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 :
:<math> \|v-Pv\|\leq (1+\|P\|)\inf_{u\in U}\|v-u\|.</math>
The proof is a one-line application of the triangle inequality: for any in , by writing as , it follows that
:<math>\|v-Pv\|\leq\|v-u\|+\|u-Pu\|+\|P(u-v)\|\leq(1+\|P\|)\|u-v\|</math>
where the last inequality uses the fact that together with the definition of the operator norm .
See also
- Lebesgue constants