subspace

A subspace is a concept used across mathematics and physics to describe a subset that itself forms a space under the same structure as the ambient space. In linear algebra, a subspace W of a vector space V over a field F is a subset that is a vector space with the same addition and scalar multiplication as V. Equivalently, W contains the zero vector and is closed under addition and scalar multiplication: for any u, v in W and any a in F, u + v and au are in W. A subspace is precisely any set that can be described as the span of a generating set.

Examples include the zero subspace {0}, the entire space V, and, in R^n, any line through the origin or any plane through the origin.

The dimension of a subspace W is the size of a basis, a linearly independent set that spans W. The span of a set S is the subspace consisting of all finite linear combinations of elements of S. The intersection of subspaces is a subspace; the sum of subspaces is a subspace, and for subspaces W and U, dim(W + U) = dim W + dim U − dim(W ∩ U).

In topology, a subspace of a topological space X carries the subspace topology, defined by open sets that are intersections with X’s open sets. In functional analysis and quantum mechanics, subspaces of a Hilbert space are central: closed subspaces, orthogonal complements, and invariant subspaces under operators describe important decompositions and physical states.

Beyond linear spaces, the term subspace is used in various mathematical contexts to denote a subset that satisfies a notion of structure preservation relative to its ambient space.