Noun
vector space (plural vector spaces)
(algebra, geometry, topology) A set of elements called vectors, together with some field and operations called addition (mapping two vectors to a vector) and scalar multiplication (mapping a vector and an element in the field to a vector), satisfying a list of constraints.
A vector space is a set of vectors which can be linearly combined.
Each vector space has a basis and dimension.
A basis, then, is a set of generators being a basis of the Lie algebra in the usual vector space sense. Source: Internet
Again from the Heine–Borel theorem, the closed unit ball of any finite-dimensional normed vector space is compact. Source: Internet
A linear functional f on a topological vector space X has either dense or closed kernel. Source: Internet
All norms on a finite-dimensional vector space are equivalent. Source: Internet
All of these concepts are usually defined as subsets of an ambient vector space (except for affine spaces, which are also considered as "vector spaces forgetting the origin"), rather than being axiomatized independently. Source: Internet
A nonempty subset W of a vector space V that is closed under addition and scalar multiplication (and therefore contains the 0-vector of V) is called a linear subspace of V, or simply a subspace of V, when the ambient space is unambiguously a vector space. Source: Internet