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.
1
vote
1answer
231 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 ...
1
vote
1answer
25 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 ...
3
votes
1answer
64 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 ...
1
vote
1answer
46 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
votes
1answer
39 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
48 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
0answers
183 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
192 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:
...
2
votes
1answer
53 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
40 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
34 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 ...
3
votes
1answer
38 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:
$$
p_\Omega:E\to ...
1
vote
1answer
44 views
About a weak topology on TVS
Let $(X,\tau)$ be a topological vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each ...
7
votes
1answer
218 views
Contractibility of convex set
Suppose that $\Omega$ is a convex open subset of an infinite dimensional vector space $E$ such that $\Omega$ is not contained in any finite dimensional subspace of $E$.
Let $Q_m\subset \Omega$ denote ...
1
vote
1answer
40 views
About Closed Unit Balls
If $(X, \|\cdot \|_X)$ is a normed vector space over $\mathbb R$, then the closed unit ball of $X$ is given by
$$B(X)=\{x\in X: \|x \|_X \le 1\}.$$
If $X^{*}$ is the set of all bounded linear ...
0
votes
1answer
47 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}$.
...
2
votes
1answer
59 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 ...
1
vote
2answers
105 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
0answers
61 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
111 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 ...
2
votes
2answers
72 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 ...
3
votes
1answer
49 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 ...
3
votes
3answers
987 views
The kernel of a continuous linear operator is a closed subspace?
If $V$ and $W$ are topological vector spaces (and $W$ is finite-dimensional) then a linear operator $L\colon V\to W$ is continuous if and only if the kernel of $L$ is a closed subspace of $V$.
...
2
votes
1answer
85 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 ...
1
vote
0answers
51 views
Finite Dimensional TVS
Let $E, F$ topological vector spaces, $E$ normable and $T: E \longrightarrow F$ linear, compact and surjective. Show that $\mbox{dim}(F)< \infty$.
5
votes
4answers
101 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?
7
votes
0answers
75 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
117 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
119 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 ...
5
votes
1answer
79 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? ...
1
vote
2answers
115 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 \|,
$$
where $ ...
3
votes
1answer
67 views
Properties of $ \text{Exp}(A) $, where $ A $ is a Banach algebra.
$ \newcommand{\Exp}{\operatorname{Exp}} $ Let $ A $ be a unital Banach algebra. For $ a \in A $, consider
$$
\Exp(A) \stackrel{\text{def}}{=} \{ e^{a_{1}} e^{a_{2}} \cdots e^{a_{n}} ~|~ n \in ...
4
votes
2answers
196 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 ...
6
votes
0answers
111 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 $E_1$, ...
5
votes
2answers
93 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 ...
8
votes
1answer
257 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 ...
2
votes
1answer
39 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 ...
5
votes
3answers
147 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.
2
votes
1answer
124 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 ...
2
votes
1answer
103 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 ...
4
votes
2answers
99 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.
0
votes
1answer
112 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
61 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
150 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 $X$ ...
0
votes
0answers
28 views
Is this question stated wrong?
I'm trying to check whether this question might be worded wrong, and here it is:
Show that if $A$ is a convex subset of a topological vector space $X$, $u \in A^o$ (the interior of $A$), $v \in ...
1
vote
1answer
43 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 ...
-2
votes
1answer
62 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
48 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
69 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
36 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 ...
