I am taking a Quantum Mechanics course not a Functional Analysis course so I have only had a very basic introduction to Hilbert Spaces.
I don't understand where the if and only if statement arises in the above theorem. How can I rewrite it so that the writing underneath the theorem makes sense?
To clarify it, I shall restate the theorem:
Suppose $f$ is a continuous linear functional on a Hilbert space $H$. Then there is a unique $y\in H$ such that $$ f(x) = (x,y) $$ for any $x\in H$.
$(x,y)$ denotes the inner product of $x$ and $y$, which you probably recognize as $\langle y|x \rangle$.
Note that for any fixed $y\in H$, the mapping $f:H \to \mathbb{C}$ defined by $f(x) = (x,y)$ is a continuous linear functional.
By the way, since it's not mentioned in your book, I want to point out that linear functionals on an infinite-dimensional space $H$ do not have to be continuous. The dual space of $H$ consists of only the continuous ones.