Definition For Vector Spaces
Given a finite or infinite-dimensional vector space over a field K (such as the real or complex numbers), the cylinder sets may be defined as
where is a Borel set in, and each is a linear functional on ; that is, the algebraic dual space to . When dealing with topological vector spaces, the definition is made instead for elements, the continuous dual space. That is, the functionals are taken to be continuous linear functionals. The article on dual spaces discusses the differences between the algebraic and the continuous dual spaces.
Read more about this topic: Cylinder Set
Famous quotes containing the words definition and/or spaces:
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“Every true man is a cause, a country, and an age; requires infinite spaces and numbers and time fully to accomplish his design;—and posterity seem to follow his steps as a train of clients.”
—Ralph Waldo Emerson (1803–1882)