Let S be a closed linear subspace of a Hilbert space $H$, and $P_S$ the associated orthogonal projection. I need to verify the following properties.

i) $||P_S(x)||\le||x||$

ii)$P_{S^\perp}=I-P_S$, where I is the identity operator on $H$

For part (ii) it says to first prove that $(K^\perp)^\perp=\overline{\text{span} K}$ in Hilbert spaces, I have done this but I am not sure what to do next.


Since $S$ is a closed linear subspace of $H$ you have $H = S \oplus S^\perp$. Thus any $x \in H$ has the form $x = s + t$ where $s \in S$ and $t \in S^\perp$. But $s$ and $t$ are orthogonal so $\|x\|^2 = \|s\|^2 + \|t\|^2$, and in particular $\|s\| \le \|x\|$.

With this notation, if $x = s + t$ then $P_S(x) = s$, and thus $\|P_S(x)\| = \|s\| \le \|x\|$.

Moreover, since $S^\perp$ is a closed linear subspace you have $H = S^\perp \oplus S^{\perp \perp}$, but (assuming $S$ was closed) you have $S^{\perp \perp} = S$. Thus $P_{S^\perp}(x) = t$ and $$ x = P_S(x) + P_{S^\perp}(x).$$


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