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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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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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Examples of continuous functions on $M_n(\mathbb R)$ [on hold]

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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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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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 ...
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Continuity implies boundness in Frechet Spaces?

Consider two Frechet spaces $(X,d_X),\;(Y,d_Y)$. Let $T:X \rightarrow Y$ be a continues linear operator. Is it true that $T$ is bounded(or Lipschitz) in the following sense: $$\exists \alpha > 0 \; ...
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Theorem 3.18, Rudin's functional analysis

Just a quick question about the the following theorem In a locally convex space $X$, every weakly bounded set is originally bounded and viceversa. Proof: Since every weak neighborhood of $0$ ...
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Banach-Alaoglu theorem, Rudin's functional analysis.

Few questions about the theorem If $V$ is a neighborhood of $0$ in a topological vector space $X$ and if $$ K = \left\{\lambda \in X^* : |\Lambda x | \leq 1 \; \text{for every} \; x \in V \right\}...
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Rudin's functional analysis theorem 3.12

Suppose $E$ is a convex subset of a locally convex space $X$. Then the weak closure $\overline{E}_w$ of $E$ is equal to its original closure $\overline{E}$. The proof starts as follows $\overline{...
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Rudin's functional analysis theorem 3.10, proof that multiplication is continuous

Suppose $X$ is a vector space and $X'$ is a separating vector space of linear functionals on $X$. Then the $X'$-topology $\tau'$ makes $X$ into a locally convex space whose dual space is $X'$. ...
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$\bigcup_{\alpha\in C}\alpha A$ is compact in the topological vector space $X$

Let $X$ be a topological vector space, with $A \subset X$ and $C \subset \mathbb{K}$ both compact subsets. Why is $\bigcup_{\alpha\in C}\alpha A$ compact in $X$?
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Compactness of a specific set in weak topology

I have the following question: Let $E$ be a polish space (that is, a topological space, which is separable and metrizable, such that $E$ would be complete if equipped with this metric). Consider the ...
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Cardinality of dense set related to cardinality of discrete set

Let $X$ be a Banach space. Let $\eta$ be the density character of $X$ (the least possible cardinality of a dense set). Does there exist $A\subset X$, which is closed and such that $\text{card} A=\eta$ ...
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Does the Polygonal Confinement Theorem hold on the set of entire functions?

The Polygonal Confinement Theorem can be found in Section 2 of this paper by Rosenthal. I am interested in a generalization of Lemma 3.1 in the paper, which states: $\textbf{Lemma 3.1:} $ If $v_1,\...
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1answer
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Do algebraically open sets define a vector space topology?

Let $X$ be a vector space. A vector space topology on $X$ is a topology such that addition and scalar multiplication are continuous. A subset $A$ of $X$ is said to be algebraically open if, for all $a\...
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Why are weak topologies useful in functional analysis?

I've been reading through chapter 3, Rudin's Functional Analysis, and an important point is the one of weak topology. From the theorems it seems to me weak topologies are somehow the result of ...
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When is a topological vector space “inner product-able”?

This is a follow-up to my question here. A topological vector space is normable, i.e. its topology is induced by some norm on the vector space, if and only if it is Hausdorff and the $0$ vector has a ...
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Is every normable topological vector space “inner productable”?

Not every norm on a vector space is induced by an inner product on that vector space. But suppose hat $X$ is a topological vector space that is normable, i.e. its topology is induced by some norm on ...
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$\ell^2$ as colimit in $\mathbf{TopVect}_{\mathbb{R}}$

Let $\mathbf{TopVect}_{\mathbb{R}}$ be the category of topological vector spaces with continuous linear maps as morphisms. Is it ineed true that $\ell^2 \cong \varinjlim_{n}\oplus_{i=1}^n\mathbb{R}$?
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Locally Convex tvs closure of $\{0\}$

Let $E$ be a topological vector space locally convex, defined by the family of seminorms $\mathcal{F}=(p_j)_{j\in J}$. I can't prove that $\underset{j\in J}\bigcap Ker(p_j)=\overline{\{0\}}$
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Balanced set in $\mathbb{K}$

Let $A$ be a balanced set in $\mathbb{K}$. Why do we have : if $A$ is not bounded then $A=\mathbb{K}$ ?
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Prove the mapping $(x,y)\mapsto x+y$ from $X \times X \to X$ is continuous when $X$ is given a weak topology.

