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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