4860:
1801:
107:
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.
5496:, 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.
4596:
4684:
5238:
1669:
2729:. 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
1955:
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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
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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
3709:
5166:
3328:
1427:
3130:(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:
968:
5309:
4924:
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
4679:
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2275:
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4855:{\displaystyle \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 .}
3926:
3793:
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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
1001:
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5587: – short exact sequence where the middle term is the direct sum of the outer ones with the structure maps being the canonical inclusion and projection
5097:
2721:, 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
2080:
1796:{\displaystyle \operatorname {Hom} _{R}{\biggl (}\bigoplus _{i\in I}M_{i},L{\biggr )}\cong \prod _{i\in I}\operatorname {Hom} _{R}\left(M_{i},L\right).}
841:. Additionally, by modifying the definition one can accommodate the direct sum of an infinite family of modules. The precise definition is as follows (
5534:
This is equivalent to the assertion that every
Hilbert space has an orthonormal basis. More generally, every closed subspace of a Hilbert space is
2008:
5546:
asserts that if every closed subspace of a Banach space is complemented, then the Banach space is isomorphic (topologically) to a
Hilbert space.
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are given, one can construct their orthogonal direct sum as above (since they are vector spaces), defining the inner product as:
2749:), then the direct sum of the modules can often be made to carry this additional structure, as well. In this case, we obtain the
1143:
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may be identified with the submodule of the direct sum consisting of those functions which vanish on all indices different from
1286:
2627:
5502:
5569:
3969:
1209:(note that this is again zero for all but finitely many indices), and such a function can be multiplied with an element
4591:{\displaystyle B\left({\left({x_{i}}\right),\left({y_{i}}\right)}\right)=\sum _{i\in I}b_{i}\left({x_{i},y_{i}}\right)}
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Every
Hilbert space is isomorphic to a direct sum of sufficiently many copies of the base field, which is either
3354:
96:
58:
1574:, so in a sense only these direct sums have to be considered. This is not true for modules over arbitrary rings.
4627:
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802:
704:
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2792:. This construction, however, does not provide a coproduct in the category of algebras, but a direct product (
1546:
are actually vector spaces, then the dimension of the direct sum is equal to the sum of the dimensions of the
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of the direct sum can be written in one and only one way as a sum of finitely many elements from the modules
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while for the direct product a scalar factor may be collected alternately with the parts, but not both:
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of being unique, and homomorphic to any other embedding of a commutative monoid in an abelian group.
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Positive-definite kernel § Connection with reproducing kernel
Hilbert spaces and feature maps
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The resulting direct sum is a
Hilbert space which contains the given Hilbert spaces as mutually
4247:
4014:
5233:{\displaystyle \langle \alpha ,\beta \rangle =\sum _{i}\langle \alpha _{i},\beta _{i}\rangle .}
686:. This parallels the extension of the scalar product of vector spaces to the direct sum above.
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is equipped with the structure of an abelian group by defining the operations componentwise:
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1626:(up to isomorphism), meaning that it doesn't matter in which order one forms the direct sum.
1087:
38:
5558: – in category theory, an object that is both product and coproduct in compatible ways
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4965:
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The norm is given by the sum above. The direct sum with this norm is again a Banach space.
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1991:
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870:
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428:. One elementary use is the reconstruction of a finite vector space from any subspace
54:
17:
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Alternatively and equivalently, one can define the direct sum of the
Hilbert spaces
5493:
4929:
3380:
2758:
2757:
of all objects carrying the additional structure. Two prominent examples occur for
1804:
1077:
125:
85:
65:
2741:
If the modules we are considering carry some additional structure (for example, a
5714:
5608:
1623:
1619:
1950:{\displaystyle \tau ^{-1}(\beta )(\alpha )=\sum _{i\in I}\beta (i)(\alpha (i))}
4865:
2797:
1987:
1634:
825:
729:
It is customary to write the elements of an ordered sum not as ordered pairs (
483:
341:
It is customary to write the elements of an ordered sum not as ordered pairs (
31:
5555:
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3362:
3065:
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can be written in exactly one way as a sum of finitely many elements of the
2327:
2002:
1602:(which are abelian groups) is naturally isomorphic to the tensor product of
1443:
50:
3337:
The construction described above, as well as
Wedderburn's use of the terms
3122:
exploited the concept of a direct sum of algebras in his classification of
4274:
749:
683:
81:
64:
The most familiar examples of this construction occur when considering
5154:{\displaystyle \sum _{i}\left\|\alpha _{(i)}\right\|^{2}<\infty .}
2718:
2715:
5163:
The inner product of two such function α and β is then defined as:
2977:{\displaystyle (x_{1}+y_{1})(x_{2}+y_{2})=(x_{1}x_{2}+y_{1}y_{2}).}
793:
can be written in one and only one way as the sum of an element of
785:
below.) With this identification, it is true that every element of
400:
can be written in one and only one way as the sum of an element of
2142:{\displaystyle i_{k}:A_{k}\mapsto A_{1}\oplus \cdots \oplus A_{n}}
4598:
in which the summation makes sense even for infinite index sets
2714:
The direct sum gives a collection of objects the structure of a
2476:-modules, which means that it is characterized by the following
1505:
is finite, then the direct sum and the direct product are equal.
1015:
is analogous but the indices do not need to cofinitely vanish.)
5485:{\textstyle \|a\|={\sqrt {\sum _{i}\left\|a_{i}\right\|^{2}}}.}
3365:, which (for commutative algebras) actually corresponds to the
2556:{\displaystyle j_{i}:M_{i}\rightarrow \bigoplus _{i\in I}M_{i}}
4404:{\displaystyle \left\{\left(M_{i},b_{i}\right):i\in I\right\}}
2070:{\displaystyle p_{k}:A_{1}\oplus \cdots \oplus A_{n}\to A_{k}}
3176:{\displaystyle \lambda (x\oplus y)=\lambda x\oplus \lambda y}
1570:
is isomorphic to a direct sum of sufficiently many copies of
2384:
is naturally isomorphic to the (external) direct sum of the
1463:, §II.1.7). The direct product is the set of all functions
1198:{\displaystyle (\alpha +\beta )_{i}=\alpha _{i}+\beta _{i}}
99:
for a way to write a module as a direct sum of submodules.
4273:
Note that not every closed subspace is complemented; e.g.
3888:{\displaystyle \sum _{i\in I}\|x(i)\|_{X_{i}}<\infty .}
3256:{\displaystyle \lambda (x,y)=(\lambda x,y)=(x,\lambda y).}
1497:, but not necessarily vanishing for all but finitely many
635:
Integral multiples are similarly defined componentwise by
2214:{\displaystyle i_{1}\circ p_{1}+\cdots +i_{n}\circ p_{n}}
701:
and is usually denoted by a plus symbol inside a circle:
313:
and is usually denoted by a plus symbol inside a circle:
2575:
to those functions which are zero for all arguments but
1581:
distributes over direct sums in the following sense: if
5574:
Pages displaying short descriptions of redirect targets
2672:{\displaystyle f:\bigoplus _{i\in I}M_{i}\rightarrow M}
1589:-module, then the direct sum of the tensor products of
5527:{\displaystyle \mathbb {R} {\text{ or }}\mathbb {C} .}
5427:
4932:. We then define the direct sum of the Hilbert spaces
4618:
because only finitely many of the terms are non-zero.
4082:
1990:
of a direct sum of vector spaces is isomorphic to the
1856:
into the direct sum). The inverse of the homomorphism
1810:
from the left hand side to the right hand side, where
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Pages displaying wikidata descriptions as a fallback
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Pages displaying wikidata descriptions as a fallback
30:
For the broader use of the term in mathematics, see
5242:This space is complete and we get a Hilbert space.
4959:to be the completion of this inner product space.
