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Let T:X->Y be a bounded linear operator. One definition of T being compact is that T maps bounded sequences to sequences with convergent subsequences. An equivalent definition is that T maps sequences of vectors with norms not exceeding 1 to sequences with convergent subsequences.
A couple of questions: What do these definitions require of X and Y. Can they be (incomplete)
normed spaces?
Also for the second definition, does it only make sense when X=Y? Every definition I've seen has assumed that but I've looked at a proof and couldn't see where they had used that assumption.
Thanks
A couple of questions: What do these definitions require of X and Y. Can they be (incomplete)
normed spaces?
Also for the second definition, does it only make sense when X=Y? Every definition I've seen has assumed that but I've looked at a proof and couldn't see where they had used that assumption.
Thanks