All Questions
Tagged with functional-analysis general-topology
3,008 questions
0
votes
2
answers
49
views
Show that the inclusion $\phi$ from $l^1$ to $c_0$ is not a topological homomorphism
Let $l^1$ be the space of absolute summable sequences, that is $(x_n)_{n \in \mathbb{N}}$ such that $\forall n \in \mathbb{N}, x_n \in \mathbb{C}$ and $\sum_{n=1}^{\infty} |x_n| \lt \infty$.
Let $c_0$ ...
- 347
-3
votes
0
answers
70
views
The closure of $c_{00}$ in $\ell^1$ [closed]
Which is the closure of
$c_{00} = \{(x_n)_n \in \mathbb{R}: \exists N \in \mathbb{N}: \forall n \geq N: x_n = 0\}$ in $\ell^1(\mathbb{R})$?
I know that in $\ell^\infty(\mathbb{F})$ the closure of $C_{...
- 1
3
votes
1
answer
72
views
On the continuity a function given by evaluating compact subsets of continuous functions
Let $B$ be a closed ball in $\mathbb{R}^n$ and write $C(B)$ for the Banach space (with respect to the supremum norm) of the continuous real-valued functions on $B$.
Now given a compact subset $K$ of $...
- 590
0
votes
0
answers
37
views
Existence of transitive set of a homeomorphism
Let $X$ be a compact Hausdorff space without isolated points and $f:X\rightarrow X$ be a homeomorphism such that the $\mathbb{Z}$-action on $X$ that is defined by $f$ is minimal. Namely, for each $x\...
- 31
2
votes
1
answer
37
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Equivalent characterization of the support of a distribution in $C_0^\infty {'} (\Omega)$.
Before presenting my question in a formal way, I should present some contextualization so my problem is well understood.
Contextualization. Let $\Omega \subset \mathbb R^n$ be a non-empty open set and ...
- 1,217
5
votes
1
answer
98
views
Motivation for Nets in topology
In Folland chapter 4 he provided an example for why sequential convergence doesn't play central role in general topological spaces as it does in metric spaces. he said consider $C^{\mathbb{R}}$, space ...
- 71
4
votes
1
answer
69
views
Is the set of probability measures with zero mean and unit variance closed under the weak topology?
Consider the set of all Borel probability measures on $\mathbb R$ that have mean zero and unit variance. Is it closed under the weak convergence topology?
For sure, it would be true if we were ...
- 3,305
2
votes
1
answer
293
views
Topology generated by seminorm, why not just take supremum
Let $\{p_{a}\}_{\alpha \in A}$ be a family of seminorms on the vector space on the vector space $\mathcal{X}$, the canonical definition of the topology generated by these seminorms is the topology ...
- 1,948
0
votes
0
answers
51
views
Metric generating a sub-$\sigma$-algebra
It is known that the space $C:=C([0, \infty), \mathbb{R}^d)$ of continuous functions is a complete separable metric space with respect to the metric:
$$
d(f, g) = \sum_{ k = 1 }^{ \infty } ( 1 \wedge \...
- 1,577
0
votes
1
answer
40
views
Continuous selection from an upper hemicontinuity correspondence
Given a u.h.c. correspondence $\phi: X\rightrightarrows Y$ (see here for a definition of u.h.c correspondence), a function $f:X\to Y$ is called a continuous selection from $\phi$ if $f$ is itself a ...
- 92
0
votes
0
answers
104
views
Determine whether the Fourier coefficients coincide.
I have the following problem:
Let $f$, $g \in L^2(\mathbb{T})$ with respect to the Lebesgue measure, with Fourier series
$
f(x) = \sum_{n=-\infty}^{n=+\infty} \alpha_n e^{inx}
$
and
$
g(x) = \sum_{n=-\...
- 67
2
votes
0
answers
41
views
Interior of the set of probability measures
$\newcommand{\X}{\mathcal X}$
$\newcommand{\R}{\mathbb R}$
$\newcommand{\M}{\mathcal M}$
$\newcommand{\PR}{\mathcal P}$
Let $\X$ be a compact subset of $\R^d$. Let $\M$ denote the space of finite ...
