Unterraums

Unterraums is not a standard term in German mathematical literature; the correct plural form of Unterraum is Unterräume. The form Unterraums may appear as the genitive singular (des Unterraums) or as a nonstandard plural in some texts. In modern mathematics, Unterräume is preferred when referring to multiple subspaces.

In linear algebra, a Unterraum (subspace) of a vector space V over a field F is a non-empty subset W ⊆ V that is closed under vector addition and scalar multiplication: for all u, v in W and all a in F, u + v is in W and a u is in W. With these operations inherited from V, W itself forms a vector space over F. The concept is central to many constructions in algebra and geometry.

Examples include the zero subspace {0}, the entire space V, any line through the origin in R^n, and any plane through the origin. These examples illustrate that subspaces must pass through the origin and be closed under linear combinations.

Key properties and constructions involve the span of a set of vectors, which is the smallest Unterraum containing that set, and the intersection and sum of subspaces, which are themselves subspaces. The dimension of a Unterraum equals the cardinality of a basis, and dim(W + U) = dim(W) + dim(U) − dim(W ∩ U) holds for subspaces W and U of V.

Subspaces are fundamental in many areas of mathematics, including the study of linear systems, projections, eigenvectors, and functional-analytic settings where closed subspaces play a crucial role. See also Unterraum (mathematics), vector space, basis, and span.