What is the cardinality of the continuous dual of $C([0,1])$ (the set of continuous functions from $[0,1]\to \mathbb{R}$)?



Cardinality of $C^*([0,1])$
The polynomial space $P([0,1])$ is dense in $C([0,1])$ by the Stone-Weierstrass theorem. Therefore $P$ and $C$ have the same (continuous) dual space. But since $P$ has a countable Hamel basis, its *algebraic* dual has the same cardinality as $\mathbb R^{\mathbb N}$. Thus $|C^\*|\leq|\mathbb R^{\mathbb N}|$. The evaluation functionals $\phi\_a\in C^\*$, $a\in[0,1]$, taking $\phi\_a(f)=f(a)$ are continuous, so $|[0,1]|\leq|C^\*|$. If I'm not mistaken, this implies that $C^\*$ has the same cardinality as the real line. (If my memory serves me well, this depends on the choice of axioms, but is true in ZFC.)


Remark: A very similar argument works for the space $C$ itself.
A continuous function is determined by its values at rational points, and, on the other hand, constant functions are continuous. This gives $|[0,1]|\leq|C^\*([0,1])|\leq|[0,1]^{\mathbb Q}|$. 


