Continuous linear extension

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In functional analysis, it is often convenient to define something on a normed vector space by defining it on a dense set and extending it to the whole space. This procedure is justified for bounded linear operators by the theorem below. The result is again linear and bounded (and thus continuous), so it is called the continuous linear extension.

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[edit] 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.

This theorem is sometimes called the BLT theorem, where BLT stands for bounded linear transformation.

[edit] Application

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

f = r_1 1_{[a_1,b_1)} + r_2 1_{[a_2,b_2)} + \cdots + r_n 1_{[a_n,b_n)}

where r1, ..., rn are real numbers, and 1X denotes the indicator function of the set X. The space of all step functions with the L∞ norm (see Lp space) is a normed vector space, which we denote by S. Define the integral of the step functions by

I\left(\sum_{i=1}^n r_i 1_{[a_i,b_i)}\right) = \sum_{i=1}^n r_i (b_i-a_i).

This defines a bounded linear operator I : S → R.

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.

[edit] 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 {xn} which converges to x. The sequence {L(xn)} 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 {xn} 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.

[edit] The Hahn-Banach theorem

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

[edit] References

  • Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis, Section I.2. Academic Press, San Diego, 1980. ISBN 0-12-585050-6.
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