Can we estimate the dimension of some function space?
e.g. if $X$ is a locally compact Hausdorff space,
what is the (linear) dimension of $C(X)$(as a linear space), which represents the set of all continuous function on $X$?
what is the dimension of $C_{c}(X)$(as a linear space), , which denote the set of continuous, complex-valued functions on $X$ with compact support?
Or can we conclude that if $X$ is a locally compact Hausdorff space, then $dimC_{c}(X) \geq 1$ ?
I'm more concerned about the second problem. I need such conclusion "if $X$ is ...,then $dimC_{c}(X)>=1$, and there exist a non-zero linear functional on $C_{c}(X)$."
Thanks a lot.