- 3,305
4
votes
0
answers
194
views
Density of a set in a sphere of a normed vector space
I have being bugged for a while with the following problem.
Consider a compact set $X$ in $\mathbb R^d$. The class of bounded functions from $X$ to $\mathbb R$ can be endowed with the sup norm $\|\...
- 3,305
2
votes
1
answer
52
views
Metrizability of point-open topology
For arbitrary topological spaces $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$, let $Y^X$ denote the space of mappings from $X$ to $Y$ endowed with the topology of point-wise convergence $\mathcal{T}_p$,...
- 880
3
votes
0
answers
100
views
Is there any Banach space other than $\ell^p$ satisfies these conditions?
Let $X$ be a Banach space.
There exists a biorthogonal system $(x_i, f_i)_{i \in \mathbb{N}},$ where $(x_i)_{i \in \mathbb{N}} \subset X$ and $(f_i)_{i \in \mathbb{N}} \subset X^*$ such that for some $...
- 489
2
votes
0
answers
65
views
Finding a convergent subnet
In Bekka, de la Harpe and Valette 's book on Property (T), there is a lemma whose proof has some parts that are a bit hard for me to understand. We work with a locally compact group $G$ and $\hat G$, ...
- 21
0
votes
0
answers
29
views
In which topology on $X^*$, $X^*$ is a barrel space where $X^*$ is the dual of a Banach space $X$.
In a locally convex linear topological space $X,$ any convex, balanced and absorbing closed set is called a barrel. $X$ is called a barrel space if each of its barrels is a neighbourhood of zero.
And ...
- 489
0
votes
0
answers
18
views
Gronwall-type Inequality for Measurable Sets
Let $f$ be a function defined on $[0,1]$, and let $A_\epsilon \subset [0,1]$ be a measurable subset. I'm interested in whether there is a Gronwall-type inequality for measurable sets of the following ...
- 169
0
votes
1
answer
41
views
Prove that $0_Y$ is an iner point of a linear mapping between Banach space.
Question:
Let $X,Y$ two Banach space and $l$ a continuous surjective linear mapping from $X$ to $Y$. Prove that it exists a $r>0$ such that $2r B_Y \subseteq Adh(l(B_X)) $.
With $B_X$ and $B_Y$ the ...
- 739
3
votes
0
answers
55
views
Does a nontrivial Hausdorff TVS always have nonzero continuous linear functionals
For the following,
Let $X$ be a vector space over $\mathbb{F}=\mathbb{R}/\mathbb{C}$, and $X^\#$ the algebraic dual, equipped with the topology of pointwise convergence $\tau_{pc}$. If a subspace $Y\...
- 2,020
2
votes
1
answer
266
views
There are closed spaces that are not complete
I am learning functional analysis and over the last week we have been introduced to some topological notions. I have seen that a space is complete when any Cauchy sequence in that space approaches ...
- 199
0
votes
1
answer
40
views
Constructing closed sets as union of open sets in topology [duplicate]
I just started learning topology as a part of functional analysis in undergrad and I have a small problem - I probably don't understand the three constructing axioms of topology correctly (...
- 11
3
votes
1
answer
120
views
Urysohn's Lemma for $C^1[0,1]$?
I need a thing like Urysohn's Lemma for $C^1[0,1]$.
In Rudin's functional analysis book in the above of page 36, it is stated that such a function there is in $ C^{\infty}(\mathbb{ R}^n)$ but one of ...
1
vote
1
answer
57
views
How to use Cauchy Schwarz Inequality for proving a Metric in the Functional space.
The question is as follows:
Show that the set $[a,b]$ of all real valued continuous functions defined on the
closed interval $[a,b]$ with function defined by $d(f,g)$, is a metric space.
$$d(f,g) = \...
- 539
0
votes
1
answer
27
views
Semicontinuity of set-valued maps
Just recently I started working with set-valued maps and things are quite "confusing" and complicated at the moment. So I would like to ask, maybe a silly question, but would appreciate any ...
