Let $U\subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $f_k\in L^p(U)$. Does weak convergence of $f_k$ to $f \in L^p$ implies $L^p$-convergence of $f_k$ to $f$? By weak convergence I mean that $$ \int_U f_kg \,dx \rightarrow \int_U fg\, dx $$ for all $g\in L^q(U)$ where $\frac{1}{p}+\frac{1}{q}=1$.
Weak convergence implies $L^p$ convergence on a smooth bounded domain?
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functional-analysis
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1For $n=1$, in $L_p([0,2\pi])$ $p\ge 1$, a standard counterexample to the claim that weakly convergent sequences are norm convergent is the sequence $(f_n)$ defined by $f_n(x)=\cos(nx)$ for each $n\in\Bbb N$. The Riemann-lebesgue Lemma shows this sequence converges weakly to $0$. One can also show that this sequence does not converge to $0$ in the $L_p$ norm. – 2012-12-14