Duality
Duality in algebraic geometry
Duality between the different cohomology spaces on algebraic varieties.
Cohomology of coherent sheaves. Let $X$ be a non-singular algebraic variety of dimension $n$ over an algebraically closed field $k$ and let $\mathcal{L}$ be a locally free sheaf on $X$. Serre's duality theorem states that the finite-dimensional cohomology (vector) spaces $H^i(X,\mathcal{L})$ and $H^{n-i}(X,\check{\mathcal{L}}\otimes\omega_X)$ are mutually dual. Here $\omega_X=\Omega_X^n$ is the sheaf of germs of regular differential forms of degree $n$ on $X$, and $\check{\mathcal{L}}=\operatorname{Hom}(\mathcal{L},\mathcal{O}_X)$ is the locally free sheaf dual to $\mathcal{L}$. If $\mathcal{L}=\mathcal{O}_X(D)$ is the invertible sheaf corresponding to a divisor $D$ on $X$, this theorem establishes the equality
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where is the canonical divisor on
. If
, a relation equivalent to the above was found as early as the 19th century. There exists a generalization of Serre's theorem to the case of cohomology of arbitrary coherent sheaves on complete algebraic varieties [1], [4]. In particular, if the variety
is a Cohen–Macaulay subvariety (e.g. a locally complete intersection) of codimension
in a non-singular projective variety
, there is duality between the
-space
and the space of global Ext's
![]() |
where is a coherent sheaf on
,
(Grothendieck's dualizing sheaf), while
. Here, the sheaf
is invertible if and only if
is a Gorenstein scheme (cf. Gorenstein ring).
Etale cohomology. Let be a complete connected non-singular algebraic variety of dimension
over an algebraically closed field
; let
be an integer which is relatively prime to the characteristic of the field
; let
be a locally free (in the étale topology) sheaf of
-modules on
; and let
be the sheaf of
-th power roots of unity. Then there exists a non-degenerate pairing of
-modules [6]:
![]() |
A more general duality theorem concerns smooth, but not necessarily complete, varieties [5]. There exists a non-degenerate pairing of -modules
![]() |
where cohomology with compact support is found on the left-hand side. If the field is the algebraic closure of a field
,
and
, then the Galois group
acts on
and the preceding pairing is a pairing of
-modules.
Poincaré's duality theorem is an analogue of the first of the theorems given for -adic cohomology: There exists a non-degenerate pairing of
-modules
![]() |
where is the Tate sheaf, which is non-canonically isomorphic to the sheaf
(cf.
-adic cohomology). Hence the isomorphism of
-spaces
![]() |
and, in particular, the equality of the Betti numbers
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As in the case of cohomology of coherent sheaves, these results can be generalized to the relative case of a proper scheme morphism, formulated in the language of derived categories [6].
Other cohomology theories. Analogues of Poincaré's theorem are valid in the theory of crystalline cohomology [7], and de Rham cohomology over a field of characteristic zero [8]. In number-theoretic applications cohomology of sheaves on the flat Grothendieck topology of arithmetic schemes are important. Duality theorems [9] are applicable to special cases of such cohomology theories.
References
[1] | A. Grothendieck, "The cohomology theory of abstract algebraic varieties" , Proc. Internat. Math. Congress Edinburgh, 1958 , Cambridge Univ. Press (1960) pp. 103–118 MR0130879 Zbl 0119.36902 |
[2] | I.V. Dolgachev, "Abstract algebraic geometry" J. Soviet Math. , 2 : 3 (1974) pp. 264–303 Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 10 (1972) pp. 47–112 Zbl 1068.14059 |
[3] | J.-P. Serre, "Faisceaux algébriques cohérents" Ann. of Math. , 61 (1955) pp. 197–258 MR0068874 Zbl 0067.16201 |
[4] | R. Hartshorne, "Residues and duality" , Springer (1966) MR0222093 Zbl 0212.26101 |
[5] | "Théorie des topos et cohomologie étale des schémas" M. Artin (ed.) A. Grothendieck (ed.) J.-L. Verdier (ed.) , Sem. Geom. Alg. , 3 , Springer (1973) MR0354654 |
[6] | J.-L. Verdier, "A duality theorem in the etale cohomology of schemes" T.A. Springer (ed.) et al. (ed.) , Proc. Conf. local fields (Driebergen, 1966) , Springer (1967) pp. 184–198 MR0230732 Zbl 0184.24402 |
[7] | P. Berthelot, "Cohomologie cristalline des schémas de caractéristique ![]() |
[8] | R. Hartshorne, "Ample subvarieties of algebraic varieties" , Springer (1970) MR0282977 Zbl 0208.48901 |
[9] | B. Mazur, "Local flat duality" Amer. J. Math. , 92 (1970) pp. 343–361 MR0271119 Zbl 0199.24501 |
[10] | A. Altman, S. Kleiman, "Introduction to Grothendieck duality theory" , Springer (1970) MR0274461 Zbl 0215.37201 |
[11] | J. Milne, "Arithmetic duality theorems" , Acad. Press (1987) MR2261462 MR0881804 Zbl 1127.14001 Zbl 0613.14019 |
Duality in algebraic topology (by G.S. Chogoshvili)
A situation in which the values of certain topological invariants determine the values of others. In algebraic topology duality manifest itself: in duality (in the sense of the theory of characters) between the homology and cohomology groups of the same dimension with dual groups of coefficients; in the isomorphism between homology and cohomology groups of complementary dimensions of a variety (Poincaré duality); in the isomorphism between the homology and cohomology groups of mutually complementary sets of a space (Alexander duality); in the mutual exchangeability, in certain situations, of homotopy and cohomotopy, as well as of homology and cohomology, groups which, in the absence of additional restrictions imposed on the dimension of the space, is valid not for ordinary, but rather for -homotopy and
-cohomotopy groups (see
-duality).
