Let $H$ be a Hilbert space, $(u_n)_{n \in N}$ an orthonormal set and $(b_n)_{n \in N}$ a sequence of non-negative numbers, prove that
$M:= \{x | x= \sum_{n} c_nu_n with |c_k| \leq b_k, \forall k \in N \}$
(i) is close (ii) is compact if and only if $\sum_n b_n^2$ converges
For the first thing, i can see that $M= \cap_{k \in N} \{x | x= \sum_{n} c_nu_n with |c_k| \leq b_k\}$ Those might be close sets, but I dont know how to prove that. And for the second point, I cant think anything useful for now.