For any natural number $n$, $V_n$ denotes a closed linear subspace of a $L_2(m)$ space, which is an Hilbert Space, where $m$ denotes a finite measure. Moreover $(V_n)$ is increasing, that is $V_n$ is a subspace of $V_{n+1}$ for any $n$. Denote by $P_n$ the orthogonal projection on $V_n$ and by $P_V$ the projection on the closure $V$ of the union of all the $V_n$. Is it true that $P_n$ converges to $P_V$ in the usual operator norm? Or does it converges just for some weaker notion of convergence? What conditions are required to ensure this?