TITLE: Fredholm operators on non-Banach spaces.
QUESTION [6 upvotes]: Apparently Fredholm operators are usually (at least in Wikipedia and my functional analysis lecture) only defined as operators $T$ between two Banach spaces.
As far as I can see, the definition can be extended to operators between arbitrary normed spaces without any problems, although then the condition that $\operatorname{im} T$ is closed is no longer independent of $\ker T < \infty$ and $\operatorname{coker} T < \infty$.
Is there some reason why this is not usually done? 
Are Fredholm operators between non-Banach spaces so much less interesting/useful than those between Banach spaces? 
If yes, why?

REPLY [3 votes]: The way I see it, Fredholm's theory is about existence; if some conditions are satisfied, then the given equation admits at least one solution. All existence results in analysis rely on completeness. These results never build a solution explicitly; rather, they prove that some approximation procedure is convergent, because it is Cauchy. Fredholm's theory is no exception.