Further Properties
The condition for L to be bounded, namely that there exists some M such that for all v
is precisely the condition for L to be Lipschitz continuous at 0 (and hence, everywhere, because L is linear).
A common procedure for defining a bounded linear operator between two given Banach spaces is as follows. First, define a linear operator on a dense subset of its domain, such that it is locally bounded. Then, extend the operator by continuity to a continuous linear operator on the whole domain (see continuous linear extension).
Read more about this topic: Bounded Operator
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