4004:{\displaystyle \bigoplus _{i\in \mathbb {N} }X_{i}}
2768:In some classical texts, the phrase "direct sum of
1072:. These functions can equivalently be regarded as
473:{\displaystyle \mathbb {R} ^{n}=W\oplus W^{\perp }}
392:below.) With this identification, every element of
5658:, University Mathematical Texts, Oliver and Boyd,
5526:
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2851:is the direct sum as vector spaces, with product
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146:can be given the structure of a vector space over
84:). The construction may also be extended to cover
1727:
1688:
897:} is then defined to be the set of all sequences
154:, §18) by defining the operations componentwise:
103:Construction for vector spaces and abelian groups
5617:Ergebnisse der Mathematik und ihrer Grenzgebiete
5349:{\displaystyle \oplus _{i\in \mathbb {N} }X_{i}}
3266:uses the three direct sums above, denoting them
3100:{\displaystyle \mathbf {H} \oplus \mathbf {H} ,}
518:
5572: – Decomposition of an algebraic structure
4129:{\textstyle \|a\|=\sum _{i}\left|a_{i}\right|.}
3057:{\displaystyle \mathbf {C} \oplus \mathbf {C} }
3013:{\displaystyle \mathbf {R} \oplus \mathbf {R} }
2737:Direct sum of modules with additional structure
832:Construction for an arbitrary family of modules
3345:follow a different convention than the one in
1986:, and so the sum is finite.In particular, the
5699:, Englewood Cliffs, NJ: Prentice Hall, Inc.,
3531:{\displaystyle \|(x,y)\|=\|x\|_{X}+\|y\|_{Y}}
1279:. In this way, the direct sum becomes a left
1268:{\displaystyle r(\alpha )_{i}=(r\alpha )_{i}}
824:This construction readily generalises to any
482:This construction readily generalizes to any
8:
5695:Dummit, David S.; Foote, Richard M. (1991),
5434:
5428:
5224:
5198:
5182:
5170:
4989:as the space of all functions α with domain
4846:
4820:
4808:
4782:
4443:is the module direct sum with bilinear form
4089:
4083:
3860:
3844:
3519:
3512:
3500:
3493:
3487:
3469:
1521:. With these identifications, every element
3462:considered as vector spaces, with the norm
3332:Clifford Algebras and the Classical Groups
2077:are the canonical projection mappings and
1970:. The key point is that the definition of
689:The resulting abelian group is called the
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4335:Direct sum of modules with bilinear forms
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2776:that is presently more commonly called a
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1322:{\displaystyle \bigoplus _{i\in I}M_{i}.}
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420:is equal to the sum of the dimensions of
318:
301:The resulting vector space is called the
45:is a construction which combines several
3711:is a module consisting of all functions
3704:{\displaystyle \bigoplus _{i\in I}X_{i}}
3622:is a collection of Banach spaces, where
1460:
842:
780:
5600:
3330:as rings of scalars in his analysis of
2394:
5245:For example, if we take the index set
3900:For example, if we take the index set
3323:{\displaystyle ^{2}R,\ ^{2}C,\ ^{2}H,}
2772:" is also introduced for denoting the
2377:
1331:It is customary to write the sequence
151:
3349:. In categorical terms, Wedderburn's
2416:if there exists some other submodule
1471:to the disjoint union of the modules
7:
5492:Comparing this with the example for
5386:which consists of all the sequences
4244:is equal to the internal direct sum
4184:if there is another closed subspace
4041:which consists of all the sequences
1982:) is zero for all but finitely many
963:{\displaystyle \alpha _{i}\in M_{i}}
813:is equal to the sum of the ranks of
5304:{\displaystyle X_{i}=\mathbb {R} ,}
4674:{\displaystyle H_{1},\ldots ,H_{n}}
3959:{\displaystyle X_{i}=\mathbb {R} ,}
2987:Consider these classical examples:
490:Construction for two abelian groups
5678:Elements of mathematics, Algebra I
5414:{\displaystyle \left(a_{i}\right)}
5145:
4871:If infinitely many Hilbert spaces
4314:
4069:{\displaystyle \left(a_{i}\right)}
3879:
2617:, then there exists precisely one
1847:) (using the natural inclusion of
1566:Every vector space over the field
505:which are written additively, the
112:Construction for two vector spaces
25:
2725:. This extension is known as the
2149:are the inclusion mappings, then
1961:in the direct sum of the modules
1637:from the direct sum to some left
1422:{\displaystyle \sum '\alpha _{i}}
432:and its orthogonal complement:
5719:Finite dimensional vector spaces
4325:{\displaystyle \ell ^{\infty }.}
3090:
3082:
3050:
3042:
3006:
2998:
2294:, and is the zero map otherwise.
2270:{\displaystyle p_{k}\circ i_{l}}
2221:equals the identity morphism of
1387:{\displaystyle \sum \alpha _{i}}
4928:and it will not necessarily be
1645:is naturally isomorphic to the
1394:. Sometimes a primed summation
5544:Lindenstrauss–Tzafriri theorem
5467:
5454:
5266:{\displaystyle I=\mathbb {N} }
5132:
5127:
5121:
5113:
5028:
5022:
3921:{\displaystyle I=\mathbb {N} }
3856:
3850:
3769:
3763:
3484:
3472:
3363:coproduct (or categorical sum)
3247:
3232:
3226:
3211:
3205:
3193:
3152:
3140:
2968:
2922:
2916:
2890:
2887:
2861:
2663:
2524:
2107:
2054:
1944:
1941:
1935:
1929:
1926:
1920:
1895:
1889:
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1880:
1351:
1338:
1256:
1246:
1234:
1227:
1160:
1147:
917:
904:
1:
3788:{\displaystyle x(i)\in X_{i}}
1994:of the duals of those spaces.
1826:-linear homomorphism sending
1357:{\displaystyle (\alpha _{i})}
996:{\displaystyle \alpha _{i}=0}
923:{\displaystyle (\alpha _{i})}
767:and is often identified with
517:is also called a direct sum (
376:and is often identified with
27:Operation in abstract algebra
5656:Elementary rings and modules
5581: – Mathematical theorem
4622:Direct sum of Hilbert spaces
2566:which sends the elements of
2277:is the identity morphism of
519:Mac Lane & Birkhoff 1999
3373:Direct sum of Banach spaces
2001:direct sum of modules is a
1803:Indeed, there is clearly a
1653:-linear homomorphisms from
1606:with the direct sum of the
1555:. The same is true for the
1283:-module, and it is denoted
5792:
5421:of reals with finite norm
5379:{\displaystyle \ell _{2},}
5034:{\displaystyle \alpha (i)}
4625:
4266:{\displaystyle A\oplus B.}
4076:of reals with finite norm
4034:{\displaystyle \ell _{1},}
3367:tensor product of algebras
2780:of algebras; that is, the
2302:
1018:It can also be defined as
828:number of abelian groups.
29:
5611:; Husemoller, D. (1973).
4435:of modules equipped with
1649:of the abelian groups of
1429:is used to indicate that
720:{\displaystyle G\oplus H}
486:number of vector spaces.
332:{\displaystyle V\oplus W}
97:decomposition of a module
5613:Symmetric Bilinear Forms
3113:William Kingdon Clifford
2790:componentwise operations
2472:in the category of left
2450:complementary submodules
877:-modules indexed by the
4302:is not complemented in
3586:{\displaystyle y\in Y.}
2613:-linear maps for every
2305:Internal direct product
1433:of the terms are zero.
