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A mapping  $  A $
of a topological vector space  $  X $
into a topological vector space  $  Y $
such that  $  A (M) $
is a bounded subset in  $  Y $
for any bounded subset  $  M $
of  $  X $. 
Every operator  $  A :  X \rightarrow Y $, 
continuous on  $  X $, 
is a bounded operator. If  $  A :  X \rightarrow Y $
is a linear operator, then for  $  A $
to be bounded it is sufficient that there exists a neighbourhood  $  U \subset  X $
such that  $  A (U) $
is bounded in  $  Y $. 
Suppose that  $  X $
and  $  Y $
are normed linear spaces and that the linear operator  $  A :  X \rightarrow Y $
is bounded. Then

$$ 
\gamma  = \ 
\sup _ {\| x \| \leq  1 }  \| A x \|  <  \infty .
$$

This number is called the norm of the operator  $  A $
and is denoted by  $  \| A \| $. 
Then

$$ 
\| A x \|  \leq   \| A  \| \cdot \|  x \| ,
$$

and  $  \| A \| $
is the smallest constant  $  C $
such that

$$ 
\| A x \|  \leq   C  \| x \|
$$

for any  $  x \in X $. 
Conversely, if this inequality is satisfied, then  $  A $
is bounded. For linear operators mapping a normed space  $  X $
into a normed space  $  Y $, 
the concepts of boundedness and continuity are equivalent. This is not the case for arbitrary topological vector spaces  $  X $
and  $  Y $, 
but if  $  X $
is bornological and  $  Y $
is a locally convex space, then the boundedness of a linear operator  $  A :  X \rightarrow Y $
implies its continuity. If  $  H $
is a Hilbert space and  $  A :  H \rightarrow H $
is a bounded symmetric operator, then the quadratic form  $  \langle  A x , x \rangle $
is bounded on the unit ball  $  K _ {1} = \{ {x } : {\| x \| \leq  1 } \} $. 
The numbers

$$ 
\beta  = \ 
\sup _ {\| x \| \leq  1 }  \langle  A x , x \rangle \ \ 
\textrm{ and } \ \ 
\alpha  =  \inf _ {\| x \| \leq  1 }  \langle  A x , x \rangle
$$

are called the upper and lower bounds of the operator  $  A $. 
The points  $  \alpha $
and  $  \beta $
belongs to the spectrum of  $  A $, 
and the whole spectrum lies in the interval  $  [ \alpha , \beta ] $. 
Examples of bounded operators are: the projection operator ([[Projector|projector]]) onto a complemented subspace of a Banach space, and an [[Isometric operator|isometric operator]] acting on a Hilbert space.

If the space  $  X $
and  $  Y $
have the structure of a partially ordered set, for example are vector lattices (cf. [[Vector lattice|Vector lattice]]), then a concept of order-boundedness of an operator can be introduced, besides the topological boundedness considered above. An operator  $  A :  X \rightarrow Y $
is called order-bounded if  $  A (M) $
is an order-bounded set in  $  Y $
for any order-bounded set  $  M $
in  $  X $. 
Examples: an isotone operator, i.e. an operator such that  $  x \leq  y $
implies  $  A x \leq  A y $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,   V.I. Sobolev,   "Elemente der Funktionalanalysis" , Akademie Verlag  (1955)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Rudin,   "Functional analysis" , McGraw-Hill  (1973)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Birkhoff,   "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1967)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.E. Taylor,   D.C. Lay,   "Introduction to functional analysis" , Wiley  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.C. Zaanen,   "Riesz spaces" , '''II''' , North-Holland  (1983)</TD></TR></table>