Proving that a subset of a Hilbert space is convex and closed

Question

Let $(X, \langle\,\cdot\,,\cdot\,\rangle)$ be a Hilbert space and let $l \in X^* = \mathcal{L}(X, \mathbb{R})$ and let $\gamma \in \mathbb{R}$ and let $C = \{x\in X \mid l(x) \leq \gamma\}$. So that's my problem. I am proving that every $x \in X$ has a unique projection in $C$. So for that I need to prove that $C$ is closed and convex. Can I say that it is convex because it is the preimage of a closed interval through a lineal continuous function and that it is closed because that same function is continuous? Am I right or too far off? Any help or confirmation of what I am saying would be greatly appreciated.