I have a question about the orthonormal set.
Let $f\in \mathcal{L}^2(\mu)$ (i.e. $f$ is measurable and $\int_X|f|^2d\mu<\infty$) on a measurable space $X$ such that $\Vert f \Vert=1$ be given. ($\Vert \cdot \Vert$ denotes inner product.)
I wonder whether I can make the orthonormal set with that given $f$.
I thought I could. My approach is this: For some orthonormal set {$\phi_n$} with a countable basis where $\phi_n\in\mathcal{L}^2(\mu)$, I can make approximation by $$ f\sim\sum_{n=1}^\infty c_n\phi_n $$ where $c_n=\int_Xf\bar{\phi_n}d\mu$.
Then I may replace $\phi_1$ with $f$, $\phi_2$ with some function made up with $\phi_2, \phi_3, \cdots$, say $g$, such that $(f,g)=0$, $\phi_3$ with some function made up with $\phi_3, \phi_4, \cdots$, say $h$, such that $(f,h)=(g,h)=0$, and so on.
By iterating the fashion above, I thought that I could finally get what I want... but I'm not sure.
Is my approach right?