Lebesgue's lemma

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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

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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:

 

The proof is a one-line application of the triangle inequality: for any u in U, by writing vPv as (vu) + (uPu) + P(uv), it follows that

 

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

See also

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References

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  • 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.