Uniform boundedness

from above (below)

A property of a family of real-valued functions $ f _ \alpha : X \rightarrow \mathbf R $, where $ \alpha \in {\mathcal A} $, $ {\mathcal A} $ is an index set and $ X $ is an arbitrary set. It requires that there is a constant $ c > 0 $ such that for all $ \alpha \in {\mathcal A} $ and all $ x \in X $ the inequality $ f _ \alpha ( x) \leq c $( respectively, $ f _ \alpha ( x) \geq - c $) holds.

A family of functions $ f _ \alpha : X \rightarrow \mathbf R $, $ \alpha \in {\mathcal A} $, 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 $ f _ \alpha : X \rightarrow Y $, where $ \alpha \in {\mathcal A} $, $ X $ is an arbitrary set and $ Y $ is a semi-normed (normed) space with semi-norm (norm) $ \| \cdot \| _ {Y} $, is called uniformly bounded if there is a constant $ c > 0 $ such that for all $ \alpha \in {\mathcal A} $ and $ x \in X $ the inequality $ \| f _ \alpha ( x) \| _ {Y} \leq c $ holds. If a semi-norm (norm) is introduced into the space $ \{ X \rightarrow Y \} $ of bounded mappings $ f: X \rightarrow Y $ by the formula

$$ \| f \| _ {\{ X \rightarrow Y \} } = \sup _ {x \in X } \ \| f ( x) \| _ {Y} , $$

then uniform boundedness of a set of functions $ f _ \alpha : X \rightarrow Y $, $ \alpha \in U $, means boundedness of this set in the space $ \{ X \rightarrow Y \} $ with the semi-norm $ \| \cdot \| _ {\{ X \rightarrow Y \} } $.

The concept of uniform boundedness from below and above has been generalized to the case of mappings $ f: X \rightarrow Y $ into a set $ Y $ that is ordered in some sense.

Comments

The uniform boundedness theorem is as follows. Let $ X $ be a linear topological space that is not a countable union of closed nowhere-dense subsets. Let $ \{ {T _ \alpha } : {\alpha \in {\mathcal A} } \} $ be a family of continuous mappings of $ X $ into a quasi-normed linear space $ Y $( cf. Quasi-norm). Assume that

$$ \| T _ \alpha ( x+ y) \| \leq \| T _ \alpha ( x) \| + \| T _ \alpha ( y) \| , $$

$$ \| T _ \alpha ( ax) \| = a \| T _ \alpha ( x) \| \ \textrm{ for } a \geq 0. $$

Now, if the set $ \{ {T _ \alpha ( x) } : {\alpha \in {\mathcal A} } \} $ is bounded for each $ x \in X $, then

$$ \lim\limits _ {\alpha \rightarrow 0 } T _ \alpha ( x) = 0 $$

uniformly in $ \alpha $. Here, the convergence to zero is strong convergence, i.e. in the quasi-norm of $ Y $.

A corollary is the resonance theorem (sometimes itself called the uniform boundedness theorem): Let $ \{ {T _ \alpha } : {\alpha \in {\mathcal A} } \} $ be a family of bounded linear operators from a Banach space $ X $ into a normed linear space $ Y $. Then the boundedness of $ \{ {\| T _ \alpha ( x) \| } : {\alpha \in {\mathcal A} } \} $ for each $ x \in X $ implies the boundedness of $ \{ {\| T _ \alpha \| } : {\alpha \in {\mathcal A} } \} $, and if $ {\mathcal A} = \mathbf N $ and $ \lim\limits _ {n\rightarrow \infty } T _ {n} ( x) = T( x) $ exists for each $ x \in X $, then $ T $ is also a bounded linear operator $ X \rightarrow Y $.

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