# What is Bounded: Definition and 536 Discussions

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.

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