Suppose M is any linear manifold in H. H is a hilbert space. Define the orthogonal complement of M to be $$M' =\{f \in H | \langle f,g\rangle= 0 ,\forall g\in M\}.$$
To see that M' is a closed linear manifold, suppose $\{u_n\}$ is a sequence in M', and that $\lim_{n\to \infty}u_n=u$ is in $H$. For every $g\in M$ $$\langle u,g\rangle =\langle\lim_{n\to \infty}u_n,g\rangle = \lim_{n\to\infty} \langle u,g\rangle =0.$$
The interchange of limits here is valid since (once again, by the Schwarz inequality) the linear operator $\langle u, g\rangle$ is a bounded linear operator for any fixed $g\in M$.
I dont understand the "interchange of limits" Where is the second limit?
It's just the English that is messing up with you. The plural is used like when you say "crossing streets without paying attention is dangerous"; where's the "second street"?
And interchange refers to exchanging the position of $\lim$ and $\langle$.