Tagged Questions
2
votes
1answer
49 views
When is a subset of $\ell^2$ compact?
I have been looking on the internet for hours now and even asking in chat without an answer. When is a set $M\subseteq\ell^2$ compact? For $L^p$, there is the Arzelà –Ascoli theorem that provides a ...
2
votes
2answers
62 views
Compact inclusion in $L^p$
Is it true that there is a compact inclusion from $L^p$ to $L^q$ whith $q<p$?
What is the counterexample if what I said is wrong?
Thank you.
1
vote
1answer
24 views
Prove that $ A\subset \ell_1 $ is compact iff $A$ satisfies the following property
$A$ is compact iff $ A $ is bounded and, given $\epsilon > 0$, there exists $ n_0$ such that $ \sum_ {k=n}^\infty |x_k|\le\epsilon $ for all $n \geq n_0 $ and for all $ x\in A $.
To prove ...
3
votes
1answer
75 views
$W^{1,p}$ compact in $L^\infty$?
Is $W^{1,p}(0,1)$ compactly contained in $L^\infty(0,1)$? Can I use this to show that I can select a sequence $(u_{n_k})$ from every bounded sequence $(u_n)$ in $W^{1,p}(0,1)$ such that $\lVert ...
5
votes
2answers
119 views
Closed set in $\ell^1$
Show that the set $$ B = \left\lbrace(x_n) \in \ell^1 : \sum_{n\geq 1} n|x_n|\leq 1\right\rbrace$$
is compact in $\ell^1$.
Hint: You can use without proof the diagonalization process to ...
3
votes
1answer
132 views
Two Real Analysis Questions
If I have $ A = \{a \in \ell_2 : |a(n)| \leqslant c(n)\}$ for $c(n)\geqslant 0$ where $ n \in N $, and I want to show that is $A$ compact in $\ell_2$ iff $\sum{c(n)^2}<\infty$. How do I go about ...
2
votes
1answer
138 views
Bounded sequences that form compact sets or not
a) Give an example of a bounded closed subset of
$$ A = \{(x_n) \in \ell^1: \sum_{n\geq1} x_n = 1\}$$
which is not compact. The metric we consider on A is induced by the normal norm on ...
