Projection onto subspaces in Hilbert space

Question

Let $M$ and $N$ be two closed subspaces of the Hilbert space $H$ and $x\in H$. Put $y=P_M(x)$ and $z=P_N(x)$. Is it true that $x$ lies in the subspace spanned by $y$ and $z$? Thanks in advanced.

Answer 1

For this to be true one needs to put some condition on the subspaces $M$ and $N$. Suppose that $H$ is not one-dimensional, and pick an arbitrary vector $v \in H$. Then set $M = N = \text{span}\{ v \}$. Now pick a vector $x \notin M$. In this case the subspace spanned by $y = P_{M}(x)$ and $z = P_{N}(x) = y$ will be either $\{ 0 \}$ or $M$, neither of which contains $x$.

A bit more generally, suppose that there is a non-zero vector $x \in H$, with $P_{M}(x) = 0$ and $P_{N}(x) = 0$, then $x$ will not lie in the span of $P_{M}(x)$ and $P_{n}(x)$.

I am afraid that the precise condition on $M$ and $N$ for your claim to hold is somewhat tautological. Namely, your claim is true if and only if \begin{equation} M + N := \{ y + z \in H| x \in M, z \in N \} \equiv H. \end{equation}

Answer 2

Hint:

consider $M= \mbox{Span}[1,0,0]^T$ and $N= \mbox{Span}[0,1,0]^T$ and the vector $x=[1,1,1]^T$