# 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.

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.

How to Cite This Entry:
Uniform boundedness. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniform_boundedness&oldid=49069
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article