Continuous linear extension

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.

Theorem

Every bounded linear transformation from a normed vector space to a complete, normed vector space can be uniquely extended to a bounded linear transformation from the completion of to In addition, the operator norm of is if and only if the norm of is
This theorem is sometimes called the BLT theorem.

Application

Consider, for instance, the definition of the Riemann integral. A step function on a closed interval is a function of the form:
where are real numbers, and denotes the indicator function of the set The space of all step functions on normed by the norm, 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 into
Let denote the space of bounded, piecewise continuous functions on that are continuous from the right, along with the norm. The space is dense in so we can apply the BLT theorem to extend the linear transformation to a bounded linear transformation from to This defines the Riemann integral of all functions in ; for every

The Hahn–Banach theorem

The above theorem can be used to extend a bounded linear transformation to a bounded linear transformation from to if is dense in If is not dense in then the Hahn–Banach theorem may sometimes be used to show that an extension exists. However, the extension may not be unique.