In functional analysis, a bounded linear operator is a linear transformation
L
:
X
→
Y
{\displaystyle L:X\to Y}
between topological vector spaces (TVSs)
X
{\displaystyle X}
and
Y
{\displaystyle Y}
that maps bounded subsets of
X
{\displaystyle X}
to bounded subsets of
Y
.
{\displaystyle Y.}
If
X
{\displaystyle X}
and
Y
{\displaystyle Y}
are normed vector spaces (a special type of TVS), then
L
{\displaystyle L}
is bounded if and only if there exists some
M
>
0
{\displaystyle M>0}
such that for all
x
{\displaystyle x}
in
X
,
{\displaystyle X,}
The smallest such
M
,
{\displaystyle M,}
denoted by
‖
L
‖
,
{\displaystyle \|L\|,}
is called the operator norm of
L
.
{\displaystyle L.}
A linear operator that is sequentially continuous or continuous is a bounded operator and moreover, a linear operator between normed spaces is bounded if and only if it is continuous.
However, a bounded linear operator between more general topological vector spaces is not necessarily continuous.