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.

Read more about this topic:  Bounded Operator

Famous quotes containing the word examples:

    “Histories are more full of examples of the fidelity of dogs than of friends.”
    —Alexander Pope (1688–1744)

    “There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
    —Bernard Mandeville (1670–1733)

    “In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
    —Michel de Montaigne (1533–1592)