Let $(H, ||.||)$ be a Hilber space with an orthornormal basis. Consider $(u_{n})$ in $H$ such that $||u_{n}|| \rightarrow \infty$. Can I affirm that $||P_{W}(u_{n})|| \rightarrow \infty$ for some subspace $W \subset H$, where dim$(W)< \infty$ ? Here $P_{W}(u_{n})$ denotes the projection of $u_{n}$ onto $W$.
Intuitively the claim seems to be true, however I'm not able to prove it formally. I've tried show that if $|| P_{W}(u_{n}) ||$ is bounded for all finite dimensional subspace $W \subset H$, then $||u_{n}||$ is bounded, but, I didn't get a uniform bound which doesn't depend on the subspace $W$.
You cannot guarantee this. Let $\{e_n: n \in \mathbb{N}\}$ be an orthonormal basis. Consider $u_n = ne_n$. Then $\|u_n\| \to \infty$. However since $\{e_n: n \in \mathbb{N}\}$ is spanning and orthogonal, for any finite dimensional space $W$ $P_W(u_n) = 0$ for large enough $n$. (To see this, use the fact that $\{e_n: n \in \mathbb{N}\}$ is spanning to express a basis for $W$ in terms of finitely many of the $e_n$. Then use orthogonality.)