Lebesgue's lemma

For Lebesgue's lemma for open covers of compact spaces in topology see Lebesgue's number lemma

In mathematics, Lebesgue's lemma is an important statement in approximation theory. It provides a bound for the projection error.

[edit] Statement

Let (V, ||·||) be a normed vector space, U_i be a subspace of V and let $P$ be a linear projector on $U$ . Then, for each v in V:

$\|v-Pv\|\leq (1+\|P\|)\inf_{u\in U}\|v-u\|.$