- 345
2
votes
0
answers
108
views
About weakly metrizable and bounded subset of a Banach space
Let $X$ be a Banach space and $w$ its weak topology. In a doctoral thesis, I have read a result, which (if I understood correctly) sound like this: "If $B$ is a bounded and $w$-metrizable subset ...
- 166
1
vote
1
answer
43
views
Condition for sequential approximate identity for $C_0(\Omega)$
Exercise 1.4.1 (p.21) of Conway's A Course in Operator Theory asks for a necessary and sufficient condition that $C_0(X)$ have a sequential approximate identity, where $X$ is assumed to be locally ...
- 2,020
5
votes
0
answers
89
views
Equivalence between weak containment and convergence in Fell Topology
We first define Fell Topology. Let $G$ be a topological group and $H$ be a Hilbert space. Let $\hat G$ be the set of equivalence classes of irreducible unitary representations of $G$. We define open ...
- 357
2
votes
0
answers
71
views
Natural way to extend Lebesgue measure concept on normed space
I will ask two questions that involve proposition 1.15 and theorem 1.16 from Follands' real analysis book.
First question:
Folland constructs the Lebesbgue Stieltjes measure in $\mathbb{R}$, and I ...
1
vote
1
answer
60
views
About convex set in normed vector spaces.
I'm trying to prove the following statement: Let E be a normed vector space. If $C \subset E$ is a convex subset, then $Int(C)$ is also convex.
First, let me explain the notations I'm using: $B(x,r)$ ...
- 147
2
votes
0
answers
94
views
Balls in metric spaces
It is well known that the unit ball $B$ in a norm space must be absorbent, symmetrical with respect to the origin (that is, $B=-B$), convex and also does not contain a subspace of dimension $1$.
More ...
- 645
3
votes
1
answer
69
views
Separation properties of weak topology on $\ell^2$
If $\ell^2$ is given weak topology, call the resulting space $X$, then it's easy to show that it's $T_4$:
Since $X$ is a Hausdorff topological vector space, it must be Tychonoff.
Moreover, $X$ is $\...
- 11.9k
1
vote
1
answer
32
views
Strong Convergence of Nets in a Norm-Closed $*$-Subalgebra of Bounded Operators
Let $\mathcal{H}$ be a Hilbert space and $\mathcal{B(H)}$ the set of all bounded operators on $\mathcal{H}$. The norm topology on $\mathcal{B(H)}$ is defined using the operator norm. For a net $(x_\...
- 2,079
4
votes
0
answers
60
views
When is a notion of convergence preserved when generating a topology?
Let $X$ be a set with no topological structure but equipped with a notion of convergence. Assuming this notion of convergence satisfies some nice consistency conditions, we can define a topology on $X$...
- 6,641
3
votes
1
answer
130
views
Initution and ideas to find counterexamples related to functions
I find it hard to construct examples to disprove certain statements, does anyone have any ideas for such examples when disproving statements?
For example:
Let $X=C[0,1]$ be set of continuous ...
- 1,151
1
vote
0
answers
58
views
Why does the weak$^*$ topology $\sigma(E^*,E)$ agree with the one induced by the completion of $E$?
Given a normed vector space $E$, its dual space $E^*$ has a standard norm itself. However, we can define the weak$^*$ topology $\sigma(E^*,E)$ over $E^*$ to be the minimal one in which the evaluation (...
- 31
0
votes
0
answers
22
views
Prove a continuous bijection between the space of probability distributions is a homeomorphism
Let $\Delta([\underline p, \overline p])$ denote the set of all cumulative distribution functions (CDF) supported on the compact set $[\underline p, \overline p]$. $G_i\in \Delta([\underline p_i, \...
0
votes
0
answers
58
views
Can all linear functionals be expressed in terms of a pairing?
Let $X$ and $Y$ be linear spaces and $\langle \cdot, \cdot \rangle:X\times Y\rightarrow \mathbb{R}$ a bilinear mapping. Often this is called a pairing. Now the weak topology is defined as the weakest ...
