If I understood it well (did I?), Byron Schmuland's solution needs no additional conditions on your space $X$, since it relies on the theorem of characterization of maximal orthonormal systems (of arbitrary cardinal), although you'd have to exhibit one such system inside the bounded and uniformly continuous functions.
An alternate solution could use Lusin's theorem, for which you need to throw in local compactness of $X$. For the relevant setup, you can follow Rudin's "Real and complex analysis", theorems 2.24 and 3.14 in the third edition. The advantage here is that you get your result cheaply (local compactness is not that weird for a base space ;-) and for any $1 \leq p < \infty$, not just for the Hilbert case.