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''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|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|Banach–Steinhaus theorem]] (also called the uniform boundedness principle) and [[Equicontinuity|Equicontinuity]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Yosida,   "Functional analysis" , Springer  (1978)  pp. 68ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,   "Real and complex analysis" , McGraw-Hill  (1966)  pp. 98</TD></TR></table>