*** Welcome to piglix ***

Ring of polynomial functions


In mathematics, the ring of polynomial functions on a vector space V over an infinite field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]. If V has finite dimension and is viewed as an algebraic variety, then k[V] is precisely the coordinate ring of V.

The explicit definition of the ring can be given as follows. If is a polynomial ring, then we can view as coordinate functions on ; i.e., when This suggests the following: given a vector space V, let k[V] be the subring generated by the dual space of the ring of all functions . If we fix a basis for V and write for its dual basis, then k[V] consists of polynomials in ; it is a polynomial ring.


...
Wikipedia

...