Inmathematics, 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 Palinear projectoronU. Then for each vinV:
The proof is a one-line application of the triangle inequality: for any uinU, by writing v − Pvas(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||.
