dimW

dimW is a common mathematical notation used to denote the dimension of a vector space W over a field F. It is typically written as dim W or dim(W). The dimension is defined as the size of a basis for W, or equivalently, the maximum number of linearly independent vectors contained in W. For finite-dimensional spaces, dim W is a nonnegative integer; for infinite-dimensional spaces, dim W is a cardinal number equal to the size of any basis of W.

Examples help illustrate the concept. In R^3, the subspace W = {(x, y, 0) : x, y ∈ R} has dim W = 2. The space of all real-coefficient polynomials has dim equal to countably infinite, since a basis is {1, t, t^2, t^3, ...}. If W is a subspace of V and V is finite-dimensional, then dim W ≤ dim V, with equality if and only if W = V. A basis for W can be extended to a basis for V, showing how lower-dimensional subspaces fit inside higher-dimensional ambient spaces.

Beyond the standard linear-algebra setting, there are other notions of dimension in mathematics (such as Krull dimension in algebraic geometry or topological dimension in topology), which use different notations and concepts. However, in many contexts dim W succinctly conveys the idea of the number of degrees of freedom or the size of a basis for a vector space.

See also: basis, linear independence, spanning set, rank, dimension theorem.