76:(modules over the ring
5528:
5486:
5415:
5380:
5350:
5305:
5267:
5234:
5155:
5088:
5087:{\displaystyle i\in I}
5062:
5035:
5006:
4983:
4953:
4918:
4917:{\displaystyle i\in I}
4892:
4856:
4675:
4612:
4592:
4457:
4429:
4405:
4326:
4295:
4267:
4238:
4218:
4198:
4172:
4152:
4130:
4070:
4035:
4005:
3960:
3922:
3889:
3815:
3814:{\displaystyle i\in I}
3789:
3747:
3725:
3705:
3662:
3636:
3616:
3587:
3558:
3557:{\displaystyle x\in X}
3532:
3456:
3436:
3416:
3396:
3379:The direct sum of two
3357:, whilst Wedderburn's
3324:
3257:
3177:
3101:
3058:
3014:
2978:
2845:
2825:
2804:Direct sum of algebras
2673:
2557:
2464:, the direct sum is a
2380:, §18). In this case,
2271:
2215:
2143:
2071:
1951:
1797:
1557:rank of abelian groups
1423:
1388:
1358:
1323:
1269:
1199:
1131:
1104:
1103:{\displaystyle i\in I}
1084:, with the fiber over
997:
964:
924:
771:; similarly for {0} ×
721:
474:
380:; similarly for {0} ×
333:
18:Complementary subspace
5579:Krull–Schmidt theorem
5570:Jordan–Hölder theorem
5565:Indecomposable module
5540:orthogonal complement
5538:because it admits an
5529:
5487:
5416:
5381:
5351:
5306:
5268:
5235:
5156:
5089:
5063:
5061:{\displaystyle H_{i}}
5036:
5007:
4984:
4982:{\displaystyle H_{i}}
4954:
4952:{\displaystyle H_{i}}
4919:
4893:
4891:{\displaystyle H_{i}}
4857:
4676:
4626:Further information:
4613:
4593:
4458:
4441:orthogonal direct sum
4430:
4406:
4327:
4296:
4294:{\displaystyle c_{0}}
4268:
4239:
4219:
4199:
4173:
4153:
4131:
4071:
4036:
4006:
3961:
3923:
3890:
3816:
3790:
3748:
3726:
3706:
3663:
3637:
3617:
3615:{\displaystyle X_{i}}
3588:
3559:
3533:
3457:
3437:
3422:is the direct sum of
3417:
3397:
3325:
3258:
3178:
3102:
3059:
3026:split-complex numbers
3015:
2979:
2846:
2826:
2770:algebras over a field
2674:
2558:
2272:
2216:
2144:
2072:
1952:
1798:
1629:The abelian group of
1424:
1389:
1359:
1324:
1270:
1200:
1132:
1130:{\displaystyle M_{i}}
1105:
998:
965:
925:
722:
475:
334:
5585:Split exact sequence
5503:
5425:
5390:
5360:
5315:
5311:then the direct sum
5277:
5249:
5167:
5098:
5072:
5045:
5016:
4993:
4966:
4936:
4902:
4875:
4685:
4639:
4602:
4467:
4447:
4419:
4343:
4306:
4278:
4248:
4228:
4208:
4188:
4162:
4142:
4080:
4045:
4015:
3970:
3966:then the direct sum
3932:
3904:
3825:
3799:
3757:
3737:
3715:
3672:
3668:then the direct sum
3649:
3626:
3599:
3568:
3542:
3466:
3446:
3426:
3406:
3386:
3270:
3187:
3134:
3128:Lectures on Matrices
3124:hypercomplex numbers
3111:, was introduced by
3078:
3038:
2994:
2858:
2835:
2815:
2798:direct sums of rings
2628:
2498:
2241:
2153:
2081:
2009:
1974:makes sense because
1864:
1670:
1635:linear homomorphisms
1508:Each of the modules
1442:The direct sum is a
1398:
1368:
1335:
1287:
1221:
1144:
1114:
1088:
974:
934:
901:
705:
436:
317:
53:. Contrast with the
5680:, Springer-Verlag,
5542:. Conversely, the
4926:inner product space
3355:categorical product
3109:split-biquaternions
2774:algebraic structure
2753:in the appropriate
2460:In the language of
2365:internal direct sum
2359:, then we say that
2299:Internal direct sum
1501:. If the index set
1080:over the index set
782:internal direct sum
390:internal direct sum
5524:
5482:
5451:
5411:
5376:
5346:
5301:
5263:
5230:
5197:
5151:
5110:
5084:
5058:
5031:
5005:{\displaystyle I,}
5002:
4979:
4949:
4914:
4888:
4852:
4671:
4608:
4588:
4544:
4453:
4425:
4401:
4322:
4291:
4263:
4234:
4214:
4194:
4168:
4158:of a Banach space
4148:
4138:A closed subspace
4126:
4104:
4066:
4031:
4001:
3990:
3956:
3918:
3885:
3843:
3811:
3785:
3743:
3721:
3701:
3690:
3661:{\displaystyle I,}
3658:
3632:
3612:
3583:
3554:
3528:
3452:
3432:
3412:
3392:
3320:
3253:
3173:
3097:
3064:is the algebra of
3054:
3010:
2974:
2841:
2821:
2796:and the remark on
2731:universal property
2727:Grothendieck group
2710:Grothendieck group
2669:
2652:
2553:
2542:
2478:universal property
2456:Universal property
2393:as defined above (
2367:of the submodules
2267:
2211:
2139:
2067:
1947:
1916:
1793:
1750:
1708:
1419:
1384:
1354:
1319:
1305:
1265:
1195:
1127:
1100:
1074:finitely supported
993:
960:
920:
797:and an element of
717:
521:, §V.6). Thus the
470:
404:and an element of
329:
5674:Bourbaki, Nicolas
5514:
5477:
5442:
5188:
5101:
5041:is an element of
4632:If finitely many
4611:{\displaystyle I}
4529:
4456:{\displaystyle B}
4428:{\displaystyle I}
4237:{\displaystyle X}
4217:{\displaystyle X}
4197:{\displaystyle B}
4171:{\displaystyle X}
4151:{\displaystyle A}
4095:
3973:
3828:
3746:{\displaystyle I}
3724:{\displaystyle x}
3675:
3635:{\displaystyle i}
3455:{\displaystyle Y}
3435:{\displaystyle X}
3415:{\displaystyle Y}
3395:{\displaystyle X}
3307:
3291:
3120:Joseph Wedderburn
3030:interval analysis
2844:{\displaystyle Y}
2824:{\displaystyle X}
2782:Cartesian product
2637:
2527:
2490:natural embedding
1988:dual vector space
1901:
1735:
1693:
1561:length of modules
1290:
763:is isomorphic to
523:Cartesian product
372:is isomorphic to
137:cartesian product
16:(Redirected from
5783:
5756:
5731:
5709:
5697:Abstract algebra
5690:
5668:
5652:Adamson, Iain T.
5643:
5642:
5623:. pp. 4–5.
5619:. Vol. 73.
5605:
5590:
5575:
5561:
5533:
5531:
5530:
5525:
5520:
5515:
5512:
5510:
5491:
5489:
5488:
5483:
5478:
5476:
5475:
5470:
5466:
5465:
5450:
5441:
5420:
5418:
5417:
5412:
5410:
5406:
5405:
5385:
5383:
5382:
5377:
5372:
5371:
5355:
5353:
5352:
5347:
5345:
5344:
5335:
5334:
5333:
5310:
5308:
5307:
5302:
5297:
5289:
5288:
5272:
5270:
5269:
5264:
5262:
5239:
5237:
5236:
5231:
5223:
5222:
5210:
5209:
5196:
5160:
5158:
5157:
5152:
5141:
5140:
5135:
5131:
5130:
5109:
5093:
5091:
5090:
5085:
5067:
5065:
5064:
