Lebesgue's lemma

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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.


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

 \|v-Pv\|\leq (1+\|P\|)\inf_{u\in U}\|v-u\|.

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