I think that $\{\text{continuous functions on } [0,1]\}$ has a different dimension from $\{\text{continuous functions on } [0,1]:F(0)=0, F\in C^1(0,1)\}$ due to the constraints in the latter. But how do I prove this?
(I realize that this question may be similar to the question Bases of spaces of continuous functions I asked previously, but this is hopefully more concise and I have thought about @Florian's enlightenment that the dimension of the set of continuous functions on a closed interval is infinite.)
Thanks loads!