The duality between homology and cohomology consists in the following. Let be an arbitrary homology theory over some admissible category of pairs of spaces and their mappings, i.e. a system which satisfies the Steenrod–Eilenberg axioms of homology theory with discrete or compact Abelian groups
. Then the system
(where
is the group of characters of
, and
and
are the homomorphisms, conjugate, respectively, with
and
) satisfies the Steenrod–Eilenberg axioms of cohomology theory and represents the cohomology theory over the same category with compact or, respectively, discrete groups
. A dual homology theory can be constructed for any cohomology theory in such a manner. Consequently, homology and cohomology theories are dual pairs; the transformation of one theory into the other, up to natural equivalences, is an involution. For any theorem of homology theory, i.e. a theorem about the system
, there exists a dual proposition about the system
, i.e. a theorem of cohomology theory, and vice versa. On passing to a dual proposition, groups are replaced by groups of characters, homomorphisms change direction, subgroups are replaced by quotient groups, and vice versa. The Steenrod–Eilenberg axioms themselves may serve as examples. For specific categories or theories the construction of this duality is realized, for example, in the following manner. Let
be a (finite) complex. The number
![]() |
is taken to be the product of the -dimensional chain
of
over a discrete or compact coefficient group
and the
-dimensional cochain
of
over the coefficient group
dual with
in the sense of the theory of characters. This product defines the multiplication of a homology class by a cohomology class, and converts
-dimensional homology and cohomology into mutual groups of characters. Two types of homology groups — projective and spectral — exist for infinite complexes. Spectral homology groups are the limits of the direct spectra of the homology groups of closed finite subcomplexes, ordered by inclusion, while the projective homology groups are the homology groups of the limits of direct spectra of the chain groups of these finite subcomplexes. Cohomology groups are obtained in a similar manner as the limits of the corresponding inverse spectra. For a discrete group of coefficients both homology groups coincide, and yield a homology group of finite cycles; if the group is compact, the cohomology groups coincide and give a cohomology group of infinite cocycles. The duality existing in finite complexes generates the mutual duality of projective groups and the mutual duality of spectral groups in infinite complexes, and these two last-named dualities (by way of singular complexes, nerves of coverings, etc.) represent the duality between an
-dimensional projective (spectral) homology group
of a space
over a discrete or a compact coefficient group
in any theory (theories of singular homology and cohomology; Aleksandrov–Čech homology and cohomology; Vietoris homology and cohomology; etc.) and an
-dimensional projective (spectral) cohomology group
in the same theory over the group
dual to
[1], [3], , [9]:
![]() |
The relations between the invariants which express the connectivities of a manifold in complementary dimensions were established by H. Poincaré in the first study on algebraic topology (1895). He showed that for an -dimensional orientable manifold, its
-dimensional and
-dimensional Betti numbers are equal, as are the
- and
-dimensional torsion coefficients. This theorem was strengthened by O. Veblen (1923) who formulated it for homology bases, while the use of cohomology groups imparted it a form expressive of the content of this duality. In order to obtain this form, it is necessary to put into correspondence each
-dimensional chain
, given on any triangulation
of an
-dimensional oriented homology manifold
and taking values in a discrete or compact coefficient group
, with an
-dimensional cochain of the cellular complex
of barycentric stars of
, which assumes, on any star, the value of
on the simplex corresponding to this star. Since the groups of the complexes
and
are identical, this correspondence defines an isomorphism of the homology and cohomology groups of complementary dimensions of
:
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Here, may also be a module, and if the manifold is not orientable, the theorem is true modulo 2. Replacing the group
by its dual group
yields the duality [1]:
![]() |
which is also of interest because its product is the intersection index of cycles, arbitrarily selected from the classes undergoing multiplication [1], [11], [12], [13], [15], [16].
J. Alexander's theorem (1922) completed a major stage (initially a set-theoretic stage) in finding the topological properties of a set that are determined by the topological properties of its complement. The theorem states that the -dimensional Betti number modulo 2 of a polyhedron situated in the
-dimensional sphere is equal to the
-dimensional Betti number modulo 2 of the complement (cf. Alexander duality).
