(i) Let $H$ be a Hilbert space, and let $M$ be a non-zero closed vector subspace of $H$. For $x \in H$, show that there is a unique closest point $P_{M}(x)$ to $x$ in $M$.

(ii) (a) Let $x \in H$. Show that $x-P_{M}(x) \in M^{\perp}$. Show also that if $y \in M$ and $x-y \in M^{\perp}$ then $y=P_{M}(x)$.

(b) Deduce that $H=M \bigoplus M^{\perp}$.

(c) Show that the map $P_{M}$ from $H$ to $M$ is a continuous linear map, with $\left\|P_{M}\right\|=1$.

(d) Show that $P_{M}$ is the projection onto $M$ along $M^{\perp}$.

Now suppose that $A$ is a subspace of $H$ that is not necessarily closed. Explain why $A^{\perp}=\{0\}$ implies that $A$ is dense in $H .$

Give an example of a subspace of $l^{2}$ that is dense in $l^{2}$ but is not equal to $l^{2}$.