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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115 views

Which are nontrivial examples of analytical functions on Frechet spaces?

Let $X$ be a real linear topological space, which is a (separable) Frechet space, such that the topology on $X$ is generated by the countable family $\{p_n:n\in\omega\}$ of norms . A real-valued ...
2
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2answers
142 views

Hahn-Banach Separation Theorem and Bishop's Theorem

I am looking at the proof of Bishop's Theorem on pages 122 and 123 of Rudin's Functional Analysis. The following quote is from the the last two sentences of the proof on pg. 123. "Every continuous ...
4
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2answers
454 views

Topology of the space of hermitian positive definite matrices

Let $\mathcal{H}_n \mathbb{C}$ be the set of hermitian $n \times n$ complex matrices. This set carries the structure of a vector space over $\mathbb{R}$ under usual addition. It also inherits the ...
3
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1answer
57 views

Some basic questions about minima of a real-valued functions

The following theorem is basically from the Fermat's Theorem page of wikipedia. Let $X$ denote a subset of $\mathbb{R}$, and suppose $f : X \rightarrow \mathbb{R}$ attains a global minimum at $x ...
2
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1answer
403 views

Are projections onto closed complemented subspaces of a topological vector space always continuous?

Suppose $X$ is a topological vector space and $X = V \oplus W$ is a decomposition of $X$ into closed subspaces. The decomposition gives rise to a projection $P$ onto $V$ (depending on the choice of ...
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4answers
283 views

Books on locally convex topological vector spaces

My friend asked me for a good book about locally convex topological vector space. I'm not familar with this. Could you give me some good references on it?
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1answer
368 views

Learning Aid for Basic Theorems of Topological Vector Spaces in Functional Analysis

I am self-teaching myself the basics of functional analysis (e.g. topological vector spaces), and frankly I am starting to get a migraine sorting out/organizing in my head all of the ...
7
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1answer
183 views

Rotation of $\mathbb{R}^3$ by using quaternion

Express the rotation of $\mathbb{R}^3$ by $\frac{\pi}{3}$ about the $x=y=z$ axis by using quaternions and identifying $\mathbb{R}^3$ with $(i,j,k)$-space. Thoughts: From my point of view, every ...
2
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2answers
508 views

$C^\infty(R^n)$ is a Banach Space when equipped with topology of uniform convergence

Prove $C^\infty(\Bbb R^n)$ is a Banach Space when equipped with topology of uniform convergence. $C^\infty(\Bbb R^n)$ is space of all continuous functions that converge to $0$ at $\infty$. And, the ...
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1answer
133 views

Properties of $ \text{Exp}(A) $, where $ A $ is a Banach algebra.

$ \newcommand{\Exp}{\operatorname{Exp}} $ Let $ A $ be a unital Banach algebra. For $ a \in A $, consider $$ \Exp(A) \stackrel{\text{def}}{=} \{ e^{a_{1}} e^{a_{2}} \cdots e^{a_{n}} ~|~ n \in ...
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2answers
1k views

Closed Bounded but not compact Subset of a Normed Vector Space

Consider $\ell^\infty $ the vector space of real bounded sequences endowed with the sup norm, that is $||x|| = \sup_n |x_n|$ where $x = (x_n)_{n \in \Bbb N}$. Prove that $B'(0,1) = \{x \in l^\infty ...
5
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2answers
124 views

Question about Topological Vector Spaces

Let $E$ be a Topological Vector Space and $U$ a bounded set of $E$ with $0\in U$, i.e. given any neighborhood $W$ of the origin, there exist $\alpha>0$ such that $\alpha U\subset W$. Is it true ...
2
votes
1answer
148 views

About sequentially complete and complete TVS

Let us use $X$ to mean a topological vector space. I know that if $X$ is a complete TVS then it is sequentially complete. I know that the converse is not true, so what I need now is to construct a TVS ...
13
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1answer
1k views

Semi-Norms and the Definition of the Weak Topology

When I was searching for the definition of the weak topology, I found two different definitions. One defines the weak topology in terms of a family of semi-norms, while the other defines it in terms ...
5
votes
3answers
697 views

Is $C([0,1])$ a compact space?

Is $C([0,1])$ (I guesss with the max-norm) a compact space? I have to know that because I want to apply Arzela Ascoli.
5
votes
1answer
380 views

An explicit construction for a “doubly weak” topology

Let $(X,s)$ be a topological vector space over $\mathbb{F}$ with linear topology $s$, which we will henceforth refer to as the strong topology. Then, as usual we can construct the continuous dual ...
4
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2answers
390 views

Continuous linear functionals

Let L be a continuous linear functional on a metric linear space X. Prove: L(S) is a bounded set for any bounded subset S of X. The metric is translation invariant.
5
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1answer
467 views

What is the topological dual of a dual space with the weak* topology?

