For $w=P_Wv$ and any $u\in W$, orthogonality gives $\|v-u\|^2=\|v-w\|^2+\|w-u\|^2$. The second term is nonnegative and vanishes only at $u=w$. Thus $w$ is the unique closest point in $W$. This argument applies to finite-dimensional subspaces.
Parents
Like how this is organised?
Build your own knowledge tree on knowen.md: notes, decisions and tasks in one connected markdown graph, shared with your team and your AI agents.