Finitedimensional

Finite-dimensional refers to a vector space whose dimension is finite. A vector space V over a field F is finite-dimensional if there exists a finite set {v1, ..., vn} that is a basis: it spans V and is linearly independent. The dimension dim V is that number n. In finite-dimensional spaces, all bases have the same cardinality by the dimension theorem. Equivalently, V is finite-dimensional iff it is isomorphic to F^n for some n, via a basis identifying coordinates with coefficients.

Key properties follow from finiteness. For a linear map T: V → W between finite-dimensional spaces, the rank-nullity theorem holds: dim Im T + dim Ker T = dim V. If dim V = n, then any linear map is represented by an n×m matrix relative to chosen bases. Consequently, every linear operator on a finite-dimensional space is continuous; all norms on V induce equivalent topologies; closed and bounded subsets are compact in finite-dimensional normed spaces, by the Heine-Borel property. The dual space V* has dimension n and can be identified with the space of linear functionals on V; the dual basis corresponds to a chosen basis of V.

Examples illustrate the concept. The space R^n with its standard basis is finite-dimensional of dimension n. The space of polynomials P_n consisting of polynomials of degree at most n is finite-dimensional with dimension n+1. In contrast, spaces such as C[0,1] or L^2[0,1] are infinite-dimensional.

Finite-dimensionality is a central organizing principle in linear algebra, shaping the behavior of linear maps, bases, duality, and topological properties of vector spaces.

---