Space of functions from [0,1] to IR and from IR to IR

Question

I am convinced that the cardinality of the set S defined as $\Re^\Re$ is the same as the set Z defined as $[0,1]^\Re$, since we can create an operator $\Phi:S\rightarrow Z$ that sends each function f to f$\circ$h where h is a bijection from [0,1] to $\Re$, and this operator seems to be bijective. My question is, if we now look at the natural vector spaces formed by S and Z, can we say the operator $\Phi$ is a linear map? And if it is, does it imply those two vector spaces are isomorphic?

Answer 1

Sure:\begin{align}\Phi(f_1+f_2)&=(f_1+f_2)\circ h\\&=f_1\circ h+f_2\circ h\\&=\Phi(f_1)+\Phi(f_2)\end{align}and, by a similar argument, $\Phi(\alpha f)=\alpha\Phi(f)$. And $\Phi^{-1}(f)=f\circ h^{-1}$.