Lebesgue's_lemma Knowpia

For Lebesgue's lemma for open covers of compact spaces in topology see Lebesgue's number 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 edit

Let $(V, ||\cdot||)$ be a normed vector space, $U$ a subspace of $V$ , and $P$ a linear projector on $U$ . Then for each $v$ in $V$ :

\|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-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 $|| P ||$ .

References edit

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.

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