podprostor

Podprostor is the Slovenian term for subspace, used in mathematics to denote a subset of a vector space that is itself a vector space under the same operations. The term is also used in topology to describe a subspace topology, where the subspace inherits open sets by intersection with the ambient space.

In linear algebra, let V be a vector space over a field F. A subset W ⊆ V is a podprostor if it is nonempty and closed under addition and scalar multiplication. Equivalently, W contains the zero vector and for any u, v ∈ W and any scalar α ∈ F, both u + v and αu belong to W. This ensures that W, with the restricted operations, satisfies the axioms of a vector space.

Examples in Euclidean space include:

- The set of all vectors (x, y, 0) in R^3, which forms a two-dimensional podprostor.

- The set of all multiples of a fixed vector, which is a one-dimensional podprostor.

- The trivial podprostor {0} and the entire space V itself.

Key properties include the existence of a basis and a well-defined dimension for any podprostor, with dim(W) ≤ dim(V). The intersection of subspaces is a subspace, as is the sum W + U of two subspaces, with the dimension formula dim(W + U) = dim(W) + dim(U) − dim(W ∩ U) holding in finite-dimensional spaces.

In topology, a podprostor Y of a topological space X carries the subspace topology: a set O ⊆ Y is open in Y if and only if O = U ∩ Y for some open set U in X. This concept is central to discussions of continuity, convergence, and topological properties inherited by subspaces.

See also: vector space, basis, dimension, linear independence, subspace topology.