This theorem in turn served as the base for a number of investigations which affected to a considerable extent the development of algebraic topology. These studies were conducted with a view to generalizing classes of spaces (plane Euclidean spaces, spheres and manifolds of arbitrary dimension, locally compact spaces, etc.), their subsets (polyhedra, closed subsets, arbitrary subsets) and domains of coefficients (integers modulo 2, the group of integers, the field of rational numbers, other specific groups and fields, arbitrary Abelian groups, topological (mainly compact) Abelian groups, etc.) to which Alexander duality applies, and also strengthening of the relations connecting mutually complementary sets (equality of Betti numbers, group isomorphism, duality of topological groups, natural and connecting homomorphisms, etc.). Several results thus obtained may be represented in the form of the following diagram [1], [3], , [5], , [7], , [9], [11]:
![]() |
where is a discrete or compact group of coefficients,
,
and
are mutually complementary sets of an
-dimensional spherical manifold
,
and
are the
-dimensional Aleksandrov–Čech homology and cohomology groups (with compact support) of the set
over
and, respectively,
, and
and
are the
-dimensional Aleksandrov–Čech spectral homology and cohomology groups of the set
over
and
, respectively. The indicated relations in the diagram, obtained by different workers and by different methods, are coordinated to the extent that the corresponding elements in the isomorphisms represent the same character of the remaining groups for vertical and horizontal dualities. They are thus various forms of the same duality theorem. The upper duality is a link duality, i.e. its product of elements is the linking coefficient of cycles, arbitrarily selected from the multiplier classes or, in the case of a compact group
, is defined by continuity of the cycle linkage. In the diagram given above, the groups of the first column may be replaced by the
-dimensional Steenrod homology and cohomology groups with compact supports, while the groups in the second column can be replaced by the
-dimensional projective Aleksandrov–Čech homology and cohomology groups. Then, for a compact
, the isomorphism of the main diagonal yields the Steenrod duality theorem in its original form if the cohomology group of the set
is replaced, in accordance with Poincaré's theorem, by the
-dimensional homology group of infinite cycles. If the group
is compact, the diagrams are isomorphic; if, in addition, the set
is compact as well, the duality of the top line of the diagram represents the theorem obtained by L.S. Pontryagin [1] in 1934 (cf. Pontryagin duality). For other generalizations and trends of study see [2], [10], [14], [15], [16].
An important form of Alexander duality, which concerns the connecting homomorphism and the exactness axiom, is the isomorphism between homology groups and between cohomology groups of adjacent dimensions. These isomorphisms, determined by P.S. Aleksandrov and A.N. Kolmogorov, state that the -dimensional homology (cohomology) group of a closed set
of a normal locally compact space
which is acyclic in dimensions
and
, over a compact (discrete) group
, is isomorphic to the
-dimensional homology (cohomology) group of the complement:
![]() |
and
![]() |
Pontryagin's theorem is deduced from these isomorphisms. Aleksandrov [2] obtained these isomorphisms from the general duality relations relating homology and cohomology groups of mutually complementary sets and the space, as well as various kernels, images and quotient groups under imbedding and excision homomorphisms. These relations also carry a large amount of other important information about the positioning of sets in space. Aleksandrov [2] obtained them with the aid of spectral homology and cohomology groups with respect to the so-called singular subcomplexes of nerves consisting of simplices, the closures of the vertices of which are non-compact. Kolmogorov proved the above duality isomorphisms by way of his functional homology and cohomology groups (cf. Kolmogorov duality). These and other dualities (e.g. Lefschetz duality) are connected by various relations. They may also be considered as consequences of some general duality in which the so-called exterior groups of a set, which are direct limits of the cohomology groups of the neighbourhoods of this set ordered by imbedding, participate [3], , [5], , [7], [12], [13]. Connections between different dualities assume a novel aspect if viewed from the point of view of sheaf theory.
References
[1] | L.S. Pontryagin, "Topological duality theorems" Uspekhi Mat. Nauk , 2 : 2 (1947) pp. 21–44 (In Russian) MR0025725 |
[2] | P.S. Aleksandrov, "On homological positioning properties of complexes and closed sets" Izv. Akad. Nauk SSSR Ser. Mat. , 6 (1942) pp. 227–282 (In Russian) |
[3] | P.S. Aleksandrov, "Fundamental duality theorems for non-closed sets" Mat. Sb. , 21 : 2 (1947) pp. 161–232 (In Russian) |
[4a] | P.S. Aleksandrov, "Topological duality theorems I" Transl. Amer. Math. Soc. (2) , 30 (1963) pp. 1–102 Trudy Mat. Inst. Steklov. , 48 (1955) pp. 1–108 MR0155320 Zbl 0127.13402 |
[4b] | P.S. Aleksandrov, "Topological duality theorems II" Transl. Amer. Math. Soc. (2) , 30 (1963) pp. 103–233 Trudy Mat. Inst. Steklov. , 54 (1959) pp. 1–136 MR0155321 Zbl 0127.13403 |
