Editing Continuous linear extension (section)

==Application==

Consider, for instance, the definition of the [[Riemann integral]]. A [[step function]] on a [[Closure (mathematics)|closed]] [[Interval (mathematics)|interval]] <math>[a,b]</math>  is a function of the form: <math>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]}</math>
where <math>r_1, \ldots, r_n</math> are real numbers, <math>a = x_0 < x_1 < \ldots < x_{n-1} < x_n = b,</math> and <math>\mathbf{1}_S</math> denotes the [[indicator function]] of the set <math>S.</math> The space of all step functions on <math>[a,b],</math> normed by the <math>L^\infty</math> norm (see [[Lp space]]), is a normed vector space which we denote by <math>\mathcal{S}.</math> Define the integral of a step function by: <math display=block>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}).</math> 
<math>I</math> as a function is a bounded linear transformation from <math>\mathcal{S}</math> into <math>\R.</math><ref> Here, <math>\R</math> is also a normed vector space; <math>\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 BLT theorem to extend the linear transformation <math>I</math> to a bounded linear transformation <math>\widehat{I}</math> from <math>\mathcal{PC}</math> to <math>\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=\widehat{I}(f).</math>