Consider the Hilbert product space $X\times X$

Question

Consider the Hilbert product space $X\times X$. In $X\times X$ define the closed convex 'diagonal' set by $$D={(x,x):x\in X}$$ Obtain a formula for projection $P_D$ and rigorously prove it.

I really struggle with finding formulas. I am hopping someone can give me some hints with where to start with finding this projection formula.

Thank you

Answer 1

Let $(u,v)\in X\times X$ be given, and find the unique $(x,x)\in D$ such that $$ \langle(u,v)-(x,x),(y,y)\rangle_{X\times X} = 0,\;\;\; y \in X. $$ The point $(x,x)$ is the orthogonal projection of $(u,v)$ onto $D$. The condition is $$ \langle u-x,y\rangle_{X}+\langle v-x,y\rangle_{X}=0, \;\;\; y \in X \\ \langle u+v-2x,y\rangle_{X} = 0,\;\;\; y \in X. $$ Therefore $x = \frac{1}{2}(u+v)$ is the unique solution. So $$ P_{D}(u,v) = \frac{1}{2}(u+v,u+v). $$ This function $P_{D}$ is automatically linear, idempotent (i.e., $P_D^{2}=P_D$), and selfadjoint on $X\times X$.

Answer 2

Hint In the case of $X = \Bbb R$, we end up with the linear transformation $$ A = \frac 1{2}\pmatrix{1&1\\1&1} $$