Case of Banach Spaces
The same question may be posed about invertible operators on any Banach space of infinite dimension. Here there are only partial results. Some classical sequence spaces have the same property, namely that the group of invertible operators is contractible. On the other hand, there are examples known where it fails to be a connected space. Where all homotopy groups are known to be trivial, the contractibility in some cases may remain unknown.
Read more about this topic: Kuiper's Theorem
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