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

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

Now, if the set is bounded for each , then

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