Is the proof of $V=U+U^\perp$ of the Cornell's teacher's handout problematic?

Question

I just searched a handout written by the professor in Cornell. However, I feel that there is some problem in it.

First, if $V$ is infinite-dimensional, then if $W$ is infinite-dimensional, does $W$ necessarily have an orthonormal basis(related question)? Even if $W$ has an orthonormal basis, say $\beta$, then how can we "extend the orthnormal basis for $W$ to $V$"? Although we can choose "one" of a vector $v\in V-W$, and we know that $v\cup\beta$ is linearly independent, how can we do the Gram-Schmidt process, turning the $v\cup\beta$ to an orthonormal set? Here the $\beta$ is infinite! Does my doubt make sense? And can this proof be modified to be correct?

Answer 1

This is not valid for infinite-dimensional spaces. If $U$ is dense in $V$ then $U^\perp=\{0\}$, so a dense but proper subspace gives a counterexample.