[5] | G.S. Chogoshvili, "The duality theorem for retracts" Dokl. Akad. Nauk SSSR , 51 : 2 (1946) pp. 87–90 (In Russian) |
[6a] | G.S. Chogoshvili, "On homology theory for non-closed sets" , General topology and its relations to modern analysis and algebra (Proc. Symp. Prague) , Acad. Press (1961) pp. 123–132 |
[6b] | G.S. Chogoshvili, "Generalized products and limits and their application in homology theory" Uspekhi Mat. Nauk , 21 : 4 (1966) pp. 23–34 (In Russian) |
[7] | S. Kaplan, "Homology properties of arbitrary subsets of Euclidean spaces" Trans. Amer. Math. Soc. , 62 (1947) pp. 248–271 MR0024128 Zbl 0034.10902 |
[8a] | K.A. Sitnikov, "Combinatorial topology of non-closed sets I" Transl. Amer. Math. Soc. (2) , 15 (1960) pp. 245–295 Mat. Sb. , 34 (1954) pp. 3–54 |
[8b] | K.A. Sitnikov, "Combinatorial topology of non-closed sets II" Transl. Amer. Math. Soc. (2) , 15 (1960) pp. 297–343 Mat. Sb. , 37 (1955) pp. 385–434 |
[8c] | K.A. Sitnikov, "Combinatorial topology of non-closed sets III" Mat. Sb. , 48 (1959) pp. 213–226 (In Russian) |
[9] | N.A. Berikashvili, "On the axiomatic theory of spectra and duality laws for arbitrary sets" Trudy Tbil. Mat. Inst. , 24 (1957) pp. 409–484 (In Russian) |
[10] | D.S. Baladze, "Steenrod's generalized homology theory" Trudy Tbil. Mat. Inst. , 41 (1972) pp. 41–83 (In Russian) |
[11] | D.G. Bourgin, "Modern algebraic topology" , Macmillan (1963) MR0160201 |
[12] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) MR0210112 MR1325242 Zbl 0145.43303 |
[13] | R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) MR0385836 Zbl 0305.55001 |
[14] | E.G. Sklyarenko, "On the theory of generalized manifolds" Math. USSR Izv. , 5 : 4–6 (1971) pp. 845–858 Izv. Akad. Nauk SSSR. Ser. Mat. , 35 : 4 (1971) pp. 831–843 |
[15] | A. Borel, J.C. Moore, "Homology theory for locally compact spaces" Mich. Math. J. , 7 (1960) pp. 137–160 MR0131271 Zbl 0116.40301 |
[16] | G.E. Bredon, "Sheaf theory" , McGraw-Hill (1967) MR0221500 Zbl 0158.20505 |
Duality in the theory of analytic spaces (by V.P. Palamodov)
Duality between the various topological vector cohomology spaces of complex spaces. There are three types of duality theorems, which correspond to Poincaré, Lefschetz and Aleksandrov–Pontryagin dualities in topology, but which concern the cohomology spaces of a complex space
with values in a coherent analytic sheaf
and supports in the family
or in a quotient space of it (see Cohomology with values in a sheaf).
Serre's duality theorem [1] belongs to the first type. Let be a complex manifold of dimension
with a countable base, let
be the sheaf of holomorphic differential forms of degree
and let
be a locally free analytic sheaf on
. For each integer
,
, one defines the bilinear mapping
![]() | (*) |
which may be written as the composition of a -multiplication
![]() |
( denotes the family of compact supports) and linear forms
on
, known as traces, of the form
![]() |
where is the form of type
with compact support which corresponds to the class
by virtue of Dolbeault's theorem (cf. Differential form). Serre's duality theorem states that if the cohomology spaces are endowed with a canonical locally convex topology (cf. Coherent analytic sheaf), then the mapping (*) is continuous with respect to the first argument and, if the space
is separable, it defines an isomorphism of vector spaces
![]() |
The roles of the sheaves and
may be interchanged, since the operation
on locally free sheaves is involutory.
In particular, if the manifold is compact,
is the canonical and
is an arbitrary divisor on
, Serre's theorem implies the equality of the dimensions of the spaces
and
, which is often used in computations with cohomology. A similar duality theorem is known for non-singular projective algebraic varieties over an arbitrary field (see Duality in algebraic geometry).
If is an arbitrary coherent analytic sheaf on the manifold
, there exists a real topological duality between the individual spaces associated with the topological vector spaces
and
, where
is the family of closed supports,
is the family of compact supports, or vice versa, while the
denote the derived functor (cf. Derived functor) of the functor
. The space
is separable if
is separable, and vice versa [2], [3]. This implies, for a compact manifold
, an isomorphism of finite-dimensional spaces
![]() |
If is a Stein manifold, one obtains topological duality between
and
, and also between
and
.
There also exists a generalization of these results to the case of complex spaces with singularities [4] and to the relative case [5], in analogy to the corresponding duality theorems in algebraic geometry.
The following duality theorem is an analogue of Lefschetz's theorem [3]: Let be a complex manifold of dimension
with a countable base; let
be a Stein compactum in
. For any coherent analytic sheaf
on
and any integer
the space
has a topology of type DFS (is strongly dual to a Fréchet–Schwartz space), and its dual space is algebraically isomorphic to
. According to another theorem of this type [6], under the same assumptions, if
is open, the space
has a topology of type QFS (is a Fréchet–Schwartz quotient space),
has a topology of type QDFS (is a quotient space of type DFS), while the associated separable spaces are in topological duality. The space
is separable if and only if
is.
