Here $c$ means the cardinal number of the continuum.
This is an exercise (4.8.7) in the book of Avner Friedman, 'foundations of modern analysis'.
I am a little bit surprised.
Just consider the $l^2$ Hilbert space. It is well-known that it has a basis $\{e_n = (0,0,\ldots, 1, 0,\ldots )\}$, which is countable. As I understand, $l^2$ is a Banach space, a Hilbert space actually, whose dimension is countably infinite.
In which place am I wrong?