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Compact Operator

Fermat

Active member
Nov 3, 2013
188
Let L be an compact operator on a compact space K , and Let I be the identity on K.
Show that Ker(I-L) is finite-dimensional.

My attempt: Let $x_{n}$ be a sequence in the unit ball. K is compact so $(I(x_{n}))=(x_{n})$ has a convergent subsequence and L is compact operator so $L(x_{n})$ has a convergent subsequence. Thus, $(I-L)(x_{n})$ has a convergent subsequence so it is compact. $I-L$ restricted to the kernel is the identity, so since the identity is compact iff the space is finite dimensional, the kernel is finite dimensional. Is this correct?