Limits of "Proper" Function Approach to Banach Spaces

[BDV]

Hello,

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?

 

[mathman]

Notation question. What is L(E,F)?

 

[BDV]

I apologize, it is the space of linear functions from E to F.

 

[Erland]

Mustn't E also be a normed space? How can one otherwise talk about a ball with a certain radius in E?

 

[mathwonk]

yes, but not necessarily complete.