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 iff the norm of is .
This theorem is sometimes called the B L T theorem, where B L T stands for bounded linear transformation.
Consider, for instance, the definition of the Riemann integral. A step function on a closedinterval 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 (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 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 B.L.T. 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 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.