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abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, the direct sum is a construction which combines several
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprodu ...
. Contrast with the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
, which is the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
notion. The most familiar examples of this construction occur when considering
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s (modules over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
) and
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s (modules over the ring Z of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s). The construction may also be extended to cover
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s and
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s. See the article
decomposition of a module In abstract algebra, a decomposition of a module is a way to write a module as a direct sum of modules. A type of a decomposition is often used to define or characterize modules: for example, a semisimple module is a module that has a decompositio ...
for a way to write a module as a direct sum of submodules.


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 generalize 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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s over the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''K''. The
cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
''V'' × ''W'' can be given the structure of a vector space over ''K'' by defining the operations componentwise: * (''v''1, ''w''1) + (''v''2, ''w''2) = (''v''1 + ''v''2, ''w''1 + ''w''2) * ''α'' (''v'', ''w'') = (''α'' ''v'', ''α'' ''w'') for ''v'', ''v''1, ''v''2 ∈ ''V'', ''w'', ''w''1, ''w''2 ∈ ''W'', and ''α'' ∈ ''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: V \oplus W It is customary to write the elements of an ordered sum not as ordered pairs (''v'', ''w''), but as a sum ''v'' + ''w''. The subspace ''V'' × of ''V'' ⊕ ''W'' is isomorphic to ''V'' and is often identified with ''V''; similarly for × ''W'' and ''W''. (See ''internal direct sum'' below.) With this identification, 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 In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of ''V'' ⊕ ''W'' is equal to the sum of the dimensions of ''V'' and ''W''. One elementary use is the reconstruction of a finite vector space from any subspace ''W'' and its orthogonal complement: \mathbb^n = W \oplus W^ This construction readily generalizes to any
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
number of vector spaces.


Construction for two abelian groups

For
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s ''G'' and ''H'' which are written additively, the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of ''G'' and ''H'' is also called a direct sum . Thus the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
''G'' × ''H'' is equipped with the structure of an abelian group by defining the operations componentwise: : (''g''1, ''h''1) + (''g''2, ''h''2) = (''g''1 + ''g''2, ''h''1 + ''h''2) for ''g''1, ''g''2 in ''G'', and ''h''1, ''h''2 in ''H''. Integral multiples are similarly defined componentwise by : ''n''(''g'', ''h'') = (''ng'', ''nh'') for ''g'' in ''G'', ''h'' in ''H'', and ''n'' an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
. 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: G \oplus H It is customary to write the elements of an ordered sum not as ordered pairs (''g'', ''h''), but as a sum ''g'' + ''h''. The
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
''G'' × of ''G'' ⊕ ''H'' is isomorphic to ''G'' and is often identified with ''G''; similarly for × ''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 Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...
of ''G'' ⊕ ''H'' is equal to the sum of the ranks of ''G'' and ''H''. This construction readily generalises to any
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
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 Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
. Additionally, by modifying the definition one can accommodate the direct sum of an infinite family of modules. The precise definition is as follows . Let ''R'' be a ring, and a
family Family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its ...
of left ''R''-modules indexed by the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
''I''. The ''direct sum'' of is then defined to be the set of all sequences (\alpha_i) where \alpha_i \in M_i and \alpha_i = 0 for
cofinitely many In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocoun ...
indices ''i''. (The
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
is analogous but the indices do not need to cofinitely vanish.) It can also be defined as functions α from ''I'' to the
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (th ...
of the modules ''M''''i'' such that α(''i'') ∈ ''M''''i'' for all ''i'' ∈ ''I'' and α(''i'') = 0 for
cofinitely many In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocoun ...
indices ''i''. These functions can equivalently be regarded as finitely supported sections of the
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
over the index set ''I'', with the fiber over i \in I being M_i. This set inherits the module structure via component-wise addition and scalar multiplication. Explicitly, two such sequences (or functions) α and β can be added by writing (\alpha + \beta)_i = \alpha_i + \beta_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 defining r(\alpha)_i = (r\alpha)_i for all ''i''. In this way, the direct sum becomes a left ''R''-module, and it is denoted \bigoplus_ M_i. It is customary to write the sequence (\alpha_i) as a sum \sum \alpha_i. Sometimes a primed summation \sum ' \alpha_i is used to indicate that
cofinitely many In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocoun ...
of the terms are zero.


