  We describe an infinitary logic for metric structures which is analogous to
$L_{\omega_1, \omega}$. We show that this logic is capable of expressing
several concepts from analysis that cannot be expressed in finitary continuous
logic. Using topological methods, we prove an omitting types theorem for
countable fragments of our infinitary logic. We use omitting types to prove a
two-cardinal theorem, which yields a strengthening of a result of Ben Yaacov
and Iovino concerning separable quotients of Banach spaces.