The third type of duality theorem is represented by the following duality theorem [8]: For any open subset , the strong dual to the space
is isomorphic to
. This theorem may be generalized as follows [7]: Let
be an
-dimensional complex manifold, countable at infinity, let
be open, let
be a coherent analytic sheaf on
, and let
be an integer. Consider canonical mappings of topological vector spaces
![]() |
![]() |
![]() |
For the separable space associated with to be isomorphic to the strong dual of
, it is necessary and sufficient for
to be closed. (An example of a non-closed
is known.) In particular, if the sheaf
is locally free and if
![]() |
then the separable spaces associated with and
are in duality.
References
[1] | J.-P. Serre, "Une théorème de dualité" Comm. Math. Helv. , 29 (1955) pp. 9–26 |
[2] | B. Malgrange, "Systèmes differentiels à coefficients constants" , Sem. Bourbaki , 246 , Benjamin (1962–1963) MR1611532 MR0304381 MR0176211 Zbl 0229.35015 Zbl 0141.27304 Zbl 0231.46073 |
[3] | C. Banica, O. Stanasila, "Algebraic methods in the global theory of complex spaces" , Wiley (1976) (Translated from Rumanian) MR0463470 Zbl 0334.32001 |
[4] | J.P. Ramis, G. Ruget, "Complex dualisantet théorèmes de dualité en géométrie algébrique complexe" Publ. Math. IHES , 38 (1970) pp. 77–91 |
[5] | J.P. Ramis, G. Ruget, "Résidus et dualité" Invent. Math. , 26 : 2 (1974) pp. 89–131 MR0352522 Zbl 0304.32007 |
[6] | V.D. Golovin, "On spaces of local cohomologies of complex analytic manifolds" Funct. Anal. Appl. , 5 : 4 (1971) pp. 320–322 Funktsional. Anal. Prilozhen. , 5 : 4 (1971) pp. 66 Zbl 0242.32009 |
[7] | V.D. Golovin, "Alexander–Pontryagin duality in complex analysis" Math. Notes , 13 : 4 (1973) pp. 339–341 Mat. Zam. , 13 : 4 (1973) pp. 561–564 Zbl 0271.32007 |
[8] | A. Grothendieck, "Sur certains espaces de fonctions holomorphes" J. Reine Agew. Math. , 192 (1953) pp. 35–64 MR0062335 MR0058865 Zbl 0051.08704 |
Duality in analytic function theory (by A.I. Markushevich and S.Ya. Khavinson)
Borel transforms
E. Borel (1895) must be credited with the idea of transforming a series
![]() |
into the series
![]() |
and conversely, under the condition that
![]() |
This is a duality relation between functions which are analytic in a neighbourhood of infinity and entire functions of exponential type
. For instance, Pólya's theorem is obtained in this manner: Let
be the supporting function of the convex envelope of the set of singularities of a function
under analytic continuation to a half-plane of the form
, and let
![]() |
be the growth indicator of the entire function ; then
![]() |
By virtue of this duality relation the problem of analytic continuation of the function to the disc
is equivalent to the study of the growth of the corresponding entire function
in different directions.
Duality in spaces of analytic functions
Let be an open set in the extended complex plane
and let
be the space of analytic functions in
with topology defined by the system of norms
![]() |
where is an increasing system of compact sets contained in
and exhausting
; thus, the convergence
in
means uniform convergence
on all compact subsets of
. Let
, let
be the subspace of
of functions for which
and let
be a compact subset of
. Consider the system
of all open sets
and the set of functions
. Two functions
and
in this set are considered to be equivalent if their restrictions to some set
coincide. The equivalence relation introduced subdivides the entire set under consideration into classes
. Each class is said to be a local analytic function on
, and the set of such functions is denoted by
. The class
is naturally converted into a linear space, with the topology of the inductive limit of sequences of normed spaces
introduced on it. This space is constructed as follows. Let
be a decreasing sequence in
such that
and
,
![]() |
Now is the space of bounded analytic functions in
with norm
![]() |
The simplest fact about the duality of spaces of analytic functions is the following. Let be an open set, let (for the sake of being specific)
, and
. The space
is dual (conjugate) to the space
in the sense of the theory of linear topological spaces. This duality is established as follows: If
is a continuous linear functional on
, then there exists a unique element
such that
![]() |
where is some (composite) contour lying in
and including
, while
, and
does not depend on
. The spaces
may be defined for arbitrary sets
, and not only for the cases considered here, when
is an open set and
is a compactum. Other generalizations include the consideration of sets on Riemann surfaces, spaces of functions of several complex variables and spaces of vector-valued analytic functions (with values in linear topological spaces).
The development of the duality theory of spaces of analytic functions was stimulated by the development of the general theory of duality of linear topological spaces and was itself a stimulus to the development of this theory by revealing deep specific relations. The applications of the duality theory of spaces of analytic functions are many, including problems of interpolation and approximation (see below), analytic continuation, subdivision and elimination of sets of singularities, and integral representations of various classes of functions.
