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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448 views

Density and closedness of $C[0,1]$ in $L^\infty[0,1]$ in norm and weak-* topologies

With results: "For convex subsets of a locally convex space, a, originally( strongly) closed equals weakly closed, and b, originally (strongly dense equals weakly dense." Could you help me solve this ...
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3answers
758 views

Return an array of evenly distributed points on a sphere give Radius and Origin. [duplicate]

Given a sphere of radius $r$, and origin $x,y,z$ what is the simplest way I can generate an evenly distributed array of points on the sphere $(x_1,y_1,z_1),(x_2,y_2,z_2),\cdots(x_n,y_n,z_n)$. Note I ...
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1answer
295 views

A surjective linear map into a finite dimensional space is open

I'm in search of different proofs of the following proposition: $\bf{Proposition}$: Suppose $X$ and $Y$ be topological vector spaces, $\text{dim }Y<\infty$, and $\Lambda:X\to Y$ is a surjective ...
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1answer
122 views

Examples of $T_0, T_1, T_3, T_4$ and Hausdorff spaces

What could be simple examples of $T_0$, $T_1$, $T_3$, $T_4$ and Hausdorff ($T_2$) topological spaces?
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249 views

Confused by proof in Rudin Functional Analysis, metrization of topological vector space with countable local base

I'm working through Rudin's Functional Analysis, and I am confused by a step in his proof for Theorem 1.24, which states that if X is a topological vector space with a countable local base, then there ...
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1answer
167 views

Barrelled space

A locally convex space is called Barrelled if each closed absorbing convex set is 0-neighborhood See. But i doubt that every absorbing set contains zero. Then is every LCV is barreled. I think, ...
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1answer
57 views

Single norm criterion

Let $E$ be a metrizable locally convex space whose topology is defined by an increasing sequence $\{p_n\}$ of seminorms. Show that the topology of $E$ can be defined by a single norm iff there ...
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75 views

Is a closed set in a TVS over $\mathbb{R}$ convex?

From Theory of Convex Structures by M. L. J. Van De Vel, on a set $X$, a topology and a convexity structure are said to be compatible, if the convexity structure is generated by the closed sets. The ...
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49 views

Two different opinions on whether a topological vector space is a uniform space

Van de Vel's Theory on Convexity Structures says a TVS is uniform iff it is locally convex: 3.10.1. Proposition. Let $X$ be a topological vector space, equipped with the standard convexity and ...
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162 views

Evaluation map is not continuous always.

Let $E$ be a not normable locally convex space, define $$F: E'\times E\to \mathbb R$$ $$(f,e)\to f(e)$$ I have to show that $F$ is not continuous when $E'\times E$ is given product topology. I was ...
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0answers
52 views

Determining Similarity of Unit Vectors

I'm seeking for an injective piecewise continuous function $f:\mathbb S^n\rightarrow[0,1]$ where $\mathbb S^N$ is the set of vectors with $L_2$ norm equals $1$. The piecewise continuity requirement ...
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1answer
96 views

local convexity of $L_p$ spaces

wiki says The spaces $L_p([0, 1])$ for $0 < p < 1$ are equipped with the F-norm they are not locally convex, since the only convex neighborhood of zero is the whole space Why is this so? ...
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2answers
104 views

Closure of opening of closure in $\mathbb R^2$

My question is somehow related to Closure of the interior of another closure However, I go a bit further. I have a closed set $X\subseteq \mathbb R^2$ and $Y:=\operatorname{cl}\operatorname{int} X$. ...
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1answer
155 views

Bounded and compact sets in a subspace of $\mathbb R^{\mathbb N}$

Let $$ X= \{u=(u_1, u_2, \ldots): u_n \ne 0 \text{ only for a finite number of terms}\}\subseteq\mathbb R^\mathbb N, $$ with the topology inherited from $\mathbb R^\mathbb N$ (the "pointwise ...
5
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1answer
91 views

Pseudonormable Product Spaces

I want to prove that a product $\prod_{i\in I}X_i$ of topological vector spaces is pseudonormable only if a finite number of the factor spaces are also pseudonormable and the rest have the trivial ...
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1answer
119 views

If every linear functional is continuous in $V$, is every linear functional continuous for $S\subseteq V$?

