Bounded Operator - Examples

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.

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