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enter image description hereUsing 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

ys wong
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    What do you mean by an orthonormal set being "closed?" – Disintegrating By Parts Sep 01 '16 at 18:26
  • where does this "exercice" comes ? it is completely unclear – reuns Sep 07 '16 at 10:39
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    so for the Fourier series : prove $\phi(n) = e^{2 i \pi n x}$ is an orthonormal family of $L^2([0,1])$, so that if $f = \sum_n (f,\phi_n) \phi_n$ then $f = \sum_n (f,\phi_n) \phi_n$ (yes this is a tautology but it is important). Then, show that $(g,\phi_n) = 0$ for every $n \implies g = 0$, i.e. that the subspace generated by ${\phi_n}$ is dense in $L^2([0,1])$ (it is enough to show that $1_{x \in [a,b]}$ has a Fourier series, and then to say the subspace generated by the step functions is dense in $L^2([0,1])$). – reuns Sep 07 '16 at 10:42
  • Finally for an arbitrary $f \in L^2([0,1])$, let $g = f - \sum_n (f,\phi_n)\phi_n$. We have $(g,\phi_n) = 0$ for every $n$, hence by the previous theorem $g= 0$. i.e. : $$f = \sum_n (f,\phi_n)\phi_n$$ voila you proved the Fourier series – reuns Sep 07 '16 at 10:43
  • The original question is posted above. Qn 9 – ys wong Sep 07 '16 at 10:59
  • can you share your pdf ? it seems there is a huge problem in it (any finite dimensional subspace is closed) – reuns Sep 07 '16 at 11:29

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