Lebesgue's lemma

In mathematics, Lebesgue's lemma is an important statement in approximation theory.

It provides a bound for the projection error, controlling the error of approximation by a linear subspace based on a linear projection relative to the optimal error together with the operator norm of the projection.

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: The proof is a one-line application of the triangle inequality: for any u in U, by writing v − Pv as (v − u) + (u − Pu) + P(u − v), it follows that where the last inequality uses the fact that u = Pu together with the definition of the operator norm ||P||.