# Rellich selection theorem

Let $D$ be an open subset of $\mathbb{R}^{n}$. If, for a sequence of functions $f_{i}\colon D\to\mathbb{R}$, $i=1,2,\ldots$ there exists a constant $B>0$ such that

 $(\forall i)\qquad\|f_{i}\|_{L^{2}(D)}=\int_{D}f_{i}^{2}\,d^{n}x

and

 $(\forall i)\,(\forall j\in\{1,\ldots n\})\qquad\int_{D}\left({\partial f_{i}% \over\partial x_{j}}\right)^{2}\,d^{n}x

then there exists a subsequence which is convergent in the $L^{2}(D)$ norm.

Title Rellich selection theorem RellichSelectionTheorem 2013-03-22 14:38:55 2013-03-22 14:38:55 rspuzio (6075) rspuzio (6075) 7 rspuzio (6075) Theorem msc 46C05