In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space X by first defining a linear transformation L on a dense subset of X and then continuously extending L 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 L from a normed vector space X to a complete, normed vector space Y can be uniquely extended to a bounded linear transformation \widehat{L} from the completion of X to Y. In addition, the operator norm of L is c if and only if the norm of \widehat{L} is c. 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 [a,b] is a function of the form: where are real numbers, and denotes the indicator function of the set S. The space of all step functions on [a,b], normed by the L^\infty norm (see Lp space), is a normed vector space which we denote by Define the integral of a step function by: I as a function is a bounded linear transformation from \mathcal{S} into \R. Let denote the space of bounded, piecewise continuous functions on [a,b] that are continuous from the right, along with the L^\infty norm. The space \mathcal{S} is dense in so we can apply the BLT theorem to extend the linear transformation I to a bounded linear transformation \widehat{I} from to \R. 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 T : S \to Y to a bounded linear transformation from \bar{S} = X to Y, if S is dense in X. If S is not dense in X, then the Hahn–Banach theorem may sometimes be used to show that an extension exists. However, the extension may not be unique.