5059:
5057:
5056:
5040:
5038:
5037:
5032:
5011:
5009:
5008:
5003:
4988:
4986:
4985:
4980:
4978:
4977:
4958:
4956:
4955:
4950:
4948:
4947:
4923:
4921:
4920:
4915:
4897:
4895:
4894:
4889:
4887:
4886:
4861:
4859:
4858:
4853:
4845:
4844:
4832:
4831:
4807:
4806:
4794:
4793:
4778:
4774:
4773:
4769:
4768:
4767:
4749:
4748:
4731:
4727:
4726:
4725:
4707:
4706:
4680:
4678:
4677:
4672:
4670:
4669:
4651:
4650:
4617:
4615:
4614:
4609:
4597:
4595:
4594:
4589:
4587:
4583:
4582:
4581:
4569:
4568:
4554:
4553:
4543:
4525:
4521:
4520:
4516:
4515:
4514:
4497:
4493:
4492:
4491:
4462:
4460:
4459:
4454:
4434:
4432:
4431:
4426:
4410:
4408:
4407:
4402:
4400:
4396:
4383:
4379:
4378:
4377:
4365:
4364:
4331:
4329:
4328:
4323:
4318:
4317:
4300:
4298:
4297:
4292:
4290:
4289:
4272:
4270:
4269:
4264:
4243:
4241:
4240:
4235:
4223:
4221:
4220:
4215:
4203:
4201:
4200:
4195:
4177:
4175:
4174:
4169:
4157:
4155:
4154:
4149:
4135:
4133:
4132:
4127:
4122:
4118:
4117:
4103:
4075:
4073:
4072:
4067:
4065:
4061:
4060:
4040:
4038:
4037:
4032:
4027:
4026:
4010:
4008:
4007:
4002:
4000:
3999:
3989:
3988:
3965:
3963:
3962:
3957:
3952:
3944:
3943:
3927:
3925:
3924:
3919:
3917:
3894:
3892:
3891:
3886:
3875:
3874:
3873:
3872:
3842:
3820:
3818:
3817:
3812:
3794:
3792:
3791:
3786:
3784:
3783:
3752:
3750:
3749:
3744:
3730:
3728:
3727:
3722:
3710:
3708:
3707:
3702:
3700:
3699:
3689:
3667:
3665:
3664:
3659:
3641:
3639:
3638:
3633:
3621:
3619:
3618:
3613:
3611:
3610:
3592:
3590:
3589:
3584:
3563:
3561:
3560:
3555:
3537:
3535:
3534:
3529:
3527:
3526:
3508:
3507:
3461:
3459:
3458:
3453:
3441:
3439:
3438:
3433:
3421:
3419:
3418:
3413:
3401:
3399:
3398:
3393:
3329:
3327:
3326:
3321:
3313:
3312:
3305:
3297:
3296:
3289:
3281:
3280:
3262:
3260:
3259:
3254:
3182:
3180:
3179:
3174:
3106:
3104:
3103:
3098:
3093:
3085:
3063:
3061:
3060:
3055:
3053:
3045:
3019:
3017:
3016:
3011:
3009:
3001:
2983:
2981:
2980:
2975:
2967:
2966:
2957:
2956:
2944:
2943:
2934:
2933:
2915:
2914:
2902:
2901:
2886:
2885:
2873:
2872:
2850:
2848:
2847:
2842:
2830:
2828:
2827:
2822:
2808:A direct sum of
2678:
2676:
2675:
2670:
2662:
2661:
2651:
2583:be an arbitrary
2562:
2560:
2559:
2554:
2552:
2551:
2541:
2523:
2522:
2510:
2509:
2440:. In this case,
2276:
2274:
2273:
2268:
2266:
2265:
2253:
2252:
2220:
2218:
2217:
2212:
2210:
2209:
2197:
2196:
2178:
2177:
2165:
2164:
2148:
2146:
2145:
2140:
2138:
2137:
2119:
2118:
2106:
2105:
2093:
2092:
2076:
2074:
2073:
2068:
2066:
2065:
2053:
2052:
2034:
2033:
2021:
2020:
1956:
1954:
1953:
1948:
1915:
1879:
1878:
1802:
1800:
1799:
1794:
1789:
1785:
1778:
1777:
1760:
1759:
1749:
1731:
1730:
1718:
1717:
1707:
1692:
1691:
1682:
1681:
1618:Direct sums are
1428:
1426:
1425:
1420:
1418:
1417:
1408:
1393:
1391:
1390:
1385:
1383:
1382:
1363:
1361:
1360:
1355:
1350:
1349:
1328:
1326:
1325:
1320:
1315:
1314:
1304:
1274:
1272:
1271:
1266:
1264:
1263:
1242:
1241:
1204:
1202:
1201:
1196:
1194:
1193:
1181:
1180:
1168:
1167:
1136:
1134:
1133:
1128:
1126:
1125:
1109:
1107:
1106:
1101:
1076:sections of the
1002:
1000:
999:
994:
986:
985:
969:
967:
966:
961:
959:
958:
946:
945:
929:
927:
926:
921:
916:
915:
852:be a ring, and {
737:), but as a sum
726:
724:
723:
718:
479:
477:
476:
471:
469:
468:
450:
449:
444:
349:), but as a sum
338:
336:
335:
330:
95:See the article
68:(modules over a
39:abstract algebra
21:
5791:
5790:
5786:
5785:
5784:
5782:
5781:
5780:
5761:
5760:
5754:
5746:, AMS Chelsea,
5734:
5729:
5713:
5707:
5694:
5688:
5672:
5666:
5650:
5647:
5646:
5631:
5621:Springer-Verlag
5607:
5606:
5602:
5597:
5588:
5573:
5559:
5552:
5501:
5500:
5457:
5453:
5452:
5423:
5422:
5397:
5393:
5388:
5387:
5363:
5358:
5357:
5336:
5318:
5313:
5312:
5280:
5275:
5274:
5247:
5246:
5214:
5201:
5165:
5164:
5116:
5112:
5111:
5096:
5095:
5070:
5069:
5048:
5043:
5042:
5014:
5013:
4991:
4990:
4969:
4964:
4963:
4939:
4934:
4933:
4900:
4899:
4878:
4873:
4872:
4836:
4823:
4798:
4785:
4759:
4740:
4739:
4735:
4717:
4698:
4697:
4693:
4692:
4688:
4683:
4682:
4661:
4642:
4637:
4636:
4630:
4624:
4600:
4599:
4573:
4560:
4555:
4545:
4506:
4501:
4483:
4478:
4473:
4465:
4464:
4445:
4444:
4417:
4416:
4369:
4356:
4355:
4351:
4350:
4346:
4341:
4340:
4337:
4309:
4304:
4303:
4281:
4276:
4275:
4246:
4245:
4226:
4225:
4206:
4205:
4186:
4185:
4160:
4159:
4140:
4139:
4109:
4105:
4078:
4077:
4052:
4048:
4043:
4042:
4018:
4013:
4012:
3991:
3968:
3967:
3935:
3930:
3929:
3902:
3901:
3864:
3859:
3823:
3822:
3797:
3796:
3775:
3755:
3754:
3735:
3734:
3713:
3712:
3691:
3670:
3669:
3647:
3646:
3624:
3623:
3602:
3597:
3596:
3566:
3565:
3540:
3539:
3518:
3499:
3464:
3463:
3444:
3443:
3424:
3423:
3404:
3403:
3384:
3383:
3375:
3347:category theory
3304:
3288:
3273:
3268:
3267:
3264:Ian R. Porteous
3185:
3184:
3132:
3131:
3076:
3075:
3036:
3035:
3028:, also used in
3022:ring isomorphic
2992:
2991:
2958:
2948:
2935:
2925:
2906:
2893:
2877:
2864:
2856:
2855:
2833:
2832:
2813:
2812:
2806:
2786:underlying sets
2739:
2712:
2701:
2691:
2653:
2626:
2625:
2604:
2595:
2574:
2543:
2514:
2501:
2496:
2495:
2488:, consider the
2462:category theory
2458:
2392:
2375:
2358:
2325:
2307:
2301:
2285:
2257:
2244:
2239:
2238:
2236:
2227:
2201:
2188:
2169:
2156:
2151:
2150:
2129:
2110:
2097:
2084:
2079:
2078:
2057:
2044:
2025:
2012:
2007:
2006:
1969:
1867:
1862:
1861:
1855:
1838:
1769:
1768:
1764:
1751:
1709:
1673:
1668:
1667:
1661:
1614:
1601:
1554:
1545:
1533:
1516:
1496:
1479:
1458:
1450:of the modules
1439:
1431:cofinitely many
1409:
1401:
1396:
1395:
1374:
1366:
1365:
1341:
1333:
1332:
1306:
1285:
1284:
1255:
1233:
1219:
1218:
1185:
1172:
1159:
1142:
1141:
1117:
1112:
1111:
1086:
1085:
