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.
Read more about this topic: Compact Operator
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“When the apple is ripe it will fall.”
—Irish proverb.
An English equivalent to this might be, “To everything there is a season.”