Continuous linear extension

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title: "Continuous linear extension" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["functional-analysis", "linear-operators"] description: "Mathematical method in functional analysis" topic_path: "general/functional-analysis" source: "https://en.wikipedia.org/wiki/Continuous_linear_extension" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical method in functional analysis ::

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: 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, \R is also a normed vector space; \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]].

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