Table of contents |
2 Construction for two vector spaces 3 Construction for arbitrarily many modules 4 Properties 5 Internal direct sums 6 Direct sum of Hilbert spaces |
The article direct sum of groups contains more specific implications of the direct sum in the group theory sense.
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:
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.
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 (+)_{'\'i∈I} 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} -> Oplus_{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 : Oplus_{i∈I} M_{i} -> M such that f o j_{i} = f_{i} for all i.
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.
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
Every Hilbert space is isomorphic to a direct sum of sufficiently many copies of the base field (either R or C).