1066:cofinitely many
1051:
1038:
1030:of the modules
1005:cofinitely many
977:
972:
971:
950:
937:
932:
931:
907:
899:
898:
896:
860:
834:
703:
702:
627:
620:
609:
602:
591:
584:
577:
570:
563:
556:
549:
542:
492:
460:
439:
434:
433:
315:
314:
285:
278:
263:
256:
213:
206:
199:
192:
185:
178:
171:
164:
114:
105:
57:, which is the
35:
28:
23:
22:
15:
12:
11:
5:
5789:
5787:
5779:
5778:
5773:
5771:Linear algebra
5763:
5762:
5759:
5758:
5752:
5732:
5727:
5711:
5705:
5692:
5686:
5670:
5664:
5645:
5644:
5629:
5599:
5598:
5596:
5593:
5592:
5591:
5582:
5576:
5567:
5562:
5551:
5548:
5523:
5519:
5513: or
5509:
5481:
5474:
5469:
5464:
5460:
5456:
5449:
5445:
5439:
5436:
5433:
5430:
5409:
5404:
5400:
5396:
5375:
5370:
5366:
5343:
5339:
5332:
5328:
5325:
5321:
5300:
5296:
5292:
5287:
5283:
5261:
5257:
5254:
5229:
5226:
5221:
5217:
5213:
5208:
5204:
5200:
5195:
5191:
5187:
5184:
5181:
5178:
5175:
5172:
5150:
5147:
5144:
5139:
5134:
5129:
5126:
5123:
5119:
5115:
5108:
5104:
5083:
5080:
5077:
5055:
5051:
5030:
5027:
5024:
5021:
5001:
4998:
4976:
4972:
4946:
4942:
4913:
4910:
4907:
4885:
4881:
4851:
4848:
4843:
4839:
4835:
4830:
4826:
4822:
4819:
4816:
4813:
4810:
4805:
4801:
4797:
4792:
4788:
4784:
4781:
4777:
4772:
4766:
4762:
4758:
4755:
4752:
4747:
4743:
4738:
4734:
4730:
4724:
4720:
4716:
4713:
4710:
4705:
4701:
4696:
4691:
4668:
4664:
4660:
4657:
4654:
4649:
4645:
4634:Hilbert spaces
4623:
4620:
4607:
4586:
4580:
4576:
4572:
4567:
4563:
4558:
4552:
4548:
4542:
4539:
4536:
4532:
4528:
4524:
4519:
4513:
4509:
4504:
4500:
4496:
4490:
4486:
4481:
4476:
4472:
4452:
4437:bilinear forms
4424:
4399:
4395:
4392:
4389:
4386:
4382:
4376:
4372:
4368:
4363:
4359:
4354:
4349:
4336:
4333:
4321:
4316:
4312:
4288:
4284:
4262:
4259:
4256:
4253:
4233:
4213:
4193:
4167:
4147:
4125:
4121:
4116:
4112:
4108:
4102:
4098:
4094:
4091:
4088:
4085:
4064:
4059:
4055:
4051:
4030:
4025:
4021:
3998:
3994:
3987:
3983:
3980:
3976:
3955:
3951:
3947:
3942:
3938:
3916:
3912:
3909:
3884:
3881:
3878:
3871:
3867:
3862:
3858:
3855:
3852:
3849:
3846:
3841:
3838:
3835:
3831:
3810:
3807:
3804:
3782:
3778:
3774:
3771:
3768:
3765:
3762:
3742:
3720:
3698:
3694:
3688:
3685:
3682:
3678:
3657:
3654:
3642:traverses the
3631:
3609:
3605:
3595:Generally, if
3582:
3579:
3576:
3573:
3553:
3550:
3547:
3525:
3521:
3517:
3514:
3511:
3506:
3502:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3477:
3474:
3471:
3451:
3431:
3411:
3391:
3374:
3371:
3360:
3359:direct product
3352:
3344:
3343:direct product
3340:
3319:
3316:
3311:
3303:
3300:
3295:
3287:
3284:
3279:
3275:
3252:
3249:
3246:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3117:
3116:
3096:
3092:
3088:
3084:
3073:
3068:introduced by
3052:
3048:
3044:
3033:
3008:
3004:
3000:
2985:
2984:
2973:
2970:
2965:
2961:
2955:
2951:
2947:
2942:
2938:
2932:
2928:
2924:
2921:
2918:
2913:
2909:
2905:
2900:
2896:
2892:
2889:
2884:
2880:
2876:
2871:
2867:
2863:
2840:
2820:
2805:
2802:
2794:see note below
2778:direct product
2763:Hilbert spaces
2738:
2735:
2711:
2708:
2697:
2689:
2680:
2679:
2668:
2665:
2660:
2656:
2650:
2647:
2644:
2640:
2636:
2633:
2600:
2591:
2570:
2564:
2563:
2550:
2546:
2540:
2537:
2534:
2530:
2526:
2521:
2517:
2513:
2508:
2504:
2457:
2454:
2432:direct sum of
2410:direct summand
2388:
2371:
2354:
2321:
2300:
2297:
2296:
2295:
2281:
2264:
2260:
2256:
2251:
2247:
2232:
2225:
2208:
2204:
2200:
2195:
2191:
2187:
2184:
2181:
2176:
2172:
2168:
2163:
2159:
2136:
2132:
2128:
2125:
2122:
2117:
2113:
2109:
2104:
2100:
2096:
2091:
2087:
2064:
2060:
2056:
2051:
2047:
2043:
2040:
2037:
2032:
2028:
2024:
2019:
2015:
1995:
1992:direct product
1965:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1914:
1911:
1908:
1904:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1877:
1874:
1870:
1860:is defined by
1851:
1834:
1792:
1788:
1784:
1781:
1776:
1772:
1767:
1763:
1758:
1754:
1748:
1745:
1742:
1738:
1734:
1729:
1724:
1721:
1716:
1712:
1706:
1703:
1700:
1696:
1690:
1685:
1680:
1676:
1657:
1647:direct product
1627:
1616:
1610:
1597:
1585:is some right
1579:tensor product
1575:
1564:
1550:
1541:
1535:
1529:
1512:
1506:
1492:
1475:
1454:
1448:direct product
1438:
1435:
1416:
1412:
1407:
1404:
1381:
1377:
1373:
1353:
1348:
1344:
1340:
1318:
1313:
1309:
1303:
1300:
1297:
1293:
1262:
1258:
1254:
1251:
1248:
1245:
1240:
1236:
1232:
1229:
1226:
1192:
1188:
1184:
1179:
1175:
1171:
1166:
1162:
1158:
1155:
1152:
1149:
1124:
1120:
1099:
1096:
1093:
1047:
1043:) ∈
1034:
1028:disjoint union
1013:direct product
992:
989:
984:
980:
957:
953:
949:
944:
940:
919:
914:
910:
906:
892:
856:
833:
830:
716:
713:
710:
660:
659:
625:
618:
607:
600:
594:
593:
589:
582:
575:
568:
561:
554:
547:
540:
507:direct product
496:abelian groups
491:
488:
467:
463:
459:
456:
453:
448:
443:
328:
325:
322:
283:
276:
261:
254:
244:
243:
215:
211:
204:
197:
190:
183:
176:
169:
162:
113:
110:
104:
101:
90:Hilbert spaces
74:abelian groups
55:direct product
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5788:
5777:
5776:Module theory
5774:
5772:
5769:
5768:
5766:
5755:
5753:0-8218-1646-2
5749:
5745:
5741:
5737:
5733:
5730:
5728:0-387-90093-4
5724:
5720:
5716:
5712:
5708:
5706:0-13-004771-6
5702:
5698:
5693:
5689:
5687:3-540-64243-9
5683:
5679:
5675:
5671:
5667:
5665:0-05-002192-3
5661:
5657:
5653:
5649:
5648:
5640:
5636:
5632:
5630:3-540-06009-X
5626:
5622:
5618:
5614:
5610:
5604:
5601:
5594:
5586:
5583:
5580:
5577:
5571:
5568:
5566:
5563:
5557:
5554:
5553:
5549:
5547:
5545:
5541:
5537:
5521:
5497:
5495:
5494:Banach spaces
5479:
5472:
5462:
5458:
5447:
5443:
5437:
5431:
5407:
5402:
5398:
5394:
