The general case is a direct sum of these two cases.

The right hand side is a direct sum of fields.

It is, of course, nothing else than the Lie algebra direct sum.

As an example, consider the direct sum , where is the set of real numbers.

The product of two elements and is in this direct sum algebra.

We conclude this section by considering characters of direct sums.

The direct sum with this norm is again a Banach space.

Just an isomorphism of B with the direct sum is not sufficient.

The finiteness condition is built into the definition of the direct sum.

Coproducts in Rng are not the same as direct sums.

Decomposition of a vector space into direct sums is not unique in general.

For a direct sum this is clear, as one can inject from or project to the summands.

In the direct sum, all but finitely many coordinates must be zero.

A finite direct sum of fields is a semi-local ring.

The wedge product endows the direct sum of these groups with a ring structure.

This means that we can form finite direct sums and direct products.

A distinction is made between internal and external direct sums, though the two are isomorphic.

Where V+V represent the direct sum of the two subspaces.

Given two objects and , their direct sum is written as .

Therefore, one gets a direct sum of vector spaces where:

The direct sum is an operation from abstract algebra, a branch of mathematics.

For example, in the category of abelian groups, direct sum is a coproduct.

More generally, one can consider finite direct sums of matrix algebras.

These direct sums also arise in the classification of composition algebras.

In this generality, one obtains a composition series, rather than a direct sum.