Using the Parseval's equation
$$\sum_{n=1}^\infty (f,\phi_{n})^2=||f||^{2}{ }$$
to show that an orthonormal set {${\phi_{n}}$} is closed in a given function space if it is complete on that space.
Also let {${\phi_{n}}$} be an orthnormal set in the space of the continous functions on the interval ${a\leq x\leq b}$ and suppose that the generalized fourier series for a function $f$ in that space converges uniformly to a sum $s(x)$ on that interval.
(a) Show that $s$ and $f$ have the same fourier coefficients wrt {${\phi_{n}(x)}$}.
(b) Use the results in $a$ to show that if {${\phi_{n}(x)}$}. is closed then $s(x)=f(x)$ on the interval ${a\leq x\leq b}$.
What i tried
The Parseval's equation can be rewritten in the form
$$\sum_{n=1}^\infty c_{n}=||f||^{2}{ }$$
then
$$||f||^{2}-\sum_{n-1}^\infty c_{n}{ }=||f-S_{n} ||^{2}=0$$ as limit of $N$ tends to infinity and that $S_{n}$ are the partial sums of a generatized Fourier series I dont quite understand this question and this thing about Parseval's equation. Could someone please explain this whole question to me. Thanks