5373:
5368:
5364:
5356:is the space
5341:
5337:
5326:
5323:
5319:
5298:
5290:
5285:
5281:
5255:
5252:
5243:
5240:
5227:
5219:
5215:
5211:
5206:
5202:
5193:
5189:
5185:
5179:
5176:
5173:
5161:
5148:
5142:
5137:
5124:
5117:
5106:
5102:
5081:
5078:
5075:
5053:
5049:
5025:
5019:
4999:
4996:
4974:
4970:
4960:
4944:
4940:
4931:
4927:
4911:
4908:
4905:
4883:
4879:
4869:
4867:
4862:
4849:
4841:
4837:
4833:
4828:
4824:
4817:
4814:
4811:
4803:
4799:
4795:
4790:
4786:
4779:
4775:
4770:
4764:
4760:
4756:
4753:
4750:
4745:
4741:
4736:
4732:
4728:
4722:
4718:
4714:
4711:
4708:
4703:
4699:
4694:
4689:
4666:
4662:
4658:
4655:
4652:
4647:
4643:
4635:
4629:
4621:
4619:
4605:
4584:
4578:
4574:
4570:
4565:
4561:
4556:
4550:
4546:
4540:
4537:
4534:
4530:
4526:
4522:
4517:
4511:
4507:
4502:
4498:
4494:
4488:
4484:
4479:
4474:
4470:
4450:
4442:
4438:
4422:
4414:
4397:
4393:
4390:
4387:
4384:
4380:
4374:
4370:
4366:
4361:
4357:
4352:
4347:
4334:
4332:
4319:
4310:
4301:
4286:
4282:
4260:
4257:
4254:
4251:
4231:
4211:
4191:
4183:
4182:
4165:
4145:
4136:
4123:
4119:
4114:
4110:
4106:
4100:
4096:
4092:
4086:
4062:
4057:
4053:
4049:
4028:
4023:
4019:
4011:is the space
3996:
3992:
3981:
3978:
3974:
3953:
3945:
3940:
3936:
3910:
3907:
3898:
3895:
3882:
3876:
3869:
3865:
3853:
3847:
3839:
3836:
3833:
3829:
3808:
3805:
3802:
3780:
3776:
3772:
3766:
3760:
3740:
3733:
3718:
3696:
3692:
3686:
3683:
3680:
3676:
3655:
3652:
3645:
3629:
3607:
3603:
3593:
3580:
3577:
3574:
3571:
3551:
3548:
3545:
3523:
3515:
3509:
3504:
3496:
3490:
3481:
3478:
3475:
3449:
3429:
3409:
3389:
3382:
3381:Banach spaces
3377:
3372:
3370:
3368:
3364:
3358:
3356:
3350:
3348:
3342:
3338:
3335:
3333:
3317:
3314:
3309:
3301:
3298:
3293:
3285:
3282:
3277:
3274:
3265:
3250:
3244:
3241:
3238:
3235:
3229:
3223:
3220:
3217:
3214:
3208:
3202:
3199:
3196:
3190:
3170:
3167:
3164:
3161:
3158:
3155:
3149:
3146:
3143:
3137:
3129:
3125:
3121:
3114:
3110:
3094:
3086:
3074:
3071:
3067:
3046:
3034:
3031:
3027:
3023:
3002:
2990:
2989:
2988:
2971:
2963:
2959:
2953:
2949:
2945:
2940:
2936:
2930:
2926:
2919:
2911:
2907:
2903:
2898:
2894:
2882:
2878:
2874:
2869:
2865:
2854:
2853:
2852:
2838:
2818:
2811:
2803:
2801:
2799:
2795:
2791:
2787:
2783:
2779:
2775:
2771:
2766:
2764:
2760:
2759:Banach spaces
2756:
2752:
2748:
2747:inner product
2744:
2736:
2734:
2732:
2728:
2724:
2723:abelian group
2720:
2717:
2709:
2707:
2705:
2700:
2696:
2692:
2685:
2666:
2658:
2654:
2648:
2645:
2642:
2638:
2634:
2631:
2624:
2623:
2622:
2620:
2616:
2612:
2609:be arbitrary
2608:
2603:
2599:
2594:
2590:
2586:
2582:
2578:
2573:
2569:
2548:
2544:
2538:
2535:
2532:
2528:
2519:
2515:
2511:
2506:
2502:
2494:
2493:
2492:
2491:
2487:
2483:
2479:
2475:
2471:
2467:
2463:
2455:
2453:
2451:
2447:
2443:
2439:
2435:
2431:
2427:
2423:
2419:
2415:
2411:
2407:
2403:
2398:
2396:
2391:
2387:
2383:
2379:
2374:
2370:
2366:
2362:
2357:
2353:
2349:
2345:
2341:
2337:
2333:
2329:
2324:
2320:
2316:
2312:
2306:
2298:
2293:
2289:
2284:
2280:
2262:
2258:
2254:
2249:
2245:
2235:
2231:
2224:
2206:
2202:
2198:
2193:
2189:
2185:
2182:
2179:
2174:
2170:
2166:
2161:
2157:
2134:
2130:
2126:
2123:
2120:
2115:
2111:
2102:
2098:
2094:
2089:
2085:
2062:
2058:
2049:
2045:
2041:
2038:
2035:
2030:
2026:
2022:
2017:
2013:
2004:
2000:
1996:
1993:
1989:
1985:
1981:
1977:
1973:
1968:
1964:
1960:
1938:
1932:
1923:
1917:
1912:
1909:
1906:
1902:
1898:
1892:
1883:
1875:
1872:
1868:
1859:
1854:
1850:
1846:
1842:
1837:
1833:
1829:
1825:
1821:
1817:
1813:
1809:
1806:
1790:
1786:
1782:
1779:
1774:
1770:
1765:
1761:
1756:
1752:
1746:
1743:
1740:
1736:
1732:
1722:
1719:
1714:
1710:
1704:
1701:
1698:
1694:
1683:
1678:
1674:
1665:
1660:
1656:
1652:
1648:
1644:
1640:
1636:
1632:
1628:
1625:
1621:
1617:
1613:
1609:
1605:
1600:
1596:
1592:
1588:
1584:
1580:
1576:
1573:
1569:
1565:
1562:
1558:
1553:
1549:
1544:
1540:
1536:
1532:
1528:
1524:
1520:
1515:
1511:
1507:
1504:
1500:
1495:
1491:
1487:
1483:
1478:
1474:
1470:
1466:
1462:
1461:Bourbaki 1989
1457:
1453:
1449:
1445:
1441:
1440:
1436:
1434:
1432:
1414:
1410:
1405:
1402:
1379:
1375:
1371:
1346:
1342:
1329:
1316:
1311:
1307:
1301:
1298:
1295:
1291:
1282:
1278:
1260:
1252:
1249:
1243:
1238:
1230:
1224:
1216:
1212:
1208:
1190:
1186:
1182:
1177:
1173:
1169:
1164:
1156:
1153:
1150:
1138:
1122:
1118:
1097:
1094:
1091:
1083:
1079:
1075:
1071:
1067:
1063:
1059:
1055:
1050:
1046:
1042:
1037:
1033:
1029:
1025:
1021:
1016:
1014:
1010:
1006:
990:
987:
982:
978:
955:
951:
947:
942:
938:
912:
908:
895:
891:
887:
883:
880:
876:
872:
868:
865: ∈
864:
861: :
859:
855:
851:
846:
844:
843:Bourbaki 1989
840:
831:
829:
827:
822:
820:
816:
812:
808:
804:
800:
796:
792:
788:
784:
783:
778:
774:
770:
766:
762:
758:
754:
751:
746:
744:
740:
736:
732:
727:
714:
711:
708:
700:
696:
692:
687:
685:
681:
677:
673:
669:
665:
657:
653:
649:
645:
641:
638:
637:
636:
633:
631:
624:
617:
613:
606:
599:
588:
581:
574:
567:
560:
553:
546:
539:
535:
534:
533:
531:
527:
524:
520:
516:
512:
508:
504:
500:
497:
489:
487:
485:
480:
465:
461:
457:
454:
451:
446:
431:
427:
423:
419:
415:
411:
407:
403:
399:
395:
391:
387:
383:
379:
375:
371:
367:
363:
360:The subspace
358:
356:
352:
348:
344:
339:
326:
323:
320:
312:
308:
304:
299:
297:
293:
289:
282:
275:
271:
267:
260:
253:
249:
241:
238:
234:
231:
227:
223:
219:
216:
210:
203:
196:
189:
182:
175:
168:
161:
157:
156:
155:
153:
149:
145:
141:
138:
134:
131:
127:
126:vector spaces
123:
119:
111:
109:
102:
100:
98:
93:
91:
87:
86:Banach spaces
83:
79:
75:
71:
67:
66:vector spaces
62:
60:
56:
52:
48:
44:
40:
33:
19:
5743:
5740:Birkhoff, G.
5736:Mac Lane, S.