Duality between completeness and uniqueness theorems
A system of elements of a locally convex space
is complete if and only if, for an arbitrary linear functional
continuous on
, it follows from
,
that
. This fact forms the connection between completeness problems in spaces of analytic functions and various uniqueness theorems for analytic functions. The functional
is connected (cf. b) above) with some analytic function
. The condition
,
renders
equal to zero at certain points or else renders the coefficients of
equal to zero. The uniqueness theorems lead to the conclusion that
, so that
as well. The following duality principle of uniqueness and completeness problems has been formulated for spaces of analytic functions in a disc. Let
and
be, respectively, the spaces of functions which are analytic in the discs
and
, where
, and let
be a function which is analytic in the bicylinder
,
. Let
and
be linear functionals defined on
and
, and let
and
be subsets of functions which can be represented, respectively, as
and
. A sequence of functions
is complete in
if and only if for each
it follows from
,
that
. In particular, if
and if
, both sets
and
coincide with the set of all entire functions of exponential type.
Duality in extremal problems of the theory of functions
It is known that the problems of best approximation in normed spaces are dually connected with certain linear extremal problems. Thus, if is a subspace in a normed space
and
is an arbitrary element of
, one has
![]() | (1) |
where is the annihilator of
, i.e. the totality of linear functionals
which vanish on the elements of
. Relation (1), which is based on the Hahn–Banach theorem, subsequently proved to be a special case of duality relations between extremal problems of mathematical programming. Let
be an
-connected domain whose boundary
consists of rectifiable contours, let
be the class of analytic functions
in
,
, let
be the class of analytic functions in
which may be represented by the Cauchy integral over their boundary values, and let
be some integrable function on
. Then:
![]() | (2) |
The left-hand side of this equation is a linear extremal problem for bounded functions (e.g. if , the resulting problem is
, i.e. the problem of Schwarz's lemma in a multiply-connected domain). The right-hand side of the equation is the problem of the best approximation of an arbitrary function
on
by boundary values of analytic functions in the integral metric. Relation (2) serves as the starting point for penetrating into each one of these two extremal problems involved: It serves to establish the characteristic properties of the extremal functions
and
, the problem of their uniqueness, etc. The function
proves to have important geometric properties: In the problem of the Schwarz lemma it maps
onto an
-sheeted disc; in other problems with an
which is analytic on
, the function
maps
into an
-sheeted disc [1]–[6].
References
[1] | A.I. Markushevitch, "Selected chapters on the theory of analytic functions" , Moscow (1976) (In Russian) |
[2] | V.P. Khavin, "Spaces of analytic functions" Itogi Nauk. Mat. Anal. 1964 (1966) pp. 76–164 (In Russian) MR1771750 MR0776964 Zbl 0601.30055 Zbl 0151.08702 |
[3] | Itogi Nauk. Mat. Anal. 1967 (1969) pp. 75–132 |
[4] | Itogi Nauk. Mat. Anal. 1963 (1965) pp. 5–80 |
[5] | A.I. Markushevitch [A.I. Markushevich] (ed.) , Fonctions d'une variable complexe. Problème contemporains , Gauthier-Villars (1962) (Translated from Russian) |
[6] | S.Ya. Khavinson, "Two papers on extremal problems in complex analysis" Transl. Amer. Math. Soc. (2) , 129 (1986) (Translated from Russian) |
Duality in the theory of topological vector spaces (by N.K. Nikol'skii)
A dual pairing is a triplet in which
are vector spaces over a field
, and
is a bilinear functional (form) on
which has the property of being non-degenerate (or separating): If
for each
,
, then
; if
for each
,
, then
. One also says that
realizes the duality and that
form a dual pair; if
is fixed, one writes
. The most important example is the natural duality:
is a locally convex topological vector space with the topology
,
is the dual space (cf. Adjoint space) of all linear
-continuous functionals on
and
if
,
; the fact that this form is non-degenerate is a consequence of, for example, the local convexity of the topology
(a corollary of the Hahn–Banach theorem). The main subject of duality theory are methods for constructing objects in
or
which are dual to given ones with respect to the form
; the correspondence between the properties of mutually dual objects; and the topologies generated by the duality. The principal tool in these studies is the apparatus of polars; if
or
, the polar of a set
,
, is the set
![]() |
The duality generates various locally convex topologies on (and also on
); for instance, the weak topology
(generated by a given duality), specified by the family of semi-norms
,
, is the weakest topology for which all the mappings
are continuous; the Mackey topology
, with a neighbourhood base of zero formed by the polars
of the absolutely convex
-compact subsets
in
; and the strong topology
, a base of which is formed by the polars of the bounded subsets
in
. For any
,
, the set
is the convex
-closed hull of the set
(the bipolar theorem). The space
is identical with
(the basic theorem of duality theory which proves that any duality may be interpreted as natural). The space
is said to be the weak conjugate (or dual) of
.
