Is every basis a frame? It's clear that every orthonormal basis is a frame, but is every basis a frame? In particular, is a non-orthogonal basis always a frame? If yes, why?
I'm asking about general Hilbert spaces, in particular infinite dimensional spaces.

Let $(e_n)_{n\in \mathbb N}$ be an ONB for a Hilbert space. $(\frac{1}{n}e_n)_{n\in \mathbb N}$ is spanning and representation is unique, i.e., it is a basis. But it is not a frame, since 
$$
\sum_{n\in \mathbb N} \left|\langle e_k, \frac{1}{n}e_n \rangle\right|^2 \leq \frac{1}{k^2}.
$$
