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:
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, .
Read more about this topic: Continuous Linear Extension
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