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An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is an abelian group under addition, and the four remaining axioms (related to the scalar multiplication) say that this operation defines a ring homomorphism from the ...
In linear algebra, the closure of a non-empty subset of a vector space (under vector-space operations, that is, addition and scalar multiplication) is the linear span of this subset. It is a vector space by the preceding general result, and it can be proved easily that is the set of linear combinations of elements of the subset.
Such an F is called a closure of f in X × Y, is denoted by f, and necessarily extends f. Additional assumptions for linear maps: If in addition, S, X, and Y are topological vector spaces and f : S → Y is a linear map then to call f closable we also require that the set D be a vector subspace of X and the closure of f be a linear map.
A linear operator : is closable in if there exists a vector subspace containing and a function (resp. multifunction) : whose graph is equal to the closure of the set in . Such an F {\displaystyle F} is called a closure of f {\displaystyle f} in X × Y {\displaystyle X\times Y} , is denoted by f ¯ , {\displaystyle {\overline {f}},} and ...
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace [1] [note 1] is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces .
A subset of a vector space over an ordered field is a cone (or sometimes called a linear cone) if for each in and positive scalar in , the product is in . [2] Note that some authors define cone with the scalar ranging over all non-negative scalars (rather than all positive scalars, which does not include 0). [3]
The definition using intersections of convex sets may be extended to non-Euclidean geometry, and the definition using convex combinations may be extended from Euclidean spaces to arbitrary real vector spaces or affine spaces; convex hulls may also be generalized in a more abstract way, to oriented matroids. [6]
The definition of a point of closure of a set is closely related to the definition of a limit point of a set.The difference between the two definitions is subtle but important – namely, in the definition of a limit point of a set , every neighbourhood of must contain a point of other than itself, i.e., each neighbourhood of obviously has but it also must have a point of that is not equal to ...