I was recently tackled by this in my class on operator theory dealing with operators on Hilbert spaces:

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> We are to find and prove the inclusion relations between the classes of operators: contractions, strict contractions, proper contractions
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I have no idea how to show this, as I have no idea how to show the inclusion or even think about it, so I am asking this here for help Thanks
Sorry forgot definitions:

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> Contraction: An operator T satisfying $ ||T|| \leq 1 $
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> Strict contraction: An operator T satisfying $ ||T|| < 1 $
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> Proper contraction: An operator on a Hilbert space satisfying $||Tx|| < ||x|| $ for all vectors x in the Hilbert space
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