Let X be a Banach space. Prove the mapping $(x,y)\mapsto x+y$ from $X \times X \to X$ is continuous when $X$ is given a weak topology. Could anybody give me some hints to start?
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Lemma 3.9 Rudin's functional analysis

I need help in understanding the proof of the following Suppose $\Lambda_1,\ldots,\Lambda_n$ and $\Lambda$ are linear functionals on a vector space $X$. Let $$ N = \left\{x : \Lambda_1x = \...
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1answer
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Rudin functional analysis, section 3.8 (Topological preliminaries)

A quote from section 3.8. of Rudin's functional analysis: Suppose next that $X$ is a set and $\mathcal{F}$ is a nonempty family of mappings $f: X \to Y_f$, where each $Y_f$ is a topological space (...
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Function spaces from a geometrical viewpoint.

I'm wondering whether there exists some geometrical theories of functional spaces. I mean; function spaces ($L^p$ spaces for example) are called topological vector spaces (TVS). I'm interested in ...
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Predual of Morrey space

The domain is $\mathbb{R}^n$ I have a question about the $H^{p,\lambda}$, the dual of Morrey space $L^{p',\lambda}$. Let $C_0$ is the space of continuous functions have the compact support. The ...
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Generated space by the harmonic oscillators $e^{iw}$.

Let $\mathbb{T}$ represent the $[-\pi,\pi)$ interval and for each $w\in\mathbb{Z}$ denote the function $$e_w:\mathbb{T}\to\mathbb{C}$$ such that $$\forall x\in\mathbb{T}, \ e_w(x)=e^{2\pi iwx}.$$ It ...
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1answer
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$C^{\infty}(\mathbb{R})$ as a Fréchet space

I think I'm misunderstanding what is being asked of me in the following question: The space $C^{\infty}(\mathbb{R})$ has a Fréchet space topology with respect to which $f_n \rightarrow f$ iff $...
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1answer
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What's the definition of proper subspace of a vector space used in Rudin's Functional analysis

I'm reading through the Rudin's functional analysis, and theorem 3.5 use the term "Proper Subspace", there's a theorem in chapter 2 that uses the same terminology. I'm reading through chapter 1 again,...
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distribution associated with a discontinuous function

Let $f\colon\mathbb{R}\to\mathbb{R}$ be such that, for every $n\in\mathbb{Z}$, $f$ is differentiable on $\left(n,n+1\right)$ and $n$ is a discontinuity of first kind of $f$. We define $$T_f(\phi)=\...
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Topological vector space completion with respect to a symplectic form?

Suppose we have an infinite-dimensional vector space $V$ with a symplectic form $\omega:V\times V\to\mathbb R$. It can be given a weak topology that makes $\omega$ continuous. Does it make sense to ...
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Convergence in Bochner spaces

I would like to know if this statement is true. If so would you like to know how to prove it or some counter-example in the negative case. Let $I$ be a interval in $\mathbb{R}$ (not necessarily ...
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Rudin functional analysis, theorem 2.11 (Open mapping theorem)

Going through the proof of such theorem, there are few bits I don't understand. I'll write down the proof and in in the middle I'll add some comments. Statement: Suppose (a) X is an F-space, ...
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Are compact sets bounded in a topological vector space?

If $X$ is a topological vector space and $K \subset X$ a compact set, can we say that $K$ is also bounded? By Rudin's functional analysis any topological vector space is Haussdorf, I read that in any ...
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Easy way to understand the definition of orbits, Rudin functional analysis.

