Construct a finite $\epsilon$-net for $X = \{u(x) \in L^2([0,2\pi]) \vert u(x) = \sum_n a_n e^{inx} \mbox{ and } |a_n| \leq (1+|n|)^{-1}\}$

Question

As the title states, I am trying to construct a finite $\epsilon$-net in $\ L^2([0,2\pi])\ $ for

$$X\ =\ \left\{\sum_{n=1}^\infty a_n e^{inx} :\ \forall_{n=1}^\infty \ |a_n| \leq (1+|n|)^{-1}\right\}.$$

I have arrived at this question because I am trying to show that $X$ is compact. I have already shown that space is complete. However, I'm not sure how to proceed.

Hints or guidance on how to think about finite $\epsilon$-nets would be appreciated.

Answer 1

Let $\ \epsilon>0\ $ be arbitrary. Select $\ N\ $ such that

$$ \sum_{n=N+1}^\infty \frac 1{(1+n)^2} \ <\ \frac{\epsilon^2}4 $$

Now, there is a finite $\frac\epsilon 2$-net $\ E\ $ in the cartesian product of closed discs around center $\ \mathbb 0$:

$$ E\ \subseteq \ \prod_{n=1}^N D_\frac 1{1+n}(\mathbb 0) $$ Let

$$ F\ :=\ E\times{\{(0\ 0\ \ldots)\}} \ \subseteq\ \ell^2 $$ Thus $F$ is an $\epsilon$-net in

$$ \prod_{n=1}^\infty D_\frac 1{1+n}(\mathbb 0) \subseteq\ \ell^2 $$

Now, so conveniently, functions $\ e^{2\cdot\pi\cdot x}\ $ are orthogonal and of norm $1$ (make sure to set the constant right :-)). This provides a canonical isometry from $\ \ell^2\ $ to $\ L^2([0;2\!\cdot\!\pi]).\ $ (The rest is obvious, I hope).