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2 Construction for two vector spaces 3 Construction for arbitrarily many modules 4 Properties 5 Internal direct sums 6 Direct sum of Hilbert spaces |
Construction for two subgroups
Let G be a group; and assume that H and K are subgroups of G with the following properties:
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.
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 Mi. The direct sum of these modules is then defined to be the set of all functions α with domain I such that α(i) ∈ Mi 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 Mi''.
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 Mi.
If the Mi are actually vector spaces, then the dimension of the direct sum is equal to the sum of the dimensions of the Mi. 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 Mi (which are abelian groups) is naturally isomorphic to the tensor product of N with the direct sum of the Mi.
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 Mi 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 ji : Mi -> Oplusi∈I Mi which sends the elements of Mi to those functions which are zero for all arguments but i. If fi : Mi -> M are arbitrary R-linear maps for every i, then there exists precisely one R-linear map f : Oplusi∈I Mi -> M such that f o ji = fi for all i.
Suppose M is some R-module, and Mi 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 Mi, then we say that M is the internal direct sum of the submodules Mi. In this case, M is naturally isomorphic to the (external) direct sum of the Mi as defined above.
If finitely many Hilbert spaces H1,...,Hn 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
If infinitely many Hilbert spaces Hi 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 Hi to be the completion of this inner product space.
Alternatively and equivalently, one can define the direct sum of the Hilbert spaces Hi as the space of all functions α with domain I, such that α(i) is an element of Hi 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).Construction for two vector spaces
The resulting vector space is called the direct sum of V and W and is usually denoted by V (+) W, with a plus symbol inside a circle.Construction for arbitrarily many modules
Properties
Internal direct sums
Direct sum of Hilbert spaces
This turns the direct sum into a Hilbert space which contains the given Hilbertspaces as mutually orthogonal subspaces.
The inner product of two such function α and β is then defined as
This space is complete and we get a Hilbert space.