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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97 views

How Does A Precessing Sphere Precess?

In particular, consider a perfectly spherical object or radius $r$, with a certain axis $L$, and this axis titled relative to a vertical axis by some amount $\theta$. Say that it's "wobbling" in a ...
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0answers
70 views

Pseudo norm-exercice

Let $f$ be a measurable function with finite values almost everywhere. We put $$N_0(f) = \displaystyle\int \dfrac{|f|}{1 + |f|} d \mu.$$ We denoted by $L^0$ the set of measurable functions $f$ such ...
2
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1answer
111 views

Smooth function which is not continuous

I have seen it mentioned that in certain infinite dimensional topological vector spaces it is possible to have a smooth curve which is not continuous, but I've never seen an explicit example. Can ...
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1answer
123 views

An counterexample of Hahn-Banach theorem in a topological vector space

Problem : Give an example of a TVS $\mathcal{X}$ that is not locally convex and a subspace $\mathcal{Y}$ of $\mathcal{X}$ such that there is a continuous linear functional $f$ on $\mathcal{Y}$ with no ...
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1answer
48 views

On Polar Sets with respect to Continuous Seminorms

In the following, $X$ is a Hausdorff locally convex topological vector space and $X'$ is the topological dual of $X$. If $p$ is a continuous seminorm on $X$ then we shall designate by $U_p$ the ...
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1answer
643 views

Question about proof that finite-dimensional subspaces of normed vector spaces are direct summands

I am reading a proof that finite-dimensional subspaces of normed vector spaces have closed direct sum complements. This is the proof: Let $\{e_1, ..., e_n\}$ be a basis for $\mathcal M$. Every $x ...
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1answer
130 views

On Absolutely Continuous Functions

I would like to know if we can extend the concept of absolute continuity to functions $f:[a,b]\to X$, where $X$ is a topological vector space. I browsed some books on Topological Vector Spaces but ...
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1answer
34 views

About Regulated Functions

Definition. Let $X$ be a Banach space. A mapping $f:[a,b]\to X$ is called regulated if it has one sided limits. In the setting of a Hausdorff topological vector space $X$, can we still define ...
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53 views

A property involving a polar of a set

Let $X$ be LCTVS and let $X'$ be its topological dual. Let $A\subseteq X$. Suppose that $x\in X$ and satisfies the following property: $$|x'(x)|<1 \mbox{ for all } x'\in A^0$$ where $A^0$ is a ...
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2answers
88 views

A separable, regular space which has cardinality of the continuum but is not first countable?

Actually the title says it all. Is there such a topological space which is separable, regular, has cardinality of the continuum but is not first countable? If so, is there also an example of a ...
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1answer
46 views

About the induced vector measure of a Pettis integrable function(part 2)

Notations: 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 ...
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180 views

About Lusin's condition (N)

We say that $f:[0,1]\to \mathbb{R}$ satisfies Lusin's condition (N) provided $$m(f(B))=0 \quad\mbox{whenever}\quad B\subseteq [0,1] \mbox{ with }m(B)=0$$ where $m$ stands for the Lebesgue measure on ...
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1answer
60 views

Checking if the induced mapping is well-defined

Let $\mathcal{M}$ be a $\sigma$-algebra of subsets of a nonempty set $T$, $X$ is Hausdorff LCTVS, $X'$ the topological dual of $X$, and $m: \mathcal{M}\to X$ a countably additive vector measure on ...
2
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1answer
53 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 ...
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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 ...
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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 ...
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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 ...
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1answer
86 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 $p\in P$ ...
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1answer
49 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 ...
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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 ...
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1answer
71 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 ...
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1answer
109 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 ...
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1answer
87 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 $V \to \mathbb{C}$. ...
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1answer
112 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 ...
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430 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 ...
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0answers
100 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 ...
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2answers
132 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 ...
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2answers
309 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
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 ...
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1answer
233 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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185 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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329 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
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
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2answers
314 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
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 ...
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votes
2answers
943 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 ...
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2answers
112 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
113 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 ...
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1answer
909 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
499 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
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1answer
281 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 ...
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2answers
286 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.
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1answer
320 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 ...
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1answer
226 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 ...
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1answer
128 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 ...
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1answer
332 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 $X$ ...
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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 $E_1$, ...
2
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1answer
147 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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170 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
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} = ...