Suppose $P_t$ is a semigroup of bounded self-adjoint operators on a Hilbert space $E$. Let $F$ be a dense subspace of $E$ such that for all $x\in F$:
$$
\lim_{t\to0}\la P_tx,x\ra=\norm x^2~.
$$
Then $P_t$ is a continuous semigroup on $E$.
By assumption we have for all $x,y\in F$:
$$
\lim_{t\to0}\norm{P_tx}^2=\norm x^2
\quad\mbox{and thus}\quad
\lim_{t\to0}\la P_tx,y\ra=\la x,y\ra~.
$$
It follows that for all $x\in F$:
$$
\lim_{t\to0}\norm{P_tx-x}^2
=\lim_{t\to0}\norm{P_tx}^2+\norm x^2-2\la P_tx,x\ra=0~.
$$