Test Functions and Distributions
In the sequel, real-valued distributions on an open subset U of Rn will be formally defined. With minor modifications, one can also define complex-valued distributions, and one can replace Rn by any (paracompact) smooth manifold.
The first object to define is the space D(U) of test functions on U. Once this is defined, it is then necessary to equip it with a topology by defining the limit of a sequence of elements of D(U). The space of distributions will then be given as the space of continuous linear functionals on D(U).
Read more about this topic: Distribution (mathematics)
Famous quotes containing the words test and/or functions:
“We do not quite say that the new is more valuable because it fits in; but its fitting in is a test of its value—a test, it is true, which can only be slowly and cautiously applied, for we are none of us infallible judges of conformity.”
—T.S. (Thomas Stearns)
“The mind is a finer body, and resumes its functions of feeding, digesting, absorbing, excluding, and generating, in a new and ethereal element. Here, in the brain, is all the process of alimentation repeated, in the acquiring, comparing, digesting, and assimilating of experience. Here again is the mystery of generation repeated.”
—Ralph Waldo Emerson (1803–1882)