# Continuous linear extension

In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space by first defining a linear transformation on a dense subset of and then extending 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

Every bounded linear transformation from a normed vector space to a complete, normed vector space can be uniquely extended to a bounded linear transformation from the completion of to . In addition, the operator norm of is iff the norm of is .

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

## Application

Consider, for instance, the definition of the Riemann integral. A step function on a closed interval is a function of the form:
where are real numbers, , and denotes the indicator function of the set . The space of all step functions on , normed by the norm (see Lp space), is a normed vector space which we denote by . Define the integral of a step function by: .
as a function is a bounded linear transformation from into .^{[1]}

Let denote the space of bounded, piecewise continuous functions on that are continuous from the right, along with the norm. The space is dense in , so we can apply the B.L.T. theorem to extend the linear transformation to a bounded linear transformation from to . This defines the Riemann integral of all functions in ; for every , .

## The Hahn–Banach theorem

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

## References

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### Footnotes

- ↑ Here, is also a normed vector space; is a vector space because it satisfies all of the vector space axioms and is normed by the absolute value function.