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, U a subspace of V, and P a linear projector on U. Then for each v in V:

‖ v − P v ‖ ≤ ( 1 + ‖ P ‖ ) inf u ∈ U ‖ v − u ‖ . {\displaystyle \|v-Pv\|\leq (1+\|P\|)\inf _{u\in U}\|v-u\|.}

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

‖ v − P v ‖ ≤ ‖ v − u ‖ + ‖ u − P u ‖ + ‖ P ( u − v ) ‖ ≤ ( 1 + ‖ P ‖ ) ‖ u − v ‖ {\displaystyle \|v-Pv\|\leq \|v-u\|+\|u-Pu\|+\|P(u-v)\|\leq (1+\|P\|)\|u-v\|}

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

See also

  • Lebesgue constants

References

  • DeVore, Ronald A.; Lorentz, George G. (1993). Constructive approximation. Grundlehren der mathematischen Wissenschaften. Vol. 303. Berlin: Springer-Verlag. ISBN 3-540-50627-6. MR 1261635. Zbl 0797.41016.