Table of contents

1 Theorem 2 Application 3 Construction of the extension 4 Reference

Theorem

Every bounded linear transformation from a normed vector space V to a complete normed vector space W can be uniquely extended to a bounded linear transformation from the completion of V to W.

Application

Consider for instance the definition of the Riemann integral. A step function is a function of the form

Let PC denote the space of bounded piecewise continuous functions, which are continuous to the right, with the L^∞ norm. The space S is dense in PC, so we can apply the BLT-theorem to extend the operator I to a bounded linear operator PC → R. This defines the Riemann integral of all functions in PC.

Construction of the extension

In fact, the extended operator can be constructed explicitly.

Let L be the bounded linear transformation from V to W that we want to extend. Denote the completion of V by V′. We want to construct the extension L′ : V′ → W. Pick an x ∈ V′. The construction of V′ implies that there is a Cauchy sequence {x_n} which converges to x. The sequence {L(x_n)} is also Cauchy, because the operator L is bounded, hence it converges to some y ∈ W. Furthermore, the limit y does not depend on the particular Cauchy sequence {x_n} we chose, so we can now define L′(x) to be y.

A proof of the theorem should additionally show that L′ inherits the linearity and boundedness of L, and that the above construction is unique.

Reference