Suppose $V$ is a finite dimensional (real) topological vector space. The first lemma in these notes says that Every vector subspace of a tvs with the induced topology is a topological vector space ...
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2answers
201 views

Is there a dumbed down version of the open mapping theorem for finite dimensional real vector spaces?

I would like to understand why any surjective linear transformation between finite dimensional topological real vector spaces, each with the natural topology, is in fact an open map. Reading around, ...
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53 views

Linear Application that is open in a TVS

Let $T: E \to F$ be a linear map between topological vector spaces $E$, $F$. If for each nonempty open set $G$, the interior of $T(G)$ is non-empty, then, $T$ is open. Proof: $$\mathrm{Int}(T(G))= ...
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1answer
43 views

Linear Application in a TVS

Let $E$ a Topological Vector Spaces, $T: E \longrightarrow \mathbb{K}$, linear. If for some $x \in E$, $Tx \neq 0$. Then, $T$ is open. I think that it is sufficient to prove that $T(G) \subset ...
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0answers
37 views

Density problem

$U$ is any open set of $\mathbb{R}$. We known that $C_0^\infty(U)$ is dense in $C^k(U)$. But what about, say $C_0^\infty((0,1))$ in $C^k([0,1])$?
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2answers
345 views

Closure of interior and interior of closure in a topological vector space

If $Y$ is a subset of topological vector space $X$ and is compact and convex show that $\overline{Y^\circ} = \overline{Y}$ and $\overline{Y}^\circ = Y^\circ$. I tried this way but I am not sure: ...
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3answers
165 views

Are vector subspaces of $\mathbb{R}^n$ always closed?

Suppose $S$ is any proper vector subspace of $\mathbb{R}^n$. Is $S$ a closed set in the usual topology on $\mathbb{R}^n$? Geometrically, I think it is clear that $S$ must be closed in $\mathbb{R}^n$ ...
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1answer
104 views

Openness of linear mapping 2

I quote a previously asked question : Let $X$ be a topological vector space over the field $K$, where $K=\mathbb{R}$ or $K=\mathbb{C}$, and let $\mathbb\{f\colon X\rightarrow K^n\}$ ($n \in ...
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1answer
32 views

Continuity and openess in quotient space

The setting: $X$ and $Y$ are topological vector spaces. $N \subset X$ is a closed subspace. $T(N)=\{0\}$ $\pi : X \rightarrow X/N$ the quotient map. $S : X/N \rightarrow Y$ uniquely determined by ...
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1answer
71 views

Set boundary preserved by an infinite union

Suppose I have a subset $U\subset\mathbb R^2$ and a real number $r>1$ with the following properties: $U$ is compact; $U\subset rU$ (self-similarity); $0\in U$; there exists an open set $H\subset ...
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1answer
115 views

Triangle inequality of a metric on a quotient space of a topological vector space

In "Functional Analysis" by Rudin, a metric $\rho$ on the quotient space $X/N$ of a topological vector space $X$ and a closed subspace $N$ is defined as follows: For $x,y \in X$, $$ \rho ...
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1answer
522 views

Is any Banach space a dual space?

Let $X$ be a Banach space. Is there always a normed vector space $Y$ such that $X$ and $Y^*$ are isometric or isomorphic as topological vector spaces (that is, there exists a linear homeomorphism ...
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2answers
76 views

Non-barreled topology compatible with the duality

Given $(X,s)$ a (real) barreled locally convex space (that is, every closed convex and absorbing set in $(X,s)$ is a neighborhood of the origin), is there a (strictly) finer, non-barreled linear ...
3
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0answers
103 views

Which (endo)functors of the category of finite-dimensional real vector spaces induce continuous maps between Hom-sets?