Properties

* The direct sum is a
submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mod ...
of the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of the modules ''M''''i'' . The direct product is the set of all functions ''α'' from ''I'' to the disjoint union of the modules ''M''''i'' with ''α''(''i'')∈''M''''i'', but not necessarily vanishing for all but finitely many ''i''. If the index set ''I'' is finite, then the direct sum and the direct product are equal. * Each of the modules ''M''''i'' may be identified with the submodule of the direct sum consisting of those functions which vanish on all indices different from ''i''. With these identifications, every element ''x'' of the direct sum can be written in one and only one way as a sum of finitely many elements from the modules ''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 In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
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
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
and
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
(up to isomorphism), meaning that it doesn't matter in which order one forms the direct sum. * The abelian group of ''R''- linear homomorphisms from the direct sum to some left ''R''-module ''L'' is naturally isomorphic to the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of the abelian groups of ''R''-linear homomorphisms from ''M''''i'' to ''L'': \operatorname_R\biggl( \bigoplus_ M_i,L\biggr) \cong \prod_\operatorname_R\left(M_i,L\right). Indeed, there is clearly a
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
''τ'' from the left hand side to the right hand side, where ''τ''(''θ'')(''i'') is the ''R''-linear homomorphism sending ''x''∈''M''''i'' to ''θ''(''x'') (using the natural inclusion of ''M''''i'' into the direct sum). The inverse of the homomorphism ''τ'' is defined by \tau^(\beta)(\alpha) = \sum_ \beta(i)(\alpha(i)) for any ''α'' in the direct sum of the modules ''M''''i''. The key point is that the definition of ''τ''−1 makes sense because ''α''(''i'') is zero for all but finitely many ''i'', and so the sum is finite.In particular, the
dual vector space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of a direct sum of vector spaces is isomorphic to the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of the duals of those spaces. *The ''finite'' direct sum of modules is a
biproduct In category theory and its applications to mathematics, a biproduct of a finite collection of objects, in a category with zero objects, is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for ...
: If p_k: A_1 \oplus \cdots \oplus A_n \to A_k are the canonical projection mappings and i_k: A_k \mapsto A_1 \oplus \cdots \oplus A_n are the inclusion mappings, then i_1 \circ p_1 + \cdots + i_n \circ p_n equals the identity morphism of ''A''1 ⊕ ⋯ ⊕ ''A''''n'', and p_k \circ i_l is the identity morphism of ''A''''k'' in the case ''l'' = ''k'', and is the zero map otherwise.


Internal direct sum

Suppose ''M'' is some ''R''-module, and ''M''''i'' is a
submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mod ...
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 submodule ''N'' of ''M'' is a direct summand of ''M'' if there exists 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 submodules.


Universal property

In the language of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, the direct sum is a
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprodu ...
and hence a
colimit In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such ...
in the category of left ''R''-modules, which means that it is characterized by the following
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
. For every ''i'' in ''I'', consider the ''natural embedding'' :j_i : M_i \rightarrow \bigoplus_ M_i which sends the elements of ''M''''i'' to those functions which are zero for all arguments but ''i''. Now let ''M'' be an arbitrary ''R''-module and ''f''''i'' : ''M''''i'' → ''M'' be arbitrary ''R''-linear maps for every ''i'', then there exists precisely one ''R''-linear map :f : \bigoplus_ M_i \rightarrow M such that ''f'' o ''ji'' = ''f''''i'' for all ''i''.


Grothendieck group

The direct sum gives a collection of objects the structure of a
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
, in that the addition of objects is defined, but not subtraction. In fact, subtraction can be defined, and every commutative monoid can be extended to an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
. This extension is known as the
Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic i ...
. The extension is done by defining equivalence classes of pairs of objects, which allows certain pairs to be treated as inverses. The construction, detailed in the article on the Grothendieck group, is "universal", in that it has the
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
of being unique, and homomorphic to any other embedding of a commutative monoid in an abelian group.


Direct sum of modules with additional structure

If the modules we are considering carry some additional structure (for example, a
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
or an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
), 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 category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprodu ...
in the appropriate
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
of all objects carrying the additional structure. Two prominent examples occur for
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s and
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s. In some classical texts, the phrase "direct sum of
algebras over a field In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition a ...
" is also introduced for denoting the
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
that is presently more commonly called a
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of algebras; that is, the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
of the
underlying set In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite se ...
s with the
componentwise operation In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
s. This construction, however, does not provide a coproduct in the category of algebras, but a direct product (''see note below'' and the remark on direct sums of rings).


