dimV

Dim V denotes the dimension of a vector space V over a field F. The dimension is defined as the cardinality of a basis: a set of vectors that spans V and is linearly independent. If V is finite-dimensional, the dimension is a nonnegative integer equal to the size of any basis. A key property is that any two bases of V have the same cardinality, so the dimension is well defined.

If V has a finite basis, every generating set contains a finite spanning set and the dimension equals the number of vectors in a basis. For subspaces W of V, dim W is at most dim V. The dimension theorem, or rank-nullity theorem, states that for a linear map T: V → W with finite-dimensional V, dim V = dim ker(T) + dim im(T).

Invariance: the dimension is preserved under vector space isomorphisms; isomorphic spaces have the same dimension. Examples include dim R^n = n; the space of polynomials of degree at most n, P_n(R), has dimension n+1 with basis {1, x, x^2, ..., x^n}.

Infinite-dimensional spaces have infinite dimension in the sense that no finite basis exists; the dimension is a cardinal number equal to the size of a basis. Examples include the space of all polynomials (countably infinite dimension) and C([0,1]), the space of continuous functions on [0,1] (uncountable dimension). The notion of dimension here is purely algebraic and distinguishes from topological notions of dimension.