Bounded operator
A mapping of a topological vector space
into a topological vector space
such that
is a bounded subset in
for any bounded subset
of
. Every operator
, continuous on
, is a bounded operator. If
is a linear operator, then for
to be bounded it is sufficient that there exists a neighbourhood
such that
is bounded in
. Suppose that
and
are normed linear spaces and that the linear operator
is bounded. Then
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This number is called the norm of the operator and is denoted by
. Then
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and is the smallest constant
such that
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for any . Conversely, if this inequality is satisfied, then
is bounded. For linear operators mapping a normed space
into a normed space
, the concepts of boundedness and continuity are equivalent. This is not the case for arbitrary topological vector spaces
and
, but if
is bornological and
is a locally convex space, then the boundedness of a linear operator
implies its continuity. If
is a Hilbert space and
is a bounded symmetric operator, then the quadratic form
is bounded on the unit ball
. The numbers
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are called the upper and lower bounds of the operator . The points
and
belongs to the spectrum of
, and the whole spectrum lies in the interval
. Examples of bounded operators are: the projection operator (projector) onto a complemented subspace of a Banach space, and an isometric operator acting on a Hilbert space.
If the space and
have the structure of a partially ordered set, for example are vector lattices (cf. Vector lattice), then a concept of order-boundedness of an operator can be introduced, besides the topological boundedness considered above. An operator
is called order-bounded if
is an order-bounded set in
for any order-bounded set
in
. Examples: an isotone operator, i.e. an operator such that
implies
.
References
[1] | L.A. Lyusternik, V.I. Sobolev, "Elemente der Funktionalanalysis" , Akademie Verlag (1955) (Translated from Russian) |
[2] | W. Rudin, "Functional analysis" , McGraw-Hill (1973) |
[3] | G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1967) |
Comments
References
[a1] | A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980) |
[a2] | A.C. Zaanen, "Riesz spaces" , II , North-Holland (1983) |
Bounded operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bounded_operator&oldid=17165