Direct sum of algebras

A direct sum of
algebras In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition a ...
X and Y is the direct sum as vector spaces, with product :(x_1 + y_1) (x_2 + y_2) = (x_1 x_2 + y_1 y_2). Consider these classical examples: :\mathbf \oplus \mathbf is ring isomorphic to
split-complex number In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
s, also used in
interval analysis Interval arithmetic (also known as interval mathematics, interval analysis, or interval computation) is a mathematical technique used to put bounds on rounding errors and measurement errors in mathematical computation. Numerical methods using ...
. :\mathbf \oplus \mathbf is the algebra of
tessarine In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^* = (w, -z), and the product of two bicomplex numbers as :(u,v)(w,z) = (u w - v z, u z ...
s introduced by
James Cockle Sir James Cockle FRS FRAS FCPS (14 January 1819 – 27 January 1895) was an English lawyer and mathematician. Cockle was born on 14 January 1819. He was the second son of James Cockle, a surgeon, of Great Oakley, Essex. Educated at Charterho ...
in 1848. :\mathbf \oplus \mathbf, called the
split-biquaternion In mathematics, a split-biquaternion is a hypercomplex number of the form :q = w + xi + yj + zk where ''w'', ''x'', ''y'', and ''z'' are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient ''w'', ''x' ...
s, was introduced by
William Kingdon Clifford William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...
in 1873.
Joseph Wedderburn Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882 – 9 October 1948) was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a fie ...
exploited the concept of a direct sum of algebras in his classification of
hypercomplex number In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group represent ...
s. See his ''Lectures on Matrices'' (1934), page 151. Wedderburn makes clear the distinction between a direct sum and a direct product of algebras: For the direct sum the field of scalars acts jointly on both parts: \lambda (x \oplus y) = \lambda x \oplus \lambda y while for the direct product a scalar factor may be collected alternately with the parts, but not both: \lambda (x,y) = (\lambda x, y) = (x, \lambda y). \! Ian R. Porteous uses the three direct sums above, denoting them ^2 R,\ ^2 C,\ ^2 H, as rings of scalars in his analysis of ''Clifford Algebras and the Classical Groups'' (1995). The construction described above, as well as Wedderburn's use of the terms and follow a different convention than the one in
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
. In categorical terms, Wedderburn's is a
categorical product In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or ring ...
, whilst Wedderburn's is a coproduct (or categorical sum), which (for commutative algebras) actually corresponds to the
tensor product of algebras In mathematics, the tensor product of two algebras over a commutative ring ''R'' is also an ''R''-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of such products is to describe the prod ...
.


Direct sum of Banach spaces

The direct sum of two
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s 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 \in X and y \in Y. Generally, if X_i is a collection of Banach spaces, where i traverses the
index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...
I, then the direct sum \bigoplus_ X_i is a module consisting of all functions x defined over I such that x(i) \in X_i for all i \in I and \sum_ \, x(i)\, _ < \infty. 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 \bigoplus_ X_i is the space \ell_1, which consists of all the sequences \left(a_i\right) of reals with finite norm \, a\, = \sum_i \left, a_i\. A closed subspace A of a Banach space X is complemented if there is another closed subspace B of X such that X is equal to the internal direct sum A \oplus B. Note that not every closed subspace is complemented; e.g. c_0 is not complemented in \ell^\infty.


Direct sum of modules with bilinear forms

Let \left\ be a
family Family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its ...
indexed by I of modules equipped with
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
s. The orthogonal direct sum is the module direct sum with bilinear form B defined by B\left(\right) = \sum_ b_i\left(\right) in which the summation makes sense even for infinite index sets I because only finitely many of the terms are non-zero.


Direct sum of Hilbert spaces

If finitely many
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s H_1, \ldots, H_n are given, one can construct their orthogonal direct sum as above (since they are vector spaces), defining the inner product as: \left\langle \left(x_1, \ldots, x_n\right), \left(y_1, \ldots, y_n\right) \right\rangle = \langle x_1, y_1 \rangle + \cdots + \langle x_n, y_n \rangle. The resulting direct sum is a Hilbert space which contains the given Hilbert spaces as mutually
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
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 In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
and it will not necessarily be
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
. 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 \alpha(i) is an element of H_i for every i \in I and: \sum_i \left\, \alpha_\right\, ^2 < \infty. The inner product of two such function α and β is then defined as: \langle\alpha,\beta\rangle=\sum_i \langle \alpha_i,\beta_i \rangle. 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 \oplus_ X_i is the space \ell_2, which consists of all the sequences \left(a_i\right) of reals with finite norm \, a\, = \sqrt. Comparing this with the example for
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s, 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, although the norm will be different. Every Hilbert space is isomorphic to a direct sum of sufficiently many copies of the base field, which is either \R \text \Complex. This is equivalent to the assertion that every Hilbert space has an orthonormal basis. More generally, every closed subspace of a Hilbert space is complemented because it admits an
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
. Conversely, the Lindenstrauss–Tzafriri theorem asserts that if every closed subspace of a Banach space is complemented, then the Banach space is isomorphic (topologically) to a Hilbert space.


See also

* * * * *


References

* . * . * . * * . {{DEFAULTSORT:Direct Sum Of Modules Linear algebra Module theory