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 (V, ||·||) be a normed vector space, Template:Mvar a subspace of Template:Mvar, and Template:Mvar a linear projector on Template:Mvar. Then for each Template:Mvar in Template:Mvar:

$\Vert v-Pv\Vert \le (1+\Vert P\Vert )\underset{u\in U}{\inf }\Vert v-u\Vert .$

The proof is a one-line application of the triangle inequality: for any Template:Mvar in Template:Mvar, by writing v − Pv as (v − u) + (u − Pu) + P(u − v), it follows that

$\Vert v-Pv\Vert \le \Vert v-u\Vert +\Vert u-Pu\Vert +\Vert P(u-v)\Vert \le (1+\Vert P\Vert )\Vert u-v\Vert$

where the last inequality uses the fact that u = Pu together with the definition of the operator norm ||P||.

See also

• Lebesgue constants