5721:, Springer,
5718:
5715:Halmos, Paul
5696:
5677:
5655:
5612:
5603:
5536:complemented
5498:
5244:
5241:
5162:
4961:
4870:
4863:
4631:
4440:
4338:
4181:complemented
4179:
4137:
3899:
3896:
3732:defined over
3594:
3378:
3376:
3336:
3331:
3127:
3118:
3070:James Cockle
2986:
2807:
2793:
2767:
2740:
2713:
2703:
2698:
2694:
2687:
2683:
2681:
2621:-linear map
2618:
2614:
2610:
2606:
2601:
2597:
2592:
2588:
2587:-module and
2584:
2580:
2576:
2571:
2567:
2565:
2489:
2485:
2481:
2480:. For every
2473:
2468:and hence a
2459:
2449:
2445:
2441:
2437:
2433:
2429:
2425:
2421:
2417:
2413:
2409:
2405:
2401:
2400:A submodule
2399:
2395:Adamson 1972
2389:
2385:
2381:
2372:
2368:
2364:
2360:
2355:
2351:
2347:
2343:
2339:
2335:
2331:
2322:
2318:
2317:-module and
2314:
2310:
2308:
2291:
2287:
2286:in the case
2282:
2278:
2233:
2229:
2222:
1998:
1983:
1979:
1975:
1971:
1966:
1962:
1958:
1857:
1852:
1848:
1844:
1840:
1835:
1831:
1827:
1823:
1819:
1815:
1811:
1807:
1805:homomorphism
1663:
1658:
1654:
1650:
1642:
1638:
1630:
1611:
1607:
1603:
1598:
1594:
1590:
1586:
1582:
1571:
1567:
1551:
1547:
1542:
1538:
1530:
1526:
1522:
1518:
1513:
1509:
1502:
1498:
1493:
1489:
1485:
1481:
1476:
1472:
1468:
1464:
1455:
1451:
1330:
1280:
1276:
1217:by defining
1214:
1210:
1206:
1139:
1081:
1078:fiber bundle
1069:
1061:
1057:
1053:
1048:
1044:
1040:
1039:such that α(
1035:
1031:
1023:
1017:
1008:
893:
889:
885:
881:
874:
866:
862:
857:
853:
849:
847:
845:, §II.1.6).
835:
823:
818:
814:
810:
806:
798:
794:
790:
786:
781:
776:
772:
768:
764:
760:
756:
752:
747:
742:
738:
734:
730:
728:
698:
694:
690:
688:
679:
675:
671:
667:
663:
661:
655:
651:
647:
643:
639:
634:
629:
622:
615:
611:
604:
597:
595:
586:
579:
572:
565:
558:
551:
544:
537:
529:
525:
514:
510:
502:
498:
493:
481:
429:
425:
421:
417:
413:
405:
401:
397:
393:
389:
385:
381:
377:
373:
369:
365:
361:
359:
354:
350:
346:
342:
340:
310:
306:
302:
300:
295:
291:
287:
280:
273:
269:
265:
258:
251:
247:
245:
239:
236:
232:
229:
225:
221:
217:
208:
201:
194:
187:
180:
173:
166:
159:
147:
143:
139:
132:
121:
117:
115:
106:
94:
77:
63:
42:
36:
4868:subspaces.
4463:defined by
4415:indexed by
3107:called the
2716:commutative
2448:are called
2378:Halmos 1974
2342:. If every
1624:associative
1620:commutative
152:Halmos 1974
5765:Categories
5639:0292.10016
5609:Milnor, J.
5595:References
5068:for every
5012:such that
4866:orthogonal
4224:such that
3753:such that
3351:direct sum
3339:direct sum
3126:. See his
3066:tessarines
2682:such that
2579:. Now let
2424:such that
2303:See also:
1437:Properties
1064:) = 0 for
886:direct sum
691:direct sum
303:direct sum
43:direct sum
32:Direct sum
5556:Biproduct
5444:∑
5435:‖
5429:‖
5365:ℓ
5327:∈
5320:⊕
5225:⟩
5216:β
5203:α
5199:⟨
5190:∑
5183:⟩
5180:β
5174:α
5171:⟨
5146:∞
5118:α
5103:∑
5079:∈
5020:α
4909:∈
4847:⟩
4821:⟨
4815:⋯
4809:⟩
4783:⟨
4754:…
4712:…
4656:…
4538:∈
4531:∑
4391:∈
4315:∞
4311:ℓ
4255:⊕
4097:∑
4090:‖
4084:‖
4020:ℓ
3982:∈
3975:⨁
3880:∞
3861:‖
3845:‖
3837:∈
3830:∑
3806:∈
3773:∈
3684:∈
3677:⨁
3644:index set
3575:∈
3549:∈
3520:‖
3513:‖
3501:‖
3494:‖
3488:‖
3470:‖
3242:λ
3215:λ
3191:λ
3168:λ
3165:⊕
3159:λ
3147:⊕
3138:λ
3087:⊕
3047:⊕
3003:⊕
2788:with the
2751:coproduct
2664:→
2646:∈
2639:⨁
2536:∈
2529:⨁
2525:→
2466:coproduct
2397:, p.61).
2334:for each
2328:submodule
2255:∘
2199:∘
2183:⋯
2167:∘
2127:⊕
2124:⋯
2121:⊕
2108:↦
2055:→
2042:⊕
2039:⋯
2036:⊕
2003:biproduct
1933:α
1918:β
1910:∈
1903:∑
1893:α
1884:β
1873:−
1869:τ
1822:) is the
1762:
1744:∈
1737:∏
1733:≅
1702:∈
1695:⨁
1684:
1444:submodule
1411:α
1403:∑
1376:α
1372:∑
1364:as a sum
1343:α
1299:∈
1292:⨁
1253:α
1231:α
1187:β
1174:α
1157:β
1151:α
1095:∈
1020:functions
979:α
948:∈
939:α
909:α
755:× {0} of
712:⊕
466:⊥
458:⊕
410:dimension
364:× {0} of
324:⊕
128:over the
51:coproduct
5742:(1999),
5717:(1974),
5676:(1989),
5654:(1972),
5550:See also
5468:‖
5455:‖
5133:‖
5114:‖
4930:complete
4776:⟩
4690:⟨
3795:for all
3538:for all
3334:(1995).
3115:in 1873.
3072:in 1848.
2810:algebras
2755:category
2702:for all
2596: :
2430:internal
2309:Suppose
1957:for any
1641:-module
1559:and the
1406:′
1275:for all
1205:for all
1068:indices
1052:for all
1007:indices
873:of left
750:subgroup
116:Suppose
82:integers
61:notion.
5744:Algebra
2784:of the
2470:colimit
2428:is the
2363:is the
1537:If the
1446:of the
1026:to the
1022:α from
1011:. (The
839:modules
779:. (See
684:integer
388:. (See
47:modules
5750:
5725:
5703:
5684:
5662:
5637:
5627:
4439:. The
4413:family
3306:
3290:
2745:or an
2719:monoid
2313:is an
2237:, and
2228:⊕ ⋯ ⊕
1999:finite
1110:being
1060:and α(
930:where
884:. The
871:family
826:finite
801:. The
678:, and
614:, and
484:finite
408:. The
290:, and
135:. The
72:) and
41:, the
5094:and:
4411:be a
3361:is a
3353:is a
2408:is a
2326:is a
2005:: If
1593:with
1480:with
1467:from
1213:from
650:) = (
564:) = (
550:) + (
228:) = (
186:) = (
172:) + (
130:field
70:field
5748:ISBN
5723:ISBN
5701:ISBN
5682:ISBN
5660:ISBN
5625:ISBN
5273:and
5143:<
4898:for
4339:Let
3928:and
3877:<
3821:and
3564:and
3442:and
3402:and
3341:and
2831:and
2761:and
2743:norm
2444:and
2436:and
1997:The
1622:and
1577:The
1003:for
970:and
888:of {
869:} a
848:Let
817:and
803:rank
775:and
748:The
697:and
662:for
596:for
513:and
501:and
494:For
424:and
384:and
309:and
246:for
124:are
120:and
88:and
59:dual
5635:Zbl
4204:of
4178:is
3024:to
3020:is
2800:).
2484:in
2420:of
2412:of
2404:of
2346:in
2338:in
2330:of
1839:to
1753:Hom
1675:Hom
1662:to
879:set
805:of
693:of
682:an
674:in
666:in
628:in
610:in
509:of
412:of
305:of
80:of
37:In
5767::
5738:;
5633:.
5615:.
3369:.
2765:.
2706:.
2693:=
2686:o
2605:→
2452:.
2446:N′
2438:N′
2418:N′
2290:=
1818:)(
1666::
1488:)∈
1137:.
1056:∈
821:.
809:⊕
789:⊕
759:⊕
745:.
741:+
733:,
670:,
656:nh
654:,
652:ng
646:,
632:.
621:,
603:,
585:+
578:,
571:+
557:,
543:,
528:×
416:⊕
396:⊕
368:⊕
357:.
353:+
345:,
298:.
294:∈
286:∈
279:,
272:,
268:,
264:∈
257:,
250:,
235:,
224:,
207:+
200:,
193:+
179:,
165:,
142:×
92:.
5757:.
5710:.
5691:.
5669:.
5641:.
5522:.
5518:C
5508:R
5480:.
