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:
“[17th-century] Puritans were the first modern parents. Like many of us, they looked on their treatment of children as a test of their own self-control. Their goal was not to simply to ensure the child’s duty to the family, but to help him or her make personal, individual commitments. They were the first authors to state that children must obey God rather than parents, in case of a clear conflict.”
—C. John Sommerville (20th century)
“Those things which now most engage the attention of men, as politics and the daily routine, are, it is true, vital functions of human society, but should be unconsciously performed, like the corresponding functions of the physical body.”
—Henry David Thoreau (1817–1862)