Having trouble with this problem. Any ideas?
Let $\Omega$ be a measure space. Let $f_n$ be a sequence in $L^p(\Omega)$ with $1
Prove that $\|f_n-f\|_p \to 0$.
Also, can you come up with a counter-example for the $L^1$ case?
Having trouble with this problem. Any ideas?
Let $\Omega$ be a measure space. Let $f_n$ be a sequence in $L^p(\Omega)$ with $1
Prove that $\|f_n-f\|_p \to 0$.
Also, can you come up with a counter-example for the $L^1$ case?
Since $1
Theorem. Let $E$ be a uniformly convex Banach space, and let $\{x_n\}$ be a weakly convergent sequence in $E$, i.e. $x_n \rightharpoonup x$ for some $x \in E$. If $$\limsup_{n \to +\infty} \|x_n\| \leq \|x\|,$$ then $x_n \to x$ strongly in $E$.
Try to construct a counter-example in the $\sigma(L^1,L^\infty)$ topology.
$p\in\{1,\infty\}$
? – 2012-06-26