Let $\pi : A \longrightarrow \mathcal L (\mathcal H)$ be a representation of a $C^{\ast}$-algebra $A$ on a Hilbert space $\mathcal H.$ Show that $\pi$ is non-degenerate i.e. $\bigcap\limits_{a \in A} \text {ker}\ \pi (a) = \{0\}$ iff $\pi (A) \mathcal H$ is dense in $\mathcal H,$ where $\pi (A) \mathcal H$ is a subspace of $\mathcal H$ given by $$\pi (A) \mathcal H : = \text {span} \left \{\pi (a) h\ \big |\ a \in A, h \in \mathcal H \right \}.$$
This result is given as a remark in one of the notes (page no. $49$) I have found online but I am having hard time in proving this equivalence. Could anyone suggest something needful?
Warm Regards.
The biggest hint would be to prove that $$\bigcap_{a\in A}\ker\pi(a)=(\pi(A)\mathcal H)^\perp.$$ Once you have this, prove both directions by proving their contrapositives.