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
This number is called the norm of the operator and is denoted by . Then
and is the smallest constant such that
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
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 .
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Bounded operator. V.I. Sobolev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bounded_operator&oldid=17165