Sequence of unitary l.i. vectors such tha the sequence converges weakly to a non-zero vector, but not strongly

Question

Let $\mathcal H$ be an infinite dimensional Hilbert space and let $\{x_{n}\}_{n=1}^{\infty}$ be a sequence of unitary linearly independent vectors. I know, using Bessel's inequality, that if the sequence is orthonormal, then the sequence converges weakly to zero, but not strongly. What will happen if orthonormal property is removed? Can the sequence converge weakly to an unitary vector, but not strongly?

Answer 1

If $x_n$ converges to $x$ weakly and $\|x_n\|\to \|x\|$, then $x_n\to x$ strongly. (Proof: expand $\|x_n-x\|^2 $ and see what happens.)

Therefore, if the weak limit of unit vectors has unit norm, it is in fact a strong limit.

But you can arrange to have a non-zero weak limit $x$ with $\|x\|<1$. Let $x_n = x+ ce_n$ where $e_n$ are standard basis vectors, and constant $c$ is appropriately chosen.