Let $u_n$ be a sequence in a Hilbert space $H$. Suppose that $u_n=u_n(p)$ depends on a parameter $p \in I \subset \mathbb{R}$.
If I have that $$\lVert u_n(p) \rVert \leq C$$ where $C$ is a constant independent of $n$ and $p$, we know that $u_n(p) \rightharpoonup u(p)$ for some $u(p)$ in the limit as $n \to \infty$.
Is it true that this weak limit is uniform in $p$??
$\newcommand\ip[2]{\langle #1,#2\rangle}$
Assuming that "$u_n(p)\rightharpoonup u(p)$ uniformly" means that $\ip{u_n(p)}x\to \ip{u_p}x$ uniformly in $p$ for every $x\in H$, then no.
For example, say $(x_n)_{n\ge0}$ is an orthonormal sequence, so $x_n\rightharpoonup0$. Let $$u_n(p)=x_{|n-p|}$$(for $p=1,2,\dots$).