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<B

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<B

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

Title	Rellich selection theorem
Canonical name	RellichSelectionTheorem
Date of creation	2013-03-22 14:38:55
Last modified on	2013-03-22 14:38:55
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	7
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 46C05