Closed Linear spaces and the direct sum

Question

Prove that if M and N are closed linear spaces and $M \perp N$, then $M \oplus N$ is a closed linear space.

I'm having trouble starting this one. Do I need to consider cauchy sequences in each M and N?

Answer 1

Let $P_M,P_N$ be orthogonal projections onto $M,N$ respectively.

Suppose $x_k \in M \oplus N$, $x_k \to x$. Then $x_k= m_k+n_k$, with $m_k \in M, n_k \in N$.

Then $m_k =P_M x_k \to P_M x, n_k = P_N x_k \to P_N x$, hence $x = P_M x + P_N x \in M \oplus N$.

Note: I should emphasise that we need $M,N$ to be orthogonal for the above to work. Since $N \subset M^\bot$, we have $P_M x_k = m_k$, and similarly for $P_N$.