any orthogonal system $\{\phi_{\alpha}\}$ in $L^2$ is countable

Question

I'm trying to follow the proof of this theorem:

any orthogonal system $\{\phi_{\alpha}\}$ in $L^2$ is countable.

the proof in the book(by Wheeden & Zygmund) is pretty simple; it just shows that $||\phi_{\alpha}-\phi_{\beta}||^2=2$ if $\alpha \neq \beta$ (after assuming the system is orthonormal) and conclude by the separability of $L^2$ space.

I only know the definition of the separable space: if a space is separable, it contains a countable dense subset. but I don't know how to use this to relate with above argument.

Answer 1

If $\|a-b\|^2=2$ then the open balls $B(a,1/\sqrt 2\;),\; B(b,1/\sqrt 2)$ are disjoint. Let $F$ be a non-empty orthonormal set and let $D$ be a dense set. Then $D$ must have a member $d_f$ in $B(f,1/\sqrt 2)$ for each $f\in F$. But $d_f\ne d_g$ when $f,g \in F$ with $f\ne g,\;$...( because $B(f,1/\sqrt 2\;)$ and $B(g,1/\sqrt 2\;)$ are disjoint)... So the cardinal of $D$ is greater than or equal to the cardinal of $F$. So if $D$ is countable then so is $F$.