  We consider the problem of learning linear operators under squared loss
between two infinite-dimensional Hilbert spaces in the online setting. We show
that the class of linear operators with uniformly bounded $p$-Schatten norm is
online learnable for any $p \in [1, \infty)$. On the other hand, we prove an
impossibility result by showing that the class of uniformly bounded linear
operators with respect to the operator norm is \textit{not} online learnable.
Moreover, we show a separation between sequential uniform convergence and
online learnability by identifying a class of bounded linear operators that is
online learnable but uniform convergence does not hold. Finally, we prove that
the impossibility result and the separation between uniform convergence and
learnability also hold in the batch setting.
