The natural question which now arises is whether the Fourier series of $f$ converges to $f$, or, more generally, whether $f$ is determined by its Fourier series. That is to say, if we know the Fourier coefficients of a function, can we find the function, and if so, how? This was on page 186 and then on page 187 ~
If $\phi_n(x)$ is orthonormal on $[a, b]$ and if
$c_n =\int_a^b f(t) \overline{\phi_n(x)}dt $
we call $c_n$ the $n$th Fourier coefficient off relative to $\phi_n(x)$. We write
$f(x)\color{red}{\sim} \sum c_n \phi_n(x)$
and call this series the Fourier series off (relative to $\phi_n(x)$ ). Note that the symbol $\sim$ implies nothing about the convergence of the series;
Then in theorem 8.11 rudin wrote this
Let {$\phi_n$} be orthonormal on $[a,b]$. Let $s_n(x) = \sum_{m=1}^n c_m\phi_m(x)$ be the nth partial sum of the Fourier series of $f$, and suppose $t_n(x) = \sum_{m=1}^n y_m\phi_m(x)$.
$$\int f \overline{t_n} =\int f \sum \overline{ y_m\phi_m(x)} \color{red}{=} \sum c_m \overline{y_m} $$
but this is only possible when $f(x)\color{red}{=} \sum c_n \phi(x)$, so If this is the case how the symbol $\sim$ implies nothing about the convergence of the series? and How could this relation be an equality ?