Lebesgue's lemma explained

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 :

\|v - Pv\|\leq ( 1+ \|P\|) inf _{u\in} \|v - u\|. The proof is a one-line application of the triangle inequality: for any in, by writing as, it follows that\|v - Pv\|\leq\|v - u\| + \|u - Pu\| + \|P(u - v)\|\leq( 1+ \|P\|)\|u - v\| where the last inequality uses the fact that together with the definition of the operator norm .

See also

• Lebesgue constants

References

• Book: 1261635. DeVore. Ronald A.. Lorentz. George G.. Constructive approximation. Grundlehren der mathematischen Wissenschaften. 303. Springer-Verlag. Berlin. 1993. 3-540-50627-6. 0797.41016. Ronald DeVore. George G. Lorentz.