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: f\equiv r_1 \mathbf{1}{[a,x_1)}+r_2 \mathbf{1}{[x_1,x_2)} + \cdots + r_n \mathbf{1}{[x{n-1},b]} where r_1, \ldots, r_n are real numbers, a = x_0 and \mathbf{1}S 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 \mathcal{S}. Define the integral of a step function by: I \left(\sum{i=1}^n r_i \mathbf{1}{ x{i-1},x_i)}\right) = \sum_{i=1}^n r_i (x_i-x_{i-1}). I as a function is a bounded linear transformation from \mathcal{S} into \R.

Let \mathcal{PC} 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 \mathcal{PC}, so we can apply the BLT theorem to extend the linear transformation I to a bounded linear transformation \widehat{I} from \mathcal{PC} to \R. This defines the Riemann integral of all functions in \mathcal{PC}; for every f\in \mathcal{PC}, \int_a^b f(x)dx=\widehat{I}(f).

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.

References

References

1. Here, $\backslash R$ is also a normed vector space; $\backslash R$ is a vector space because it satisfies all of the [[Vector space#Formal definition. vector space axioms]] and is normed by the [[Absolute value. absolute value function]].