I'm trying to understand a claim I heard in class. To be concrete, suppose $X$ is a compact, hausdorff topological space, and let $C(X)$ be the space of continuous functions on $X$ with the supremum ...
0
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1answer
280 views

When is $\| f \|_\infty$ a norm of the vector space of all continuous functions on subset S?

Let S be any subset of $\mathbb{R^n}$. Let $C_b(S)$ denote the vector space of all bounded continuous functions on S. For $f \in C(S)$, define $\| f \|_\infty = \sup_{x \in S} |f(x)|$ When is this a ...
0
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1answer
141 views

About the filtering family of seminorms

We first start with following definitions. Definition 1. A family $\mathcal{P}$ of seminorms on a real vector space $X$ is called filtering if for any $p_1,p_2\in \mathcal{P}$ there exsist $q\in ...
10
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1answer
562 views

If weak topology and weak* topology on $X^*$ agree, must $X$ be reflexive?

Let $X$ be a Banach space and suppose that the weak topology on $X^*$ agrees with the weak* topology on $X^*$. Must $X$ be reflexive? To prove the contrapositive, it will suffice to assume that $X$ ...
6
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0answers
154 views

Isomorphism between spaces of sections.

Let $M$ be a compact manifold and let $E_i \xrightarrow{\Large \pi_i} M$, $i = 1, 2$, be two (real or complex) vector bundles of the same rank $k$ over $M$. Assume we have metrics $g_1, g_2$ on $E_1$, ...
2
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1answer
242 views

Questions regarding internal and interior points for a convex subset of a topological vector space

Suppose that $X$ is a topological vector space, with a convex subset $A$. How do we show that if the vector $u$ is in the interior of $A$, then $u$ is an internal point of $A$ and if the interior of ...
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1answer
271 views

Locally convex space which is not Banach.

I know that all Banach spaces are (Hausdorff)locally convex spaces. I would like to verify that the converse is not true by giving an example of a space which is locally convex but not Banach. I am ...
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2answers
61 views

Prove $\overline{x+A} = x+\bar{A}$ and $\alpha\bar{A} = \overline{\alpha A}$

If $A$ is a subset of $(V,\parallel.\parallel)$, then let $\bar{A}$ denote its closure. Show that if $x\in V$ and $\alpha \in \mathbb{R}$, then $\overline{x+A} = x+\bar{A}$ and $\alpha\bar{A} = ...
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1answer
105 views

show $\langle f,g \rangle _w = \int^b_a f(x)g(x)dx$ is an inner product

Let $w(x)$ be a strictly positive continuous function on [a,b]. Define a form on $C[a,b]$ by the formula $\langle f,g \rangle _w = \int^b_a f(x)g(x)dx$ for $f,g \in C[a,b]$. Show that it is an inner ...
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0answers
55 views

Show that the norm of the derivitive of a $C^1$ function over a vector space is non-negative, homogeneous and satisfies the triangle ineq

For $f$ in $C^1[a,b]$, define $p(f)= \parallel f'\parallel _{\infty}$. Show that $p$ is non-negative, homogeneous, and satisfies the triangle inequality. Why is it not a norm? -I can easily show the ...
3
votes
2answers
252 views

Left topological zero-divisors in Banach algebras.

Let $ A $ be a unital Banach algebra. Define $ \zeta: A \longrightarrow [0,\infty) $ by $$ \forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|, $$ where $ ...
1
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4answers
474 views

Nonconstant linear functional on a topological vector space is an open mapping

In the middle of another proof (Theorem 3.4, p. 60) in his Functional Analysis book, Rudin says that "every nonconstant linear functional on $X$ (topological vector space) is an open mapping." Is ...
2
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2answers
153 views
1
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1answer
252 views

Problem 5. ( chap3. p.87, functional analysis, W.Rudin)

I had done part a, b, and d. But I cannot breakthrough part c, and part e. I restate entire problem in the following: For $0<p<\infty$, let $l^p$ be the space of all functions $x$ (real or ...
3
votes
1answer
74 views

About the continuity of $B$ (problem 12 chap.2, p.55, functional analysis, W.Rudin)

Let $X$ be the normed space of all real polynomails in one variable, with $||f||=\int_0^1 |f(t)|dt$. Put $B(f,g)=\int_0^1 f(t)g(t)dt$, and show that $B$ is a bilinear functional on $X\times X$ which ...
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1answer
712 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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vote
3answers
2k 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 ...
3
votes
2answers
437 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 ...
0
votes
1answer
138 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?
6
votes
1answer
313 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
286 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
58 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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votes
0answers
80 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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0answers
51 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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1answer
241 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
54 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 ...
6
votes
1answer
106 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
112 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 $Y:=\operatorname{cl}\operatorname{int} X$. ...
5
votes
1answer
171 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
votes
1answer
93 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
131 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 ...
3
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2answers
249 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, ...
0
votes
0answers
58 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))= ...