Solution
= Solution
The upper frame bound gives $\|Ux\|_{\ell^2}\leq\sqrt B\|x\|$. For $c\in\ell^2$, the adjoint is the norm-convergent synthesis series
$$
U^*c=\sum_i c_i e_i,
$$
because its inner product with $x$ is $\sum_i c_i\langle e_i,x\rangle$.