Tagged Questions
2
votes
2answers
50 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 ...
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 ...
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 ...
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 ...
5
votes
4answers
102 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?
4
votes
2answers
201 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 ...
8
votes
1answer
258 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 ...
0
votes
1answer
113 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 ...
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} = ...
1
vote
2answers
116 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 $ ...
0
votes
1answer
125 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
40 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
131 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
2answers
128 views
Return an array of evenly distributed points on a sphere give Radius and Origin.
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 ...
1
vote
1answer
53 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 ...
2
votes
2answers
71 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
88 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 ...
1
vote
1answer
21 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 ...
1
vote
1answer
39 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 ...
4
votes
1answer
215 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 ...
0
votes
0answers
48 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
2
votes
3answers
111 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} ...
2
votes
0answers
45 views
Topology of $(\mathcal{A},*)$ determined by $\mathcal{A}_{sa}$?
Let $(\mathcal{A},*)$ be a $*$-algebra, we have the following observation:
Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\mathcal{A}$ such that the involution is an isometry with respect to ...
2
votes
0answers
140 views
Definition of a topological module
A topological universal algebra of type $\Omega$ is a universal algebra $A$ of type $\Omega$ that is also a topological space, such that for any $n\!\in\!\mathbb{N}$ and any operation ...
4
votes
4answers
125 views
Question on Topological vector space 1
I have numbered this question as (1) because I will be posting series of questions where I don't understand. I hope its allowed.
I want to prove the following :
If $X$ is a topological vector ...
3
votes
1answer
177 views
Extreme boundary of a compact, convex, metrizable set is $G_\delta$
Let $X$ be a topological vector space (no assumptions about local convexity are made in the question, though I am worried they might be required). Suppose $K\subset X$ is a compact, convex, metrizable ...
2
votes
1answer
73 views
Limit on a topological vector space
in the Wikipedia article on Gâteaux derivative , the limit of a function between two topological vector spaces is taken. How is the limit defined on a topological space for a function ? I find ...
0
votes
1answer
151 views
$X^*$ with its weak*-topology is of the first category in itself
Let $X$ be an infinite-dimensional Fréchet space. Prove that $X^*$,with its weak*-topology is of the first category in itself.
4
votes
1answer
108 views
Connected components that are relatively open in $\sigma(T)$
Let $T$ be an bounded linear operator on a Banach space $X$. Suppose the spectrum of $T$, $\sigma(T)$ has infinitely many connected components, then $\sigma(T)$ must contain infinitely many ...
17
votes
3answers
578 views
When do weak and original topology coincide?
Let $X$ be a topological vector space with topology $T$.
When is the weak topology on $X$ the same as $T$? Of course we always have $T_{weak} \subset T$ by definition but when is $T \subset ...
0
votes
1answer
76 views
Topological fields questions
From Wikipedia:
"Let $K$ be a topological field, namely a field with a topology such that addition, multiplication, and division are continuous. In most applications $K$ will be either the field of ...
1
vote
0answers
109 views
Are topological vector spaces completely regular?
Every uniformizable space is a completely regular topological
space.
topological vector spaces are uniform spaces.
every Hausdorff topological vector space is completely regular.
From ...
1
vote
2answers
116 views
Completability of a uniform space, metric space and topological vector space?
From Wikipedia
In particular, topological vector spaces are uniform spaces and one
can thus talk about completeness, uniform convergence and uniform
continuity. (This implies that every ...
2
votes
3answers
168 views
Boundedness in a topological space?
I was wondering if there is a concept of boundedness for subsets of
a topological space?
If yes to 1, is it this one from Wiki
Elements of a Bornology B on a set X are called bounded sets and the ...
3
votes
1answer
304 views
Meaning of “a mapping preserves structures/properties”
Sometimes I see something like "a mapping preserves the structures
of its domain and of its codomain". From Wiki about morphisms in category theory:
a morphism is an abstraction derived from ...
1
vote
1answer
145 views
Relationship between Convergence and Open sets
If you show that convergence of nets in a topological vector space $V$ with topology $\tau$ is equivalent to convergence of nets in a topological vector space $V$ With topology $\sigma$, does it ...
7
votes
3answers
401 views
Do continuous linear functions between Banach spaces extend?
Just wondering... Let $E$, $G$ be Banach spaces, let $U\subset E$ be a subset of $E$, and let $f:U\rightarrow G$ be a continuous linear function. Can $f$ be extended to a continuous linear function on ...
3
votes
0answers
234 views
Understanding examples - metric spaces, Minkowski functionals and topologies
I'm teaching myself a course on functional analysis but having trouble understanding the notes I've been using. I was hoping I could write out a section of the content and you might be able to help me ...
4
votes
0answers
161 views
Strong topology vs Natural topology
Let $X$ be a locally convex space and $\left< X, X^{\prime} \right>$ stands for the dual pair. The bidual of $X$ is denoted by $X^{\prime \prime}$ and this is a dual of $X^{\prime}$
with a ...
10
votes
0answers
256 views
Differential forms on fuzzy manifolds
This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me.
Question
In the usual setting of open subsets of ...
4
votes
1answer
186 views
Why This Map is Closed?
Consider the following definition of closed maps, defined in the book Nonlinear Programming by Bazaraa et al.:
Let $X$ and $Y$ be nonempty closed sets in $\mathbb{R}^p$ and $\mathbb{R}^q$, ...
16
votes
2answers
607 views
When is a notion of convergence induced by a topology?
I'm interested in sufficient conditions for a notion of sequential convergence to be induced by a topology. More precisely: Let $V$ be a vector space over $\mathbb{C}$ endowed with a notion $\tau$ of ...
0
votes
2answers
425 views
Topology: Proof that a finitely generated cone is closed
Looking for the proof of the lemma asserting that the conical surface (envelope) is a closed space.
Thank you.
