For any natural number n, V(n) denotes a closed linear subspace of a L2(m) Space, which is an Hilbert Space, where m denotes a finite mesure. 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 PV the projection on the closure V of the union of all the V(n). Is it true that p(n) converges to PV in the usual operator norm  ? Or does it converges just for some weaker notion of convergence  ? Which conditions are required to ensure this  ?

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?