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.
We can also form direct sums with any finite number of summands, for example
are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spaces).
This relies on the fact that the direct sum is associative up to isomorphism.
In the case 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 the two 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.
The most familiar examples of this construction occur when 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, direct sum is a coproduct.
So for this category, a categorical direct sum is often simply called a coproduct to avoid any possible confusion.
The direct sum of group representations generalizes the direct sum of the underlying modules, adding a group action to it.
Another equivalent way of defining the direct sum is as follows: Given two representations
is the natural map obtained by coordinate-wise action as above.
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.)
Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If
is an infinite collection of nontrivial rings, then the direct sum of the underlying additive groups can be equipped with termwise multiplication, but this produces a rng, that is, a ring without a multiplicative identity.
if both are square matrices (and to an analogous block matrix, if not).
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
This 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.