Fredholm operators on non-Banach spaces.
$begingroup$
Apparently Fredholm operators are usually (at least in Wikipedia and my functional analysis lecture) only defined as operators $T$ between to Banach spaces $X$ and $Y$.
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?
functional-analysis soft-question banach-spaces
asked Jan 18 at 18:01
0x5390x539
1,393518
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add a comment |
$begingroup$
Apparently Fredholm operators are usually (at least in Wikipedia and my functional analysis lecture) only defined as operators $T$ between to Banach spaces $X$ and $Y$.
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?
functional-analysis soft-question banach-spaces
asked Jan 18 at 18:01
0x5390x539
1,393518
$endgroup$
add a comment |
$begingroup$
Apparently Fredholm operators are usually (at least in Wikipedia and my functional analysis lecture) only defined as operators $T$ between to Banach spaces $X$ and $Y$.
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?
functional-analysis soft-question banach-spaces
asked Jan 18 at 18:01
0x5390x539
1,393518
$endgroup$
Apparently Fredholm operators are usually (at least in Wikipedia and my functional analysis lecture) only defined as operators $T$ between to Banach spaces $X$ and $Y$.
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?
functional-analysis soft-question banach-spaces
functional-analysis soft-question banach-spaces
asked Jan 18 at 18:01
0x5390x539
1,393518
asked Jan 18 at 18:01
0x5390x539
1,393518
edited Jan 18 at 22:20
0x539
asked Jan 18 at 18:01
0x5390x539
1,393518
asked Jan 18 at 18:01
0x5390x539
1,393518
asked Jan 18 at 18:01
0x5390x539
1,393518
1,393518
add a comment |
add a comment |
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