Let $H$ be an infinite dimensional Hilbert space over $\mathbb{C}$

Let $\{v_n\}_{n \in \mathbb{N}} \subset H$ be a sequence of linearly independent vectors in $H$ such that $v_n \to u$

Let $\forall m \in \mathbb{N}: V_m = \operatorname{span} \{v_n\}_{n \geq m}$ and $P_m$ be the orthogonal projection on $V_m$

My question is if it is true that: $$ \forall v \in V_1: \lim_{m \to \infty} P_m(v)= a \cdot u $$ in $H$-norm and with $a \in \mathbb{C}$

Thanks.

closedspan of $\{v_n: \, n \ge m\}$? Since otherwise there is no orthogonal projection onto $V_m$, in general. $\endgroup$ – Jochen Glueck Feb 24 at 11:31