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.

learn more… | top users | synonyms

2
votes
1answer
52 views

About the induced vector measure of a Pettis integrable function

In what follows, $X$ stands for a Hausdorff LCTVS and $X'$ its topological dual. Let $(T,\mathcal{M},\mu)$ be a finite measure space, i.e., $T$ is a nonempty set, $\mathcal{M}$ a $\sigma$-algebra of ...
0
votes
2answers
58 views

Composition of a vector measure and a linear functional

Assume that $X$ is topological vector space over the field $\mathbb{R}$. Let $\mathcal{M}$ be a $\sigma$-algebra of subsets of a nonempty set $T$. We say that $$m: \mathcal{M}\to X$$ is a vector ...
3
votes
1answer
220 views

About Henstock integrable vector-valued function

In what follows, $X$ is a Hausdorff locally convex topological vector space over the reals whose topology is generated by a family $P$ of all continuous seminorms on $X$. We consider the following ...
2
votes
1answer
76 views

About Equicontinuous and Boundedness

Let $X$ be a TVS and $X'$ denotes the space of all continuous linear functionals on $X$. Let us denote the $weak^*$-topology on $X'$ by $\sigma(X',X).$ My question is this. Why does every ...
2
votes
1answer
85 views

About a Weak Topology of a Vector Space

Let $X$ be a real vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each $p\in P$ ...
1
vote
1answer
48 views

About a Weak Topology on TVS(part 2)

Let $(X,\tau)$ be a topological vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each ...
4
votes
1answer
55 views

Homeomorphism via Minkowski functional?

Suppose $E$ is an infinite dimensional topological vector space and $\Omega\subset E$ is open, convex and $0\in \Omega$. The Minkowski-functional of $\Omega$ is defined by: $$ p_\Omega:E\to ...
1
vote
1answer
70 views

About a weak topology on TVS

Let $(X,\tau)$ be a topological vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each ...
1
vote
1answer
105 views

About Closed Unit Balls

If $(X, \|\cdot \|_X)$ is a normed vector space over $\mathbb R$, then the closed unit ball of $X$ is given by $$B(X)=\{x\in X: \|x \|_X \le 1\}.$$ If $X^{*}$ is the set of all bounded linear ...
0
votes
1answer
86 views

$V$ is finite dimensional iff $V'$ with the weak topology is normable

Why is the following statement valid? Note, $V$ is locally convex Hausdorff topological vector space over $\mathbb{C}$ and $V'$ is the space of all continuous linear maps from $V \to \mathbb{C}$. ...
3
votes
1answer
111 views

The weak topology of a product

Let $E$ and $F$ be normed vector spaces and let $E^{\sigma}$, resp. $F^{\sigma}$ be $E$, resp. $F$ with the weak topology associated with the elements of the duals $E^*$, resp. $F^*$. Then, for ...
7
votes
1answer
420 views

Contractibility of convex set

Suppose that $\Omega$ is a convex open subset of an infinite dimensional vector space $E$ such that $\Omega$ is not contained in any finite dimensional subspace of $E$. Let $Q_m\subset \Omega$ denote ...
2
votes
0answers
98 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
votes
2answers
131 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
votes
2answers
302 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
votes
1answer
56 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
votes
1answer
224 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 ...
5
votes
4answers
181 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?
18
votes
1answer
327 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
votes
1answer
165 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
votes
2answers
294 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 ...
1
vote
1answer
114 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 ...
4
votes
2answers
911 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
votes
2answers
111 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
111 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 ...
10
votes
1answer
874 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
485 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.
4
votes
1answer
277 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
votes
2answers
283 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.
4
votes
1answer
318 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
votes
1answer
222 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
votes
1answer
127 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 ...
9
votes
1answer
328 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
votes
0answers
140 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 $E_1$, ...
1
vote
1answer
131 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 ...
-1
votes
1answer
159 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 ...
1
vote
2answers
57 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} = ...
0
votes
1answer
101 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 ...
1
vote
0answers
48 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
203 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
vote
3answers
310 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
votes
2answers
124 views
0
votes
1answer
197 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 entired problem in the following: For $0<p<\infty$, let $l^p$ be the space of all functions $x$ (real or ...
2
votes
1answer
59 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 ...
0
votes
1answer
445 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 ...
1
vote
3answers
752 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 ...
2
votes
1answer
293 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
120 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?
5
votes
1answer
248 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 ...
1
vote
1answer
164 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, ...