let K be a non-empty closed convex subspace of an hilbert H, $$ \quad \forall u \in H \quad\exists \overline{u} \in K , ~~ \|u-\overline{u}\|= \min_{v ~\in K}\|u-v\| $$ $$ ~~ ( ~~ \|.\| ~~ : ~~ norm ~~ on ~~ H ~~ and ~~ \overline{u}= \pi_ku ~~ \text{ called the projection of u into K}~~).$$ How can i Prove the following characterization : $$\langle u-\pi_ku,w-\pi_ku\rangle\leq 0. $$ $$\forall w \in K,~~ \langle.,.\rangle ~~\text{ the scalar product in H}$$
2026-05-15 00:06:23.1778803583
How can I prove the following characterization : $\langle u-\pi_ku,w-\pi_ku\rangle\leq 0$?
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