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 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\|.

See also

Lebesgue constant (interpolation)

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