5473:2
5463:i
5459:a
5448:i
5438:=
5432:a
5408:)
5403:i
5399:a
5395:(
5374:,
5369:2
5342:i
5338:X
5331:N
5324:i
5299:,
5295:R
5291:=
5286:i
5282:X
5260:N
5256:=
5253:I
5228:.
5220:i
5212:,
5207:i
5194:i
5186:=
5177:,
5149:.
5138:2
5128:)
5125:i
5122:(
5107:i
5082:I
5076:i
5054:i
5050:H
5029:)
5026:i
5023:(
5000:,
4997:I
4975:i
4971:H
4945:i
4941:H
4912:I
4906:i
4884:i
4880:H
4850:.
4842:n
4838:y
4834:,
4829:n
4825:x
4818:+
4812:+
4804:1
4800:y
4796:,
4791:1
4787:x
4780:=
4771:)
4765:n
4761:y
4757:,
4751:,
4746:1
4742:y
4737:(
4733:,
4729:)
4723:n
4719:x
4715:,
4709:,
4704:1
4700:x
4695:(
4667:n
4663:H
4659:,
4653:,
4648:1
4644:H
4606:I
4585:)
4579:i
4575:y
4571:,
4566:i
4562:x
4557:(
4551:i
4547:b
4541:I
4535:i
4527:=
4523:)
4518:)
4512:i
4508:y
4503:(
4499:,
4495:)
4489:i
4485:x
4480:(
4475:(
4471:B
4451:B
4423:I
4398:}
4394:I
4388:i
4385::
4381:)
4375:i
4371:b
4367:,
4362:i
4358:M
4353:(
4348:{
4320:.
4287:0
4283:c
4261:.
4258:B
4252:A
4232:X
4212:X
4192:B
4166:X
4146:A
4124:.
4120:|
4115:i
4111:a
4107:|
4101:i
4093:=
4087:a
4063:)
4058:i
4054:a
4050:(
4029:,
4024:1
3997:i
3993:X
3986:N
3979:i
3954:,
3950:R
3946:=
3941:i
3937:X
3915:N
3911:=
3908:I
3883:.
3870:i
3866:X
3857:)
3854:i
3851:(
3848:x
3840:I
3834:i
3809:I
3803:i
3781:i
3777:X
3770:)
3767:i
3764:(
3761:x
3741:I
3719:x
3697:i
3693:X
3687:I
3681:i
3656:,
3653:I
3630:i
3608:i
3604:X
3581:.
3578:Y
3572:y
3552:X
3546:x
3524:Y
3516:y
3510:+
3505:X
3497:x
3491:=
3485:)
3482:y
3479:,
3476:x
3473:(
3450:Y
3430:X
3410:Y
3390:X
3318:,
3315:H
3310:2
3302:,
3299:C
3294:2
3286:,
3283:R
3278:2
3251:.
3248:)
3245:y
3239:,
3236:x
3233:(
3230:=
3227:)
3224:y
3221:,
3218:x
3212:(
3209:=
3206:)
3203:y
3200:,
3197:x
3194:(
3171:y
3162:x
3156:=
3153:)
3150:y
3144:x
3141:(
3095:,
3091:H
3083:H
3051:C
3043:C
3032:.
3007:R
2999:R
2972:.
2969:)
2964:2
2960:y
2954:1
2950:y
2946:+
2941:2
2937:x
2931:1
2927:x
2923:(
2920:=
2917:)
2912:2
2908:y
2904:+
2899:2
2895:x
2891:(
2888:)
2883:1
2879:y
2875:+
2870:1
2866:x
2862:(
2839:Y
2819:X
2704:i
2699:i
2695:f
2690:i
2688:j
2684:f
2667:M
2659:i
2655:M
2649:I
2643:i
2635::
2632:f
2619:R
2615:i
2611:R
2607:M
2602:i
2598:M
2593:i
2589:f
2585:R
2581:M
2577:i
2572:i
2568:M
2549:i
2545:M
2539:I
2533:i
2520:i
2516:M
2512::
2507:i
2503:j
2486:I
2482:i
2474:R
2442:N
2434:N
2426:M
2422:M
2414:M
2406:M
2402:N
2390:i
2386:M
2382:M
2376:(
2373:i
2369:M
2361:M
2356:i
2352:M
2348:M
2344:x
2340:I
2336:i
2332:M
2323:i
2319:M
2315:R
2311:M
2292:k
2288:l
2283:k
2279:A
2263:l
2259:i
2250:k
2246:p
2234:n
2230:A
2226:1
2223:A
2207:n
2203:p
2194:n
2190:i
2186:+
2180:+
2175:1
2171:p
2162:1
2158:i
2135:n
2131:A
2116:1
2112:A
2103:k
2099:A
2095::
2090:k
2086:i
2063:k
2059:A
2050:n
2046:A
2031:1
2027:A
2023::
2018:k
2014:p
1984:i
1980:i
1978:(
1976:α
1972:τ
1967:i
1963:M
1959:α
1945:)
1942:)
1939:i
1936:(
1930:(
1927:)
1924:i
1921:(
1913:I
1907:i
1899:=
1896:)
1890:(
1887:)
1881:(
1876:1
1858:τ
1853:i
1849:M
1845:x
1843:(
1841:θ
1836:i
1832:M
1830:∈
1828:x
1824:R
1820:i
1816:θ
1814:(
1812:τ
1808:τ
1791:.
1787:)
1783:L
1780:,
1775:i
1771:M
1766:(
1757:R
1747:I
1741:i
1728:)
1723:L
1720:,
1715:i
1711:M
1705:I
1699:i
1689:(
1679:R
1664:L
1659:i
1655:M
1651:R
1643:L
1639:R
1633:-
1631:R
1615:.
1612:i
1608:M
1604:N
1599:i
1595:M
1591:N
1587:R
1583:N
1572:K
1568:K
1563:.
1552:i
1548:M
1543:i
1539:M
1534:.
1531:i
1527:M
1523:x
1519:i
1514:i
1510:M
1503:I
1499:i
1494:i
1490:M
1486:i
1484:(
1482:α
1477:i
1473:M
1469:I
1465:α
1459:(
1456:i
1452:M
1415:i
1380:i
1352:)
1347:i
1339:(
1317:.
1312:i
1308:M
1302:I
1296:i
1281:R
1277:i
1261:i
1257:)
1250:r
1247:(
1244:=
1239:i
1235:)
1228:(
1225:r
1215:R
1211:r
1207:i
1191:i
1183:+
1178:i
1170:=
1165:i
1161:)
1154:+
1148:(
1123:i
1119:M
1098:I
1092:i
1082:I
1070:i
1062:i
1058:I
1054:i
1049:i
1045:M
1041:i
1036:i
1032:M
1024:I
1009:i
991:0
988:=
983:i
956:i
952:M
943:i
918:)
913:i
905:(
894:i
890:M
882:I
875:R
867:I
863:i
858:i
854:M
850:R
819:H
815:G
811:H
807:G
799:H
795:G
791:H
787:G
777:H
773:H
769:G
765:G
761:H
757:G
753:G
743:h
739:g
735:h
731:g
715:H
709:G
699:H
695:G
680:n
676:H
672:h
668:G
664:g
658:)
648:h
644:g
642:(
640:n
630:H
626:2
623:h
619:1
616:h
612:G
608:2
605:g
601:1
598:g
592:)
590:2
587:h
583:1
580:h
576:2
573:g
569:1
566:g
562:2
559:h
555:2
552:g
548:1
545:h
541:1
538:g
536:(
530:H
526:G
515:H
511:G
503:H
499:G
462:W
455:W
452:=
447:n
442:R
430:W
426:W
422:V
418:W
414:V
406:W
402:V
398:W
394:V
386:W
382:W
378:V
374:V
370:W
366:V
362:V
355:w
351:v
347:w
343:v
327:W
321:V
311:W
307:V
296:K
292:α
288:W
284:2
281:w
277:1
274:w
270:w
266:V
262:2
259:v
255:1
252:v
248:v
242:)
240:w
237:α
233:v
230:α
226:w
222:v
220:(
218:α
214:)
212:2
209:w
205:1
202:w
198:2
195:v
191:1
188:v
184:2
181:w
177:2
174:v
170:1
167:w
163:1
160:v
158:(
150:(
148:K
144:W
140:V
133:K
122:W
118:V
78:Z
34:.
20:)
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