Direct sum

abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. In a sense, the direct sum of vector spaces is the "most general" vector space that contains the given ones as subspaces.

Construction for two subgroups

Let G be a group; and assume that H and K are normal subgroups of G

• H ∩ K = E, the trivial subgroup of G
• G = H * K.

Then we say that G is the direct sum of subgroups H and K, written as G = H + K. In this case, for all h in H and k in K, h*k = k*h, and for every element g in G, there are unqiue h in H, k in K, such that g = h*k. This in turn is roughly equivalent to saying that G is isomorphic to the direct product H × K, and so the direct sum is an "internal" direct sum.

The article direct sum of groups contains more specific implications of the direct sum in the group theory sense.

Construction for two vector spaces

Suppose V and W are vector spaces over the field K. We can turn the cartesian product V × W into a vector space over K by defining the operations componentwise:

• (v₁, w₁) + (v₂, w₂) = (v₁ + v₂, w₁ + w₂)
• α (v, w) = (α v, α w)

The resulting vector space is called the direct sum of V and W and is usually denoted by a plus symbol inside a circle: V ⊕ W.

The subspace V × {0} of V ⊕ W is isomorphic to V and is often identified with V; similar for {0} × W and W. With this identification, it is true that every element of V ⊕ W can be written in one and only one way as the sum of an element of V and an element of W. The dimension of V ⊕ W is equal to the sum of the dimensions of V and W.

Construction for arbitrarily many modules

The direct sum can also be defined for abelian groups and for modules over arbitrary rings. Note that abelian groups are modules over the ring Z of integers, and vector spaces are modules over fields. So we only need to consider the case of modules in the sequel.

Assume R is some ring, I some set, and for every i in I we are given a left R-module M_i. The direct sum of these modules is then defined to be the set of all functions α with domain I such that α(i) ∈ M_i for all i ∈ I and α(i) = 0 for all but finitely many indices i.

Two such functions α and β can be added by writing (α + β)(i) = α(i) + β(i) for all i (note that this is again zero for all but finitely many indices), and such a function can be multiplied with an element r from R by writing (rα)(i) = r(α(i)) for all i. In this way, the direct sum becomes a left R module. We denote it by

<math> \bigoplus_{i \in I} M_i <math>

Properties

With the proper identifications, we can again say that every element x of the direct sum can be written in one and only one way as a sum of finitely many elements of the M_i.

If the M_i are actually vector spaces, then the dimension of the direct sum is equal to the sum of the dimensions of the M_i. The same is true for the rank of abelian groups and the length of modules.

Every vector space over the field K is isomorphic to a direct sum of sufficiently many copies of K, so in a sense only these direct sums have to be considered. This is not true for modules over arbitrary rings.

The tensor product distributes over direct sums in the following sense: if N is some right R-module, then the direct sum of the tensor products of N with M_i (which are abelian groups) is naturally isomorphic to the tensor product of N with the direct sum of the M_i. Direct sums are also commutative and associative, meaning that it doesn't matter in which order one forms the direct sum.

The group of R-linear homomorphisms from the direct sum to some left R-module L is naturally isomorphic to the direct product of the groups of R-linear homomorphisms from M_i to L.

In the language of category theory, the direct product is a coproduct and hence a colimit in the category of left R-modules, which means that it is characterized by the following universal property. For every i in I, consider the natural embedding j_i : M_i -> ⊕_i∈I M_i which sends the elements of M_i to those functions which are zero for all arguments but i. If f_i : M_i -> M are arbitrary R-linear maps for every i, then there exists precisely one R-linear map f : ⊕_i∈I M_i -> M such that f o j_i = f_i for all i.

Internal direct sums

Suppose M is some R-module, and M_i is a submodule of M for every i in I. If every x in M can be written in one and only one way as a sum of finitely many elements of the M_i, then we say that M is the internal direct sum of the submodules M_i. In this case, M is naturally isomorphic to the (external) direct sum of the M_i as defined above.

A direct summand of M is a submodule N such that there is some other submodule N′ of M such that M is the internal direct sum of N and N′.

Direct sum of Banach spaces

The direct sum of two Banach spaces X and Y is the direct sum of X and Y considered as vector spaces, with the norm ||(x,y)|| = ||x||_X ||y||_Y for all x ∈ X and y ∈ Y.

Generally, if X_i, where i traverses the index set I, is a collection of Banach spaces, then the direct sum ⊕_i∈I X_i consists of all functions x with domain I such that x(i) ∈ X_i for all i ∈ I and

<math> \sum_{i \in I} \| x(i) \|_{X_i} \mbox{ is finite.} <math>

The norm is given by the sum above. The direct sum with this norm is again a Banach space.

For example, if we take the index set I = N and X_i = R, then the direct sum ⊕_i∈N is the space l₁, which consists of all the sequences (a_i) of reals with finite norm ||a|| = ∑_i |a_i|.

Direct sum of Hilbert spaces

If finitely many Hilbert spaces H₁,...,H_n are given, one can construct their direct sum as above (since they are vector spaces), and then turn the direct sum into a Hilbert space by defining the inner product as

<(x₁,...,x_n), (y₁,...,y_n)> = <x₁, y₁> + ... + <x_n, y_n>

This turns the direct sum into a Hilbert space which contains the given Hilbert spaces as mutually orthogonal subspaces.

If infinitely many Hilbert spaces H_i for i in I are given, we can carry out the same construction; notice that when defining the inner product, only finitely many summands will be non-zero. However, the result will only be an inner product space and it won't be complete. We then define the direct sum of the Hilbert spaces H_i to be the completion of this inner product space.

Alternatively and equivalently, one can define the direct sum of the Hilbert spaces H_i as the space of all functions α with domain I, such that α(i) is an element of H_i for every i in I and

Σ_i || α(i) ||² < ∞

The inner product of two such function α and β is then defined as

<α, β> = Σ_i <α(i), β(i)>

This space is complete and we get a Hilbert space.

For example, if we take the index set I = N and X_i = R, then the direct sum ⊕_i∈N is the space l₂, which consists of all the sequences (a_i) of reals with finite norm ||a|| = ∑_i |a_i|². Comparing this with the example for Banach spaces, we see that the Banach space direct sum and the Hilbert space direct sum are not necessarily the same. But if there are only finitely many summands, then the Banach space direct sum is isomorphic to the Hilbert space direct sum.

Every Hilbert space is isomorphic to a direct sum of sufficiently many copies of the base field (either R or C).