Bounded linear functional on a subspace of a Hilbert space implies uniformly continuous?

Question

If $f: M \to \mathbb{R}$ is a bounded linear functional where $M$ is a subspace of a Hilbert space, then is it the case that $f$ is uniformly continuous? I definitely believe that $f$ is continuous since its bounded but I feel like uniform continuity is a bit much to expect. This result was used in a proof I am reading and it just seemed to good to be true.