Search results
Results from the WOW.Com Content Network
A topological vector space (TVS) , such as a Banach space, is said to be a topological direct sum of two vector subspaces and if the addition map (,) + is an isomorphism of topological vector spaces (meaning that this linear map is a bijective homeomorphism), in which case and are said to be topological complements in .
This parallels the extension of the scalar product of vector spaces to the direct sum above. The resulting abelian group is called the direct sum of G and H and is usually denoted by a plus symbol inside a circle: It is customary to write the elements of an ordered sum not as ordered pairs (g, h), but as a sum g + h.
This vector space is the coproduct (or direct sum) of countably many copies of the vector space F. Note the role of the finiteness condition here. One could consider arbitrary sequences of elements in F, which also constitute a vector space with the same operations, often denoted by F N - see below. F N is the product of countably many copies of F.
The vector space is said to be the algebraic direct sum (or direct sum in the category of vector spaces) when any of the following equivalent conditions are satisfied: The addition map : is a vector space isomorphism. [1] [2] The addition map is bijective.
In particular, the direct sum of square matrices is a block diagonal matrix. The adjacency matrix of the union of disjoint graphs (or multigraphs) is the direct sum of their adjacency matrices. Any element in the direct sum of two vector spaces of matrices can be represented as a direct sum of two matrices. In general, the direct sum of n ...
An -graded vector space, often called simply a graded vector space without the prefix , is a vector space V together with a decomposition into a direct sum of the form V = ⨁ n ∈ N V n {\displaystyle V=\bigoplus _{n\in \mathbb {N} }V_{n}}
In mathematics, a group G is called the direct sum [1] [2] of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract algebra, this method of construction of groups can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information.
When the scalar field is the real numbers, the vector space is called a real vector space, and when the scalar field is the complex numbers, the vector space is called a complex vector space. [4] These two cases are the most common ones, but vector spaces with scalars in an arbitrary field F are also commonly considered.