All Questions
Tagged with lp-spaces general-topology
108
questions
0
votes
0
answers
44
views
Trying to understand the proof for the criterion of compactness in $l_p$ space
I have the following theorem about the criterion of compactness in $l_p$ space
For the set $K\subset (l_p,||.||_p), p\geq 1$, following conditions
are equivalent:
i) $K$-totally bounded in $(l_p,||.||...
- 249
3
votes
0
answers
90
views
How to use Lebesgue Number for Complex Analysis Holder Inequality
Problem: Let $f$ be analytic in an open set $U\subset \mathbb{C}$ and let $K\subset U$ be compact. Show that there exists a constant $C$ depending on $U$ and $K$ such that $$|f(z)| \leq C \left( \...
- 619
2
votes
0
answers
111
views
Correct condition for $L_p$ to be separable.
I am a physics undergraduate so I only had two lectures about measure theory. Could someone explain to me what is the most correct statement between these two ($p<\infty$):
From Is $L^p$ separable?...
- 71
1
vote
0
answers
120
views
Possible Frechet-Kolmogorov compactness theorem equivalent in $L^\infty$?
The Frechet-Kolmogorov(-Sudakov) compactness theorem states that in $L^p(\mathbb{R}^n)$ for $1\leq p<+\infty$, a set of functions $\mathcal{F}$ is totally bounded if and only if $\mathcal{F}$ is
...
- 131
1
vote
1
answer
83
views
Weak-$\star$ Convergence in $L_1$
Let
$(X,d)$ be a complete separable metric space;
$\mu$ be a Borel probability measure on $X$;
$(f_n),(g_n)\subset L_1(X,\mu)$ be sequences of non-negative uniformly bounded sequences, bounded by $1$,...
- 310
2
votes
1
answer
142
views
$\sigma$-finiteness of measures and separability of $L^p$ spaces
The fact that a measure $\mu$ is $\sigma$-finite determines or not the separability of $L^2(\mu)$? I proved that $L^2([0,1])$ endowed with the counting measure $m$ is not separable since it admits an ...
- 45
4
votes
0
answers
113
views
Relationship between the diameter of a set and the radius of a closed ball that contains it, in the $L_1$ space
What is the minimal radius $r(d)$ such that for every finite set of diameter $d$ there is a closed ball of radius $r(d)$ that contains it? This in the $L^1$ space with $n$ dimensions.
In formulas, ...
- 141
2
votes
0
answers
36
views
Intersection of infinitely many $L^p$ spaces over $\mathbb{R}$: reference-request
Consider $$X:=\bigcap_{1\le p<+\infty} L^p(\mathbb{R}).$$
Due to the interpolation property, one can easily write $$X=\bigcap_{n\in\mathbb{N}} L^n(\mathbb{R}),$$ so it is not difficoult to see $X$ ...
- 6,630
1
vote
1
answer
65
views
How does completion affect the topology?
Given a set $X$ there are two ways to turn it into a topological space, first, specify the convergence (of nets), second, specify the topology. The two ways are equivalent.
Let $X=C(0,1)$ equipped ...
- 327
0
votes
0
answers
19
views
Is $\{ \psi\in L^\infty(\sigma) \mid \operatorname{ess\,inf} \vert\psi\vert >0 \}$ meagre/nonmeagre/comeagre?
Let $\sigma\subset[0,1]$ be a measurable set with positive Lebesgue-measure. The set
$$
\{ \psi\in L^\infty(\sigma) \mid \operatorname{ess inf} \vert\psi\vert >0 \}
$$
is dense in $L^2(\sigma)$, ...
- 11
1
vote
1
answer
304
views
Let $X$ be a locally compact Polish space. Is the space of continuous functions with compact support dense in that of $\mu$-integrable functions?
I'm reading this question for which I would like to clarify the theorem mentioned there. We have
(S1) Let $X$ be a locally compact Hausdorff space. Then the space of continuous functions with compact ...
- 17.6k
1
vote
0
answers
41
views
Closed subset in different norms
Let us say I have a space of functions $X$ (e.g. $X=C([0,1],[0,1])$ continuous from the unit interval into itself) and two subspaces $S_1$ and $S_2$ such that $S_2\subset S_1(\subset X)$ (e.g. $S_i=L^...
- 1,139
2
votes
0
answers
81
views
How to prove a claim about mapping closed sets to subsets of open sets of $\ell^2$
In class, I have seen the claim below, but for $\mathbb{R^n}$. The question is, whether it still applies for $\ell^2$ and how to prove it.
The claim for $\ell^2$
Let $E, F$ be closed, disjoint sets in ...
- 2,452
0
votes
0
answers
70
views
How is the topology on $\ell^2$ and $\ell^p$ spaces called?
I am looking for a specific name for the topology induced by metric on $\ell^2$. This space has its "standard" topology generated by $\sqrt{\sum_{i=1}^{\infty}(x_i-y_i)^2}$. With such ...
- 2,452
0
votes
2
answers
52
views
A question on the size of the topology with respect to the sequential convergence.
Let $\tau_1$ and $\tau_2$ be two topologies on a non-empty set $X.$ Let the sequential convergence with respect to one of the topologies be equivalent to the sequential convergence with respect to the ...
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