Let be a locally convex space over
or
. Each one of the following conditions is necessary and sufficient for a set
,
, to be bounded: a)
is bounded in the weak topology; and b)
is an absorbing set. If
is a neighbourhood of zero,
is
-compact. A metric space
is complete if and only if a set
,
, is closed in the topology
whenever all intersections
are closed in the same topology, where
runs through the set of neighbourhoods of zero in
(Krein–Shmul'yan theorem). If
is a complete separable space and
is a linear functional on
, then
if and only if
in the topology
implies that
(Grothendieck's theorem). A subset
of a complete space
is relatively
-compact if it is relatively
-sequentially compact (Eberlein's theorem). A convex subset
of a Fréchet space over
is
-compact if and only if for any
there exists an
such that
(James' theorem).
is the finest and
is the coarsest among the topologies
for which
(the Mackey–Arens theorem, which yields a description of duality-preserving topologies of importance in applications). Each one of the following conditions concerning the space
suffices for
to coincide with the Mackey topology: a)
is a barrelled space; and b)
is a bornological space (in particular, a metric space). The strong topology
, generally speaking, does not preserve the duality; if
is locally convex and
, the space
is said to be the strong dual of
, and if, in addition,
preserves the duality (i.e. if
), the space
is said to be semi-reflexive (
is a reflexive space if
).
Let be a subspace of
;
and
will then be dual pairs with respect to natural factorizations of the form
. If a family of dualities
is given, the duality of the product space
and the subspace
is realized by the form
![]() |
where
![]() |
The dualities of the inductive and the projective limits ,
are described in a similar manner. The presence of duality-preserving topologies in the spaces
,
makes it possible to interpret these statements as the description of natural dualities for
(the Tikhonov topology),
(the quotient topology),
(the induced topology),
and
, respectively. In the case of a normed space
the natural isomorphism of
and
is an isometry
![]() |
The use of duality in specific problems of linear analysis is proportional to the role played in such problems by linear (continuous) functionals. Especially essential (and possibly even crucial) are the ideas of duality theory in the following branches of analysis: in the study of linear topological (metric) properties of locally convex spaces and, in particular, the description of the natural duality for a given space [1], [2], [3], [5]; in the theory of generalized functions ; in the theory of extremal problems [6], [7]; in the spectral and structure theory of linear operators [1], [2]; in the completeness and uniqueness theorems in the theory of analytic functions; in the Fantappié theory of analytic functionals [8]; see also Duality in analytic function theory.
References
[1] | N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French) MR0583191 Zbl 1106.46003 Zbl 1115.46002 Zbl 0622.46001 Zbl 0482.46001 |
[2] | W.J. Robertson, "Topological vector spaces" , Cambridge Univ. Press (1964) MR0162118 Zbl 0123.30202 |
[3] | H.H. Schaefer, "Topological vector spaces" , Macmillan (1966) MR0193469 Zbl 0141.30503 |
[4] | N. Dunford, J.T. Schwartz, "Linear operators" , 1–3 , Interscience (1958–1971) MR1009164 MR1009163 MR1009162 MR0412888 MR0216304 MR0188745 MR0216303 MR1530651 MR0117523 Zbl 0635.47003 Zbl 0635.47002 Zbl 0635.47001 Zbl 0283.47002 Zbl 0243.47001 Zbl 0146.12601 Zbl 0128.34803 Zbl 0084.10402 |
[5] | M.M. Day, "Normed linear spaces" , Springer (1958) MR0094675 Zbl 0082.10603 |
[6] | A.D. Ioffe, V.M. Tikhomirov, "Theory of extremal problems" , North-Holland (1979) (Translated from Russian) MR0528295 MR0527119 Zbl 0407.90051 |
[7] | R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970) MR0274683 Zbl 0193.18401 |
[8] | V.P. Khavin, "Spaces of analytic functions" Itogi Nauk. Mat. Anal. 1964 (1966) pp. 76–164 (In Russian) MR1771750 MR0776964 Zbl 0601.30055 Zbl 0151.08702 |
[9] | S.Ya. Khavinson, "Extremal problems for bounded analytic functions with interior side conditions" Russian Math. Surveys , 18 (1963) pp. 23–96 Uspekhi Mat. Nauk , 18 : 2 (1963) pp. 25–98 Zbl 0149.03302 |
[10] | J. Diestel, "Geometry of Banach spaces - selected topics" , Springer (1975) MR0461094 Zbl 0307.46009 |
Comments
There is a well-used topology on the dual of a topological vector space
: the weak-
-topology. It is the weakest topology on
in which all mappings
,
, are continuous.
Duality in extremal problems and convex analysis (by V.M. Tikhomirov)
A property of convex sets, convex functions and convex extremal problems, viz. that they can be posed in a dual manner — in the basic and in the dual (conjugate) spaces. Closed convex sets in a locally convex topological vector space may be described in a dual way: they are identical with the intersection of the closed half-spaces which contain them. This makes it possible for any convex set in a vector space
to be connected with a dual object in the conjugate space — its polar
. Closed convex functions (i.e. functions with convex and closed supergraphs) in a locally convex topological vector space also permit a dual description (cf. also Dual functions; Conjugate function): they are pointwise least upper bounds of the affine functions which do not exceed them in size. Such a duality permits one to establish a connection between a convex function
and the dual object — the conjugate function given on the conjugate space
and defined by the formula
![]() |
Pointwise least upper bounds of linear functions in a locally convex topological vector space are convex closed homogeneous functions. This fact forms the base for the duality between convex sets and convex homogeneous functions. The dualities just described are based on the Hahn–Banach theorem about the extension of linear functionals and the theorem of separability of convex sets.
