Continuous linear extension: Difference between revisions

→‎Application: B.L.T. to BLT, to match the other appearance of this acronym.
(→‎Application: B.L.T. to BLT, to match the other appearance of this acronym.)
 
<math>\mathsf{I}</math> as a function is a bounded linear transformation from <math>\mathcal{S}</math> into <math>\mathbb{R}</math>.<ref>Here, <math>\mathbb{R}</math> is also a normed vector space; <math>\mathbb{R}</math> 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]].</ref>
 
Let <math>\mathcal{PC}</math> denote the space of bounded, [[piecewise]] continuous functions on <math>[a,b]</math> that are continuous from the right, along with the <math>L^\infty</math> norm. The space <math>\mathcal{S}</math> is dense in <math>\mathcal{PC}</math>, so we can apply the B.L.T.BLT theorem to extend the linear transformation <math>\mathsf{I}</math> to a bounded linear transformation <math>\tilde{\mathsf{I}}</math> from <math>\mathcal{PC}</math> to <math>\mathbb{R}</math>. This defines the Riemann integral of all functions in <math>\mathcal{PC}</math>; for every <math>f\in \mathcal{PC}</math>, <math>\int_a^b f(x)dx=\tilde{\mathsf{I}}(f)</math>.
 
==The Hahn–Banach theorem==