In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space
by first defining a linear transformation
on a dense subset of
and then continuously extending
to the whole space via the theorem below.
The resulting extension remains linear and bounded, and is thus continuous, which makes it a continuous linear extension.
This procedure is known as continuous linear extension.
Every bounded linear transformation
from a normed vector space
to a complete, normed vector space
can be uniquely extended to a bounded linear transformation
In addition, the operator norm of
This theorem is sometimes called the BLT theorem.
Consider, for instance, the definition of the Riemann integral.
A step function on a closed interval
is a function of the form:
are real numbers,
denotes the indicator function of the set
The space of all step functions on
norm (see Lp space), is a normed vector space which we denote by
Define the integral of a step function by:
as a function is a bounded linear transformation from
denote the space of bounded, piecewise continuous functions on
so we can apply the BLT theorem to extend the linear transformation
to a bounded linear transformation
This defines the Riemann integral of all functions in
The above theorem can be used to extend a bounded linear transformation
to a bounded linear transformation from
then the Hahn–Banach theorem may sometimes be used to show that an extension exists.
However, the extension may not be unique.