I was reading a textbook on Analysis from H.Amann & J.Escher. The Definition of differentiability
for functions with a single variable
for functions with "multivariable"
attracted my attention, for in the first case one just needs the target space to be a normed (vector) space while in the other case it is a Banach space. I can't see why the completeness in the second case is necessary (if it really is) for the definition, the authors didn't explain it (directly), neither.
So, could somebody tell me whether the completeness of the target space is needed? If it isn't always needed, when, I mean when we consider some theorem about differentiability, should we assume that completeness?
Sorry for my poor English, I hope the questions would be still clear enough to You.
(the screenshots are taken on p.301, Analysis I and p.149, Analysis II respectively)