>> The dimensionality of L^2(R) is still countable.
At this point I was implicitly talking about Linf(R). Does it still have countable dimension? With the sup norm I'm pretty sure the answer is no, but with Lp taken at p->inf maybe it does?
In any case, thanks for cleaning up the rough edges.
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.
At this point I was implicitly talking about Linf(R). Does it still have countable dimension? With the sup norm I'm pretty sure the answer is no, but with Lp taken at p->inf maybe it does?
In any case, thanks for cleaning up the rough edges.