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.

Statement

Let be a normed vector space, a subspace of, and a linear projector on. Then for each in :
The proof is a one-line application of the triangle inequality: for any in, by writing as, it follows that
where the last inequality uses the fact that together with the definition of the operator norm.