We want to use that every Hilbert space has a Hamel basis (where a Hamel basis of $H$ is a set $V \subseteq H$ such that $V$ is linearly independent and such that every element of $H$ can be written uniquely as a finite linear combination of V's elements) in order to provide an unbounded linear functional, $f$ on $H$.
My understanding from linear functionals is that we want to find a mapping, $f$, from $H$ to $C$. And, finding an unbounded function would require us to prove that there is no constant $M$ such that $||f(x)|| \leq M(x)$ for all $x \in H$, or that for every such $M$ there would be another such $Q > M$.
How can I do this using the information about every Hilbert space has a Hamel basis? I am just beginning to learn about Hilbert spaces and this is the first I've heard of a Hamel basis, so please help me to understand? Thank you.
Choose a countable subset of the Hamel basis, $b_n$ and define $f(b_n) = n \|b_n\|$ and zero for all the other elements of the basis.