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The most familiar examples of this construction occur when considering Vector Space s (modules over a Field ) and Abelian Group s (modules over the ring Z of Integer s). The construction can also be extended to cover Banach Space s and Hilbert Space s. CONSTRUCTION FOR VECTOR SPACES AND ABELIAN GROUPS We give the construction first in these two cases, under the assumption that we have only two objects. Then we generalise to an arbitrary family of arbitrary modules. The key elements of the general construction are more clearly identified by considering these two cases in depth. Construction for two vector spaces Suppose ''V'' and ''W'' are Vector Space s over the Field ''K''. We can turn the Cartesian Product ''V'' × ''W'' into a vector space over ''K'' by defining the operations componentwise:
for ''v'', ''v''1, ''v''2 in ''V'', ''w'', ''w''1, ''w''2 in ''W'', and α in ''K''. The resulting vector space is called the ''direct sum'' of ''V'' and ''W'' and is usually denoted by a plus symbol inside a circle: : The subspace ''V'' × {0} of ''V'' ⊕ ''W'' is isomorphic to ''V'' and is often identified with ''V''; similarly for {0} × ''W'' and ''W''. (See ''internal direct sum'' below.) 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''. This construction readily generalises to any Finite number of vector spaces. Construction for two abelian groups For Abelian Group s ''G'' and ''H'' which are written additively, the Direct Product is also called direct sum. Thus we turn the Cartesian Product ''G'' × ''H'' into an abelian group by defining the operations componentwise:
for ''g''1, ''g''2 in ''G'', and ''h''1, ''h''2 in ''H''. Note that we can also extend the operation of taking integral multiples to the direct sum:
for ''g'' in ''G'', ''h'' in ''H'', and ''n'' an Integer . This parallels the extension of the scalar product of vector spaces to the direct sum above. The resulting abelian group is called the ''direct sum'' of ''G'' and ''H'' and is usually denoted by a plus symbol inside a circle: : The subspace ''G'' × {0} of ''G'' ⊕ ''H'' is isomorphic to ''G'' and is often identified with ''G''; similarly for {0} × ''H'' and ''H''. (See ''internal direct sum'' below.) With this identification, it is true that every element of ''G'' ⊕ ''H'' can be written in one and only one way as the sum of an element of ''G'' and an element of ''H''. The Rank of ''G'' ⊕ ''H'' is equal to the sum of the ranks of ''G'' and ''H''. This construction readily generalises to any Finite number of abelian groups. CONSTRUCTION FOR AN ARBITRARY FAMILY OF MODULES One should notice a clear similarity between the definitions of the direct sum of two vector spaces and of two abelian groups. In fact, each is a special case of the construction of the direct sum of two modules. Additionally, by modifying the definition one can accommodate the direct sum of an infinite family of modules. The precise definition is as follows. Assume ''R'' is some ring, and {''M''''i'' : ''i'' in ''I''} is a Family of left ''R''-modules indexed by the Set ''I''. The ''direct sum'' of {''M''''i''} is then defined to be the set of all Function s α with domain ''I'' such that α(''i'') ∈ ''M''''i'' for all ''i'' ∈ ''I'' and α(''i'') = 0 for all but Finite ly 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 (''r''α)(''i'') = ''r''(α(''i'')) for all ''i''. In this way, the direct sum becomes a left ''R''-module. We denote it by : 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''. INTERNAL DIRECT SUM 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′''. In this case, ''N'' and ''N′'' are '''complementary subspaces'''. CATEGORICAL INTERPRETATION In the language of Category Theory , the direct sum 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'' : 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 : such that ''f'' o ''ji'' = ''f''''i'' for all ''i''. DIRECT SUM OF MODULES WITH ADDITIONAL STRUCTURE If the modules we are considering carry some additional structure (e.g. a Norm or an Inner Product ), then the direct sum of the modules can often be made to carry this additional structure, as well. In this case, we obtain the Coproduct in the appropriate Category of all objects carrying the additional structure. The two most prominent examples occur for Banach Space s and Hilbert Space s. Direct sum of Banach spaces | ||
| For Example, If We Take The Index Set ''I'' | '''N''' and ''X''<sub>''i''</sub> = '''R''', then the direct sum &oplus<sub>''i''&isin'''N'''</sub> is the space ''l''<sub>1</sub>, which consists of all the sequences (''a''<sub>''i''</sub>) of reals with finite norm ''a'' = &sum<sub>''i''</sub> ''a''<sub>''i''</sub> |
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| For Example, If We Take The Index Set ''I'' | '''N''' and ''X''<sub>''i''</sub> = '''R''', then the direct sum &oplus<sub>''i''&isin'''N'''</sub> is the space ''l''<sub>2</sub>, which consists of all the sequences (''a''<sub>''i''</sub>) of reals with finite norm <math>\left\ a
ight\ = \sqrt{\sum_i \left\ a_i
ight\^2}</math> 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 |
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