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.

Statement

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

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

See also

• Lebesgue constant (interpolation)