Let $B_1$ en $B_2$ be Banach spaces. Consider an open subset $G$ of $B_1$. A sequence of functions $(f_n)_{n \in \mathbb{N}}$ from $G$ to $B_2$ is called locally uniformly convergent if for every $z \in G$ there exists a $\delta > 0$ such that $B(z, \delta) \subseteq G$ and $(f_n\vert_{B(z, \delta)})_{n \in \mathbb{N}}$ converges uniformly.
Using a finite cover argument, one can see that this implies that the sequence $(f_n\vert_C)_{n \in \mathbb{N}}$ actually converges uniformly for every compact subset $C$ of $G$.
I was wondering if the converse is also true, i.e. if the sequence $(f_n\vert_C)_{n \in \mathbb{N}}$ converges uniformly for every compact subset $C$ of $G$, does the sequence $(f_n)_{n \in \mathbb{N}}$ converge locally uniformly? If $B_1$ is of finite dimension, then $B_1$ is locally compact and this is trivial. My gut feeling says that something could go wrong if $B_1$ is of infinite dimension, but I can't find a counterexample.