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Bounded operator

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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 .

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)
How to Cite This Entry:
Bounded operator. V.I. Sobolev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bounded_operator&oldid=17165
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098