Lebesgue's 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
Let $(V, ·)$ be a normed vector space, $U$ a subspace of $V$, and $P$ a linear projector on $U$. Then for each $v$ in $V$:
 * $$ \|v-Pv\|\leq (1+\|P\|)\inf_{u\in U}\|v-u\|.$$

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 $P$.