The meaning of the dual specification of convex sets and convex functions is reflected in the involutory nature of the polar operator and the conjugation operator
, which exists for convex closed sets containing zero and convex closed functions which are everywhere larger than
. This last result, which concerns functions (the Fenchel–Moreau theorem), generates many duality theorems for the extremal problems of linear and convex programming. An example of a pair of dual problems in linear programming is the following:
![]() | (1) |
Here
![]() |
![]() |
The following alternative is valid for a pair of dual problems in linear programming: The values of the problems are either finite and equal and both problems have a solution, or else the set of permissible values of one of the problems is empty or the solution of the problem equals infinity. The usual method for constructing a dual problem is as follows. The problem of minimization:
![]() | (2) |
where is a linear space,
, is included in a class of similar problems which depend on a parameter:
![]() |
where is some other linear space,
,
(the function
is known as a perturbation of
). As a rule,
is assumed to be convex. The problem which is dual to the problem in relation to a given perturbation is the problem
![]() | (2ast) |
where is the function dual to (conjugate with)
in the sense of Legendre–Young–Fenchel (cf. Dual functions). For the simplest problems in convex programming, of the type
![]() | (3) |
where is a linear space,
are convex functions on
and
is a convex set in
(linear programming problems are special cases of (3)), the following standard perturbations, which depend on the parameters
,
,
,
,
,
, are usually employed. The duality theorems for the general classes of linear problems state that, if certain assumptions regarding the perturbation
are made, the values of the problems (2) and (2ast) coincide and, in addition, the solution of one of the problems is a Lagrange multiplier for the other.
References
[1] | H. Minkowski, "Geometrie der Zahlen" , Chelsea, reprint (1953) MR0249269 Zbl 0050.04807 |
[2] | H. Minkowski, , Gesammelte Abhandlungen , 1–2 , Teubner (1911) Zbl 42.0023.03 |
[3] | W. Fenchel, "On conjugate convex functions" Canad. J. Math. , 1 (1949) pp. 73–77 MR0028365 Zbl 0038.20902 |
[4] | R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970) MR0274683 Zbl 0193.18401 |
[5] | I. Ekeland, R. Téman, "Analyse convexe et problèmes variationnels" , Dunod (1974) MR0463993 MR0463994 Zbl 0281.49001 |
[6] | A.D. Ioffe, V.M. Tikhomirov, "Theory of extremal problems" , North-Holland (1979) (Translated from Russian) MR0528295 MR0527119 Zbl 0407.90051 |
Duality of finite Abelian groups (by A.I. Kostrikin)
The classical prototype of general Pontryagin duality and of its various subsequent modifications. It concerns the properties of the isomorphic correspondence between a finite Abelian group $A$ and the group $\hat{A}=Hom(A,k^*)$ of its characters with values in the multiplicative group $k^*$ of an algebraically closed field $k$ of a characteristic which does not divide the order of $A$ (cf. Character group). The natural mapping $$ f\colon A\to \hat{\hat{A}} $$ defined by the rule $$ f(a)(\chi)=\chi(a) $$ for all $a\in A$, $\chi\in\hat{A}$, is also an isomorphism, while for any subgroup $B\subseteq A$ one has $f(B)=(B^\perp)^\perp$, where $$ B^\perp=\{\chi\in\hat{A}\colon\chi(b)=1\quad\text{for all}\quad b\in B\}\cong\widehat{A/B}. $$ The correspondence $B\to B^\perp$ establishes a duality between the lattices of subgroups of $A$ and $\hat{A}$. This is a one-to-one correspondence and it has the properties $$ (BC)^\perp=B^\perp\cap C^\perp,\quad (B\cap C)^\perp=B^\perp C^\perp. $$
References
[1] | L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104 |
[2] | B. Huppert, "Endliche Gruppen" , 1 , Springer (1979) pp. 482–490 MR0224703 Zbl 0412.20002 |
Comments
Duality is a very pervasive and important concept in (modern) mathematics. Besides in various articles mentioned above such as Alexander duality; Kolmogorov duality; Lefschetz duality; S-duality, still more duality-type material can be found in the articles Hypergraph (dual hypergraph), Graph, planar (dual graph), Algebra of logic (duality principle in logic, dual operations), Induced representation (Frobenius reciprocity or duality), Projective plane and Projective space (duality principle in projective geometry), Duality principle (in geometry and in logic), Linear programming (dual linear programs and the dual simplex method), Unitary representation (the dual space of irreducible representations of a group), Partially ordered set (same set with opposite order), Thom space (for Atiyah's S-duality theorem), Dual category; Pontryagin duality (also for Tanaka–Krein duality), Topological vector space (for more on duality of locally convex spaces), Symmetric space (a duality between symmetric Riemannian homogeneous spaces of noncompact type and of compact type), Formal group (Cartier duality between formal groups and commutative unipotent algebraic groups), Steenrod duality (cohomological), Convex set (duality of convex bodies), Code with correction of arithmetical errors (for the idea of a dual code), Vector space (dual vector space and dual linear operator), Adjoint module.
Let $\hat{A}$ denote the Picard variety (of linear divisor classes) of an Abelian variety $A$. Then the duality theorem for Abelian varieties states that $\hat{\hat{A}} = A$.
Duality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Duality&oldid=43404