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
Famous quotes containing the word application:
“We will not be imposed upon by this vast application of forces. We believe that most things will have to be accomplished still by the application called Industry. We are rather pleased, after all, to consider the small private, but both constant and accumulated, force which stands behind every spade in the field. This it is that makes the valleys shine, and the deserts really bloom.”
—Henry David Thoreau (1817–1862)
“May my application so close
To so endless a repetition
Not make me tired and morose
And resentful of man’s condition.”
—Robert Frost (1874–1963)
“It is known that Whistler when asked how long it took him to paint one of his “nocturnes” answered: “All of my life.” With the same rigor he could have said that all of the centuries that preceded the moment when he painted were necessary. From that correct application of the law of causality it follows that the slightest event presupposes the inconceivable universe and, conversely, that the universe needs even the slightest of events.”
—Jorge Luis Borges (1899–1986)