Compact Operator - Equivalent Formulations

Equivalent Formulations

A bounded operator T is compact if and only if any of the following is true

  • Image of the unit ball in X under T is relatively compact in Y.
  • Image of any bounded set under T is relatively compact in Y.
  • Image of any bounded set under T is totally bounded in Y.
  • there exists a neighbourhood of 0, and compact set such that .
  • For any sequence from the unit ball in X, the sequence contains a Cauchy subsequence.

Note that if a linear operator is compact, then it is easy to see that it is bounded, and hence continuous.

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