Let $\operatorname{Vect-fin}$ be a category of finite-dimensional vector spaces over $\mathbb{R}$. In this category Hom-sets $\operatorname{Hom}(V,W)$ are themselves finite-dimensional vector spaces ...
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1answer
228 views

Locally Convex Space Via Seminorms

Suppose that we have a Hausdorff locally convex space with its topology $\tau$ and let $P(X)$ be a separating family of $\tau$-continuous semi-norms so that $\tau$ is generated by $P(X)$. How do we ...
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127 views

Curiosities about the content of a rare book: Topological Vector Spaces by A. Grothendieck

The book is a celebrated and highly influential book by A. Grothendeck, which was published in 1954, in French and for various reasons, it has been out of print since 1973. I am very much interested ...
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2answers
264 views

Equation of a line on a plane…

Hi this question belongs to camera projections but i cannot understand the mathematics... i am not getting how the cross product of two vectors (underlined in red) gives the equation of a ...
3
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0answers
102 views

Density of operators

I am interested in operators on non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{**})$ by ...
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0answers
50 views

functions from the sphere

Can we assign in a continuous manner to each point of the sphere $S^2$ a two point subset of S^2? I think this would contradict in some way "The Poincare theorem" Thanks
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1answer
484 views

Are smooth functions with compact support weakly-* dense in $L^\infty$?

My question is this : given $f \in L^\infty(\mathbb{R}^2)$, can we find a sequence $\phi_n$ of smooth, compactly supported functions (test functions) such that for any $g \in L^1(\mathbb{R}^2)$, ...
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0answers
173 views

The converse of James's Theorem

The famous James theorem states that: Theorem. Let $X$ be a (Hausdorff separated) locally convex space (LCS for short) with topological dual $X^*$ and let $B\subset X$ be weakly-closed. If $X$ is ...
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3answers
327 views

What is the appropriate topology on $C_c^\infty (\mathbb{R}^d)$?

Let $\{ U_k:k\in \mathbb{N}\}$ be an increasing sequence of open subsets of $\mathbb{R}^d$ whose union is $\mathbb{R}^d$ and such that each $K_k:=\overline{U_k}$ is compact and $K_k\subseteq U_{k+1}$. ...
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1answer
313 views

Banach-Alaoglu theorem

If we have a Hilbert space $H$, (so it is reflexive) then by Banach-Alaoglu's theorem, the closed unit ball $B\subset H$ is weakly-compact. My question is, Is there any corollary or similar theorem ...
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1answer
201 views

Seminorms in locally convex spaces

This is a theorem in Rudin's functional analysis: Theorem. Suppose $\mathcal{P}$ is a separating family of seminorms on a real vector space $X$. Associate to each $p\in \mathcal{P}$ and to each $n\in ...
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68 views

Topological vector spaces question regarding dual spaces

there's a question that I've be puzzling over for a little while now and I haven't made much progress, so I thought I'd better ask for some help. It is as follows: Suppose that $\langle E,F\rangle$ ...
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1answer
87 views

Translate of a closed set is closed(part2)

Previously, I raised a question whether $$ (a+F)^c=a+F^c.$$ Jonas Meyer pointed out that it is true. After which, I was able to prove the first inclusion. The details are as follows: let $y\in a+F^c$. ...
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1answer
174 views

Translate of a closed set is closed

We assume that $X$ is a topological vector space with a topology $\tau$. I want to show that if $F$ is a closed subset of $X$, then its translate $a+F$ is also closed in $X$. Am I right to say that $$ ...
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67 views

Inductive limits

Let $E_n$ be a family of Banach spaces. Under which conditions imposed on $(E_n)$ can we represent the $\ell_\infty$-sum $(\bigoplus_{n\in \mathbb{N}} E_n)_{\ell_\infty}$ as a complemented subspace of ...
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1answer
155 views

Locally convex space characterization in terms of duality

Let $(X,\tau)$ be a topological vector space (TVS for short). Denote by $X^*$ the topological dual of $(X,\tau)$. If there exists a locally convex topology $\mu$ on $X$ compatible with the duality ...
3
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1answer
584 views

How to plot N points on the surface of a D-dimensional sphere roughly equidistant apart?

Let's say I have a D-dimensional sphere with a radius R. I want to plot N number of points evenly distributed (equidistant apart from each other) on the surface of the sphere. It doesn't matter where ...
6
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2answers
74 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 $\lambda>0$ ...
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2answers
338 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 ...
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3answers
231 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} ...
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1answer
59 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} ...
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1answer
1k 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: ...