- 507
1
vote
1
answer
76
views
Help with Proof Related to Space of Codimension One
Let $H$ be a infinite dimensional Hilbert space and $\{e_{i}\}$, $i\in\mathbb{N}$ be an orthogonal basis of it. Let the span of $e_{1}$ be $U$ and $span(e_{i}:i>1)=V$, so $U^{\perp}=\overline{V}\...
- 1,399
2
votes
1
answer
153
views
$C(K_1)\cong C(K_2)$ if and only if $K_1\cong K_2$
I had read the following statement in a book without proof. One of the directions is trivial, however the other is not:
Let $C(K_j)$ the Banach space of continuous functions $f:K_j\to \mathbb{R}$ ...
user173262
1
vote
1
answer
77
views
Does $L^p$ convergence preserve the $L^1 $ norm of the sequence
I have the following question which I could not figure out an answer for.
Suppose you have a sequence of non-negative functions $f_n$ in $L^p(U)$ with $U \subset \mathbb{R}^n$ bounded, $1 \leq p \leq \...
- 615
3
votes
3
answers
125
views
Examples of topological spaces that closed+boundness implies compactness
One familiar example of such one is the Euclideans. Recently I learnt that the space of holomorphic functions over an open set $U\subset\mathbb{C}^n$ also has this property. The formal statement is as ...
- 803
1
vote
0
answers
85
views
On the intersection of nested sequence of nice sets [closed]
Let $\{H_n\}$ be a sequence of non-empty closed, bounded and convex sets in a Banach space $X$ with $H_{n+1}\subseteq H_n$ for all $n \in \mathbb{N}.$ Denote $\delta(A)= \sup\{\|x-y\|: x, y\in A\},$ ...
- 29
0
votes
1
answer
45
views
Compactess of a set in $\mathbb{R}^d$ defined as the union of compact sets
Let $f:[a,b]\to \mathbb{R}^d$ be a function of class $C^1$. Let us consider the following set:
$$
\mathcal{A}:=\bigcup_{x\in [a,b]}\{y\in \mathbb{R}^d, \quad ||y- f(x)||\leq 1/2 \}
$$
I think that ...
- 169
2
votes
1
answer
103
views
Mollifiers make partition of unity
I am working through Evan's PDE book and I found these nice exercises. Problem 5 is not that hard. I am having trouble with problem 6:
Let $U$ and $V$ be open sets with $V \subset \subset U$. Show ...
- 2,691
1
vote
0
answers
58
views
Is the set of $\varepsilon$-discontinuities closed if it is defined without $\limsup$?
Let $f_n \in C([0,1])$ be a sequence of functions with $f_n(x) \to f(x)$ for all $x$. Consider the set of discontiuities of $f$, which I denote by $\mathcal{D}$. Then, we can define:
$$\mathcal{D} = \...
- 611
0
votes
0
answers
52
views
Intuition abount countable compactness
In the context of direct method for calculus of variations, it is useful to introduce the notion of countably compact space, more explicitly:
Def: Let $K$ topological space, $K$ is countably compact ...
- 365
4
votes
1
answer
95
views
Are left-inverses of continuous functions pointwise limits of continuous functions?
Let $f : X \to Y$ be a surjective continuous map and suppose $g: Y \to X$ satisfies $g \circ f = \text{id}_X$. If we have that $Y$ is compact and $X$ and $Y$ are both subsets of $\mathbb{R}^n$, does ...
- 611
0
votes
1
answer
44
views
Sum of topological interior and topological interior of the sum
In Banach spaces, or on a generic TVS of infinite dimension, i think that doesn't exists any inclusion relation beetween the topological interior of the sum of two subset and the sum of the ...
- 365
0
votes
0
answers
39
views
What is the difference between these separation theorems?
Theorem 1: Let $A$ and $B$ be convex sets of real normed $X$. If $A$ is compact in $X$ and $B$ is closed, then there is $f\in X^*$ and $\lambda_1,\lambda_2\in \mathbb{R}$ such that $f(x)\leq\lambda_1&...
- 1,018