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## Homework Statement

Consider the functions [tex]f_n(x)=e^{inx},n\in\mathbb{Z},-\pi\leq x\leq\pi[/tex] viewed as points in [tex]\mathscr{L}^2[-\pi,\pi][/tex]. Prove that this set of functions is closed and bounded, but not compact.

**2. The attempt at a solution**

I'm first trying to prove that the set of functions is closed. Let

*g*be a limit point of the set, so given any positive number epsilon, there exists

*f*

_{n}such that [tex]\|f_n-g\|_2<\sqrt{\epsilon}[/tex]. Squaring both sides, I get on the left hand side

[tex]

2\pi -2\text{Re}\int_{-\pi}^{\pi}g(x)e^{-inx}dx+\int_{-\pi}^{\pi}|g(x)|^2 dx

[/tex]

So

[tex]

g\sim\sum\limits_{-\infty}^{\infty}c_n e^{inx}

[/tex]

and by Parseval's Theorem,

[tex]

\sum\limits_{-\infty}^{\infty}|c_n|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|g|^2 dx

[/tex]

I don't know where to go from here.