4
$\begingroup$

Let $H$ be an complex infinite dimensional Hilbert space

Let $\{H_n\}_{n \in \Bbb N}$ be a sequence of subspaces of $H$ such that $\cap_{n=1}^\infty H_n = H_0$, where $H_0$ is a one dimensional subspace of $H$ and such that $H_{n+1}\subsetneq H_n$

Let, for each $n \in \Bbb N$, be $v_n \in H_n$.

My questions are: is $\{v_n\}$ convergent ? if it is convergent, is $v_n \to v_0 \in H_0$ ?

Thanks for any suggestion

  • 2
    The sequence $\{v_n\}$ need not converge: set $v_n:=n\cdot v_0$ for $n\ge 1$. – 2017-01-14
  • 0
    Even if $H_0=\{0\}$ using a similar idea. – 2017-01-14
  • 0
    It's not true even if you require $\|v_n\|\leq C<\infty$ $\forall n$. Take $H=\ell^2$ with basis $\{e_1,e_2,\ldots\}$, $H_n=span\{e_0,e_n,e_{n+1},\ldots\}$ and $v_n=e_n$. – 2017-01-18

0 Answers 0