  We introduce the operators "modified limit" and "accumulation" on a Banach
space, and we use this to define what we mean by being internally computable
over the space. We prove that any externally computable function from a
computable metric space to a computable Banach space is internally computable.
We motivate the need for internal concepts of computability by observing that
the complexity of the set of finite sets of closed balls with a nonempty
intersection is not uniformly hyperarithmetical, and thus that approximating an
externally computable function is highly complex.
