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.

TheoremEdit

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, for bounded linear transformation.

ApplicationEdit

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 BLT 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 theoremEdit

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.

ReferencesEdit

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

FootnotesEdit

  1. ^ 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.