In the theorem of Banach-Steinhaus the following defintion of orbits is given. Assuming $X,Y$ are topological vector spaces and $\Gamma$ is a collection of linear mappings from $X$ to $Y$ we define ...
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Does we have separation theorem for closed subsets in a topological vector space?

In Rudin's Functional Analysis, page $10,$ he stated the following separation theorem for topological vector space. Theorem $1.10:$ Suppose $K$ and $C$ are subsets of a topological vector space $X,$...
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Neighborhood of zero on weak topology

I have a homework, I have tried so much on this question, but I couldn't solve it. How can I prove it? Can you help me? Thank you. $E$ is a normed space and $E'$ is the topological dual of $E$. If $...
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Is there a $\gamma>1$ such that $\frac{\binom{4p}{2p-1}}{\binom{4p}{0}+\binom{4p}{1}+\cdots+\binom{4p}{p-1}}>\gamma^{p}$

In a proof of the Larman-Rogers conjecture (there is $\gamma>1$ such that $\chi(\mathbb{R}^{d})>\gamma^d) $ they used that there is a $\gamma>1$ such that $\frac{\binom{4p}{2p-1}}{\binom{4p}{...
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The Proof of Absolutely Convex Hull?

A set is absolutely convex if and only if it is convex and balanced. Can you prove that ''A set ${\displaystyle C}$ is absolutely convex if and only if for any points $x_1,x_2\in C$ and any numbers $\...
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Theorem 2.9 rudin functional analysis (application of Baire's theorem)

I'm reading through the theorem 2.9 (Rudin functional analysis) which states Suppose $X$ and $Y$ are topological vector spaces, $K$ is a compact convex set in $X$, $\Gamma$ is a collection of ...
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1answer
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Existence of an injective continuous linear form. On a normed vector space $E$ [closed]

Let $(E,N)$ Be a normed vector space. The dimension of $E$ could be infinite. Does there exist a linear form $f$ continuous: $$\exists C>0 \quad \text{such that}\quad |f(x)|\leq C\lVert x\rVert\...
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Completion of topological vector spaces

It is known that if you have a Hausdorff topological vector space $X$, then it embeds bi-continuously in a complete topological vector space. I was wondering if someone knew an example of a non-...
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distribution-valued integral in Grafakos

In the book Modern Fourier Analysis by Grafakos in the proof of the T(1) Theorem in the implication that shows $L^2$ boundedness (in the 3rd edition this is on page 243), Grafakos considers a ...
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Definition of strictly convex topological vector space.

Is there a definition of strictly convex topological vector space? I searched on the internet and in some books also. But I did not find any term like this. Does this definition exist? If we try to ...
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How to get density of a region (subspace) in a vector space?

I have a simple problem, which I think must have an easy solution. I have a vector space say with a 1000 dimensions for each vector. Now, I have a large number of sample vectors from this vector ...
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About the equivalence of two definitions of topological linking

If $E$ is a Banach space, denote with $I$ the identity map of $E$, denote the space of continuous functions from $E$ to $E$ with $\operatorname{C}(E,E)$ and denote the space of homeomorphism from $E$ ...
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Is there a dense convex cone which is not a subspace?

In a finite-dimensional space any dense convex subset is the whole space, and (by the Stone-Weierstrass theorem) there are many examples of dense convex cones which are subspaces in infinite-...
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Rudin functional analysis theorem 2.5.

I'm reading through the Banach-Steinhaus, there are few bits I don't get. The proof starts as Pick balanced neighborhoods $W,U$ of $0$ in $Y$ such that $\bar{U} + \bar{U} \subset W$. I can pick ...
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Local minima are global for convex functions

Let $f:\mathbb R^n\to \mathbb R$ be a convex function. Show that any local minimum point is also a global minimum point. I've seen a proof by contradiction here(end of pp.2), but I'd like to prove ...
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First countable dual space of a von Neumann Algebra.

Let A be a C* algebra on a Hilbert space H and R=A'', the respective von Neumann algebra. If A is separable, the set of its states is a first countable topological space in the weak* topology. Does ...