# Limits of "Proper" Function Approach to Banach Spaces

• BDV
In summary, the conversation discusses a proof in H Cartan's differential calculus that shows how L(E,F), the space of linear functions from E to F, is also a Banach space if F is a Banach space. The proof relies on the concept of a function being "proper" or continuous on a ball of arbitrary radius, and extends this property to the entire space. It is mentioned that this type of argument may fail in certain cases and that E must also be a normed space in order to talk about a ball with a certain radius in E. However, it is not necessary for E to be complete.
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?

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.

## What is the "proper" function approach to Banach spaces?

The "proper" function approach to Banach spaces is a method used to study the behavior of Banach spaces under certain mathematical operations. It involves defining a "proper" function, which is a function that satisfies certain conditions and is used to analyze the behavior of the Banach space.

## What are the limitations of the "proper" function approach?

One of the limitations of the "proper" function approach is that it only applies to certain types of Banach spaces, such as reflexive and strictly convex spaces. It also cannot be used to study the behavior of Banach spaces under all types of operations.

## How does the "proper" function approach differ from other methods of studying Banach spaces?

The "proper" function approach differs from other methods, such as the geometric approach or the operator approach, in that it focuses on the behavior of Banach spaces under specific operations and relies on the use of "proper" functions rather than geometric or operator properties.

## Can the "proper" function approach be used to study infinite-dimensional Banach spaces?

Yes, the "proper" function approach can be applied to study infinite-dimensional Banach spaces. However, it may be more limited in its applicability compared to other methods, as some operations may not have a well-defined "proper" function in infinite-dimensional spaces.

## What are some potential areas of further research in the "proper" function approach to Banach spaces?

Some potential areas of further research in the "proper" function approach include exploring its applicability to more general types of Banach spaces, developing new techniques for constructing "proper" functions, and investigating its connections to other methods of studying Banach spaces.

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