Continuous linear extension explained

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

[a,b] is a function of the form: f\equiv r _{1} 1 _{[a,x _{1)}} + r _{2} 1 _{[x _{1,x} _{2)}} + … + r _{n} 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 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 l{S}. Define the integral of a step function by: $$I\; \backslash left(\backslash sum\_^n\; r\_i\; \backslash mathbf\_$$