This page lists some examples of vector spaces. See vector space for the definitions of terms used on this page. See also: dimension, basis. Notation. Let F denote an arbitrary field such as the real numbers R or the complex numbers C. Trivial or zero vector space The simplest example of a vector space is the trivial one: {0}, which contains only the zero vector (see the third axiom in the Vector space article). Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one. The zero vector space is conceptually different from the null space of a linear operator L, which is the kernel of L. (Incidentally, the null space of L is a zero space if and only if L is injective.) Field The next simplest example is the field F itself. Vector addition is just field addition, and scalar multiplication is just field multiplication. This property can be used to prove that a field is a vector space. Any non-zero element of F serves as a basis so F is a 1-dimensional vector space over itself. The field is a rather special vector space; in fact it is the simplest example of a commutative algebra over F. Also, F has just two subspaces: {0} and F itself. Coordinate space A basic example of a vector space is the following. For any positive integer n, the set of all n-tuples of elements of F forms an n-dimensional vector space over F sometimes called coordinate space and denoted Fn. An element of Fn is written where each xi is an element of F. The operations on Fn are defined by Commonly, F is the field of real numbers, in which case we obtain real coordinate space Rn. The field of complex numbers gives complex coordinate space Cn. The a + bi form of a complex number shows that C itself is a two-dimensional real vector space with coordinates (a,b). Similarly, the quaternions and the octonions are respectively four- and eight-dimensional real vector spaces, and Cn is a 2n-dimensional real vector space. The vector space Fn has a standard basis: where 1 denotes the multiplicative identity in F. Infinite coordinate space Let F∞ denote the space of infinite sequences of elements