Let $X$ be a Banach space. A subset $S ⊂ X$ is called a Hamel basis of $X$ if $S$ is linearly independent and every element of X is a finite linear combination of elements of $S$.
(i) Prove that if $X$ is infinite dimensional, then $S$ is uncountable. (Use the fact that every finite dimensional linear subspace of $X$ is closed.)
(ii) Give an example of an infinite dimensional normed linear space which has a countable Hamel basis.
My approach for part (i): I'm trying to prove that $S$ is complete, and $S$ has no isolated points, but failed at any of these. I was trying to use the fact that let dim $(S)=k$, then exist a unique representation of L.I. elements $x_1, x_2,...,x_n \in X$ such that for every $x$ in $X$, $x = a_1x_1+...+a_nx_n + y$ where $y$ in $S$. By setting $x=0$, I can show that the space $X$ has dimension $n+k$, which is finite, so the statement is not true for finite case:P. But I don't understand why when $X$ infinite, this statement can be true. Hope someone can help me with this problem.
For part (ii), completely stacked at these kinds of questions:P