real). Such a metric $d$ has the following properties: $$d(x+z,y+z)=d(x,y),\ d(0,\alpha ...
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1answer
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In a complex metrizable topological vector space, $d(0,\alpha x)\neq |\alpha|d(0,x), \ \alpha \in \mathbb C.$

Let $(X,\tau)$ be a complex metrizable topological vector space with the metric $d$. Does the following hold: $$d(0,\alpha x)=d(0,x),\ \forall \alpha \in \mathbb C, |\alpha|=1 \ ?$$ In general, the ...
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1answer
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Examples of continuous functions on $M_n(\mathbb R)$ [closed]

What are some examples of continuous functions on $M_n(\mathbb R)$ and its subspaces (equipped with the usual topology), which are useful in proving topological properties of these spaces?
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1answer
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Open dense subset of $\mathbb{R}$.

I found this question in one of the posts Let $A\subseteq\mathbb{R}$ be open and dense. Show that $$\mathbb{R}=\{x+y:x,y\in A\}$$ The poster stated that it is not too hard to prove but I can't ...
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Rudin's functional analysis Theorem 3.18, second part.

Just a follow up to the following two questions: Rudin's functional analysis 3.18, every originally bounded subset of a locally convex space is weakly bounded. Theorem 3.18, Rudin's ...
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1answer
30 views

Rudin's functional analysis 3.18, every originally bounded subset of a locally convex space is weakly bounded.

Reading through the proof of the following In a locally convex space $X$, every weakly bounded set is originally bounded, and viceversa The trivial part of the proof. Since every weak ...