Define the notions of basis and dimension of a vector space. Prove that two finitedimensional real vector spaces with the same dimension are isomorphic.

In each case below, determine whether the set $S$ is a basis of the real vector space $V:$

(i) $V=\mathbb{C}$ is the complex numbers; $S=\{1, i\}$.

(ii) $V=\mathbb{R}[x]$ is the vector space of all polynomials in $x$ with real coefficients; $S=\{1,(x-1),(x-1)(x-2),(x-1)(x-2)(x-3), \ldots\} .$

(iii) $V=\{f:[0,1] \rightarrow \mathbb{R}\} ; S=\left\{\chi_{p} \mid p \in[0,1]\right\}$, where

$$\chi_{p}(x)= \begin{cases}1 & x=p \\ 0 & x \neq p\end{cases}$$