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.



A sequence of orthogonal projection in Hilbert space
The answer is **no**, in general.


As a counterexample, let $H = L^2([0,1])$, let $(q\_n)\_{n \in \mathbb{N}}$ be your favourite enumeration of $[0,1] \cap \mathbb{Q}$ and define
\begin{align\*}
 v\_n := 1 + \frac{1}{n} 1\_{[0,q\_n]}
\end{align\*}
for each $n \in \mathbb{N}$.


Then the span of $\{v\_n: \, n \ge m\}$ is dense in $L^2([0,1])$ for each $m$ (since $\{q\_n: \, n \ge m\}$ is dense in $[0,1]$) and hence $P\_m$ is the identity operator. However, $v\_n$ converges to the constant function with value $1$ as $n \to \infty$.


