scalar multiplication are defined in the obvious manner. If the degree of the polynomials is unrestricted then the dimension of F[x] is countably infinite. If instead one restricts to polynomials with degree less than or equal to n, then we have a vector space with dimension n + 1. One possible basis for F[x] is a monomial basis: the coordinates of a polynomial with respect to this basis are its coefficients, and the map sending a polynomial to the sequence of its coefficients is a linear isomorphism from F[x] to the infinite coordinate space F∞. The vector space of polynomials with real coefficients and degree less than or equal to n is often denoted by Pn. Several variables The set of polynomials in several variables with coefficients in F is vector space over F denoted F[x1, x2, ..., xr]. Here r is the number of variables. Function spaces See main article at Function space, especially the functional analysis section. Let X be a non-empty arbitrary set and V an arbitrary vector space over F. The space of all functions from X to V is a vector space over F under pointwise addition and multiplication. That is, let f : X → V and g : X → V denote two functions, and let α in F. We define where the operations on the right hand side are those in V. The zero vector is given by the constant function sending everything to the zero vector in V. The space of all functions from X to V is commonly denoted VX. If X is finite and V is finite-dimensional then VX has dimension |X|(dim V), otherwise the space is infinite-dimensional (uncountably so if X is infinite). Many of the vector spaces that arise in mathematics are subspaces of some function space. We give some further examples. Generalized coordinate space Let X be an arbitrary set. Consider the space of all functions from X to F which vanish on all but a finite number of points in X. This space is a vector subspace of FX, the space of all possible functions from X to F. To see this, note that the union of two finite sets is finite, so that the sum of two functions in this space will still vanish outside a finite set. The space described above is commonly denoted (FX)0 and is called generalized coordinate space for the