Limits of "Proper" Function Approach to Banach Spaces

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Discussion Overview

The discussion revolves around the properties of Banach spaces, specifically focusing on the implications of a function being "proper" and continuous within the context of linear function spaces. Participants explore when such arguments may fail, particularly in relation to the definitions and conditions of the spaces involved.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant references a proof that if F is a Banach space, then L(E,F) is also a Banach space, contingent on the behavior of a function being "proper" on a ball of arbitrary radius.
  • Another participant seeks clarification on the notation L(E,F), which is identified as the space of linear functions from E to F.
  • A question is raised regarding whether E must also be a normed space to discuss the concept of a ball with a certain radius.
  • It is noted that E does not necessarily need to be complete, suggesting a nuance in the conditions under which the original argument holds.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the conditions under which the argument about "proper" functions fails, indicating that multiple views and clarifications are present without a consensus.

Contextual Notes

The discussion highlights the dependence on definitions of normed and Banach spaces, as well as the implications of completeness and the nature of "proper" functions, which remain unresolved.

BDV
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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?
 
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Notation question. What is L(E,F)?
 
I apologize, it is the space of linear functions from E to F.
 
Mustn't E also be a normed space? How can one otherwise talk about a ball with a certain radius in E?
 
yes, but not necessarily complete.
 

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