Uniform boundedness
from above (below)
A property of a family of real-valued functions , where
,
is an index set and
is an arbitrary set. It requires that there is a constant
such that for all
and all
the inequality
(respectively,
) holds.
A family of functions ,
, is called uniformly bounded if it is uniformly bounded both from above and from below.
The notion of uniform boundedness of a family of functions has been generalized to mappings into normed and semi-normed spaces: A family of mappings , where
,
is an arbitrary set and
is a semi-normed (normed) space with semi-norm (norm)
, is called uniformly bounded if there is a constant
such that for all
and
the inequality
holds. If a semi-norm (norm) is introduced into the space
of bounded mappings
by the formula
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then uniform boundedness of a set of functions ,
, means boundedness of this set in the space
with the semi-norm
.
The concept of uniform boundedness from below and above has been generalized to the case of mappings into a set
that is ordered in some sense.
Comments
The uniform boundedness theorem is as follows. Let be a linear topological space that is not a countable union of closed nowhere-dense subsets. Let
be a family of continuous mappings of
into a quasi-normed linear space
(cf. Quasi-norm). Assume that
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Now, if the set is bounded for each
, then
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uniformly in . Here, the convergence to zero is strong convergence, i.e. in the quasi-norm of
.
A corollary is the resonance theorem (sometimes itself called the uniform boundedness theorem): Let be a family of bounded linear operators from a Banach space
into a normed linear space
. Then the boundedness of
for each
implies the boundedness of
, and if
and
exists for each
, then
is also a bounded linear operator
.
Cf. also Banach–Steinhaus theorem (also called the uniform boundedness principle) and Equicontinuity.
References
[a1] | K. Yosida, "Functional analysis" , Springer (1978) pp. 68ff |
[a2] | W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98 |
Uniform boundedness. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniform_boundedness&oldid=14233