Direct sum of modules
The direct sum is a construction which combines several vector spaces (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 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 V (+) W, with a plus symbol inside a circle.
The subspace V × {0} of V oplus 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 oplus 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 arbitrary 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 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.
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 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.
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.
Internal direct sums
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.
Direct sum of Hilbert spaces
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
- <(x1,...,xn), (y1,...,yn)> = <x1, y1> + ... + <xn, yn>
This turns the direct sum into a Hilbert space which contains the given Hilbertspaces as mutually orthogonal subspaces.
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
- Σi || α(i) ||2 < ∞
The inner product of two such function α and β is then defined as
- <α, β> = Σi <α(i), β(i)>
This space is complete and we get a Hilbert space.