Continuous linear extension

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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 \mathsf{T} on a dense subset of X and then extending \mathsf{T} to the whole space via the theorem below. The resulting extension remains linear and bounded (thus continuous).

This procedure is known as continuous linear extension.

Theorem[edit]

Every bounded linear transformation \mathsf{T} from a normed vector space X to a complete, normed vector space Y can be uniquely extended to a bounded linear transformation \tilde{\mathsf{T}} from the completion of X to Y. In addition, the operator norm of \mathsf{T} is c iff the norm of \tilde{\mathsf{T}} is c.

This theorem is sometimes called the B L T theorem, where B L T stands for bounded linear transformation.

Application[edit]

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 \mathit{1}_{[a,x_1)}+r_2 \mathit{1}_{[x_1,x_2)} + \cdots + r_n \mathit{1}_{[x_{n-1},b]} where r_1, \ldots, r_n are real numbers, a=x_0<x_1<\ldots <x_{n-1}<x_n=b, and \mathit{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: \mathsf{I} \left(\sum_{i=1}^n r_i \mathit{1}_{ [x_{i-1},x_i)}\right) = \sum_{i=1}^n r_i (x_i-x_{i-1}). \mathsf{I} as a function is a bounded linear transformation from \mathcal{S} into \mathbb{R}.[1]

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 B.L.T. theorem to extend the linear transformation \mathsf{I} to a bounded linear transformation \tilde{\mathsf{I}} from \mathcal{PC} to \mathbb{R}. This defines the Riemann integral of all functions in \mathcal{PC}; for every f\in \mathcal{PC}, \int_a^b f(x)dx=\tilde{\mathsf{I}}(f).

The Hahn–Banach theorem[edit]

The above theorem can be used to extend a bounded linear transformation T:S\rightarrow 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[edit]

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

Footnotes[edit]

  1. ^ Here, \mathbb{R} is also a normed vector space; \mathbb{R} is a vector space because it satisfies all of the vector space axioms and is normed by the absolute value function.