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## Main Question or Discussion Point

Hello,

I have seen (in H Cartan's differential calculus) a proof that if F is a Banch space,

I was wondering when/how does this type of argument

I have seen (in H Cartan's differential calculus) a proof that if F is a Banch space,

*L*(E,F) where E is some vector space, is also a Banach space. One of the main points of the proof is based on the behaviour of a function being "proper" (continuous) on a ball of arbitrary radius "n" and by such being able to extend the property to the entire space.I was wondering when/how does this type of argument

__fail__?