Let $\big(H,\langle.,.\rangle \big)$ be a separable Hilbert space, with an orthonormal basis $\{u_n\}_{n \in \mathbb{N}}$, and set $V_N = <u_0, \dots, u_N >$ (the set of all the linear combinations generated by $\{u_n\}_{n=1}^{N}$).
Let $P_N:X \rightarrow V_N$ be the projector over $V_N$.
How can be proved that $P_N$ does NOT converge (as $N \rightarrow \infty$) in the operator norm?
I have proved that $P_N$ is linear and bounded for every $N$. But I still miss the more important part. Thank you :)