Don't know off the top of my head. I suspect it does have countable dimension but I don't know how to prove it.
The space of continuous functions with the sup norm is actually a much smaller space than the space of L^\infty - the former is not even dense in the latter.
I suspect actual functions on R with the max norm probably is uncountable, but that's also a very weird space. The overwhelming majority of functions in there are unmeasurable.
The space of continuous functions with the sup norm is actually a much smaller space than the space of L^\infty - the former is not even dense in the latter.
I suspect actual functions on R with the max norm probably is uncountable, but that's also a very weird space. The overwhelming majority of functions in there are unmeasurable.