Examples
- Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix.
- Many integral transforms are bounded linear operators. For instance, if
-
- is a continuous function, then the operator defined on the space of continuous functions on endowed with the uniform norm and with values in the space with given by the formula
- is bounded. This operator is in fact compact. The compact operators form an important class of bounded operators.
- The Laplace operator
-
- (its domain is a Sobolev space and it takes values in a space of square integrable functions) is bounded.
- The shift operator on the l2 space of all sequences (x0, x1, x2...) of real numbers with
-
- is bounded. Its operator norm is easily seen to be 1.
Read more about this topic: Bounded Operator
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