The direct sum is an operation between structures in abstract algebra, a branch of mathematics.
is real coordinate space, is the Cartesian plane,
A similar process can be used to form the direct sum of two vector spaces or two modules.
Direct sums can also be formed with any finite number of summands; for example,
are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spaces).
That relies on the fact that the direct sum is associative up to isomorphism.
That is false, however, for some algebraic objects like nonabelian groups.
In the that where infinitely many objects are combined, the direct sum and direct product are not isomorphic even for abelian groups, vector spaces, or modules.
In this direct sum, the x and y axes intersect only at the origin (the zero vector).
the phrase "direct sum" is used, while if the group operation is written
A distinction is made between internal and external direct sums though both are isomorphic.
is expressible uniquely as an algebraic combination of an element of
For an example of an internal direct sum, consider
This definition generalizes to direct sums of finitely many abelian groups.
is the subgroup of the direct product that consists of the elements
The most familiar examples of that construction occur in considering vector spaces, which are modules over a field.
[4][5] In such a category, finite products and coproducts agree, and the direct sum is either of them: cf.
For example, in the category of abelian groups, the direct sum is a coproduct.
Therefore, for that category, a categorical direct sum is often called simply a coproduct to avoid any possible confusion.
The direct sum of group representations generalizes the direct sum of the underyling modules by adding a group action.
Another equivalent way of defining the direct sum is as follows: Given two representations
is not a coproduct in the category of rings, and should not be written as a direct sum.
[7] In the category of rings, the coproduct is given by a construction similar to the free product of groups.)
The use of direct sum terminology and notation is especially problematic in dealing with infinite families of rings.
is an infinite collection of nontrivial rings, the direct sum of the underlying additive groups may be equipped with termwise multiplication, but that produces a rng, a ring without a multiplicative identity.
such as a Banach space, is said to be a topological direct sum of two vector subspaces
is an isomorphism of topological vector spaces (meaning that this linear map is a bijective homeomorphism) in which case
That is true if and only if when considered as additive topological groups (so scalar multiplication is ignored),
For example, every vector subspace of a Hausdorff TVS that is not a closed subset is necessarily uncomplemented.
Every closed vector subspace of a Hilbert space is complemented.