# fa.functional analysis – An orthogonal projection sequence in the Hilbert space

Let $$H$$ to be an infinite dimensional space of Hilbert on $$mathbb {C}$$

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

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

My question is to know 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 $$a in mathbb {C}$$

Thank you.