Duality

Duality in algebraic geometry

$\newcommand{\calO}{\mathcal{O}}$ $\newcommand{\calF}{\mathcal{F}}$ $\newcommand{\tensor}{\otimes}$ $\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\End}{End}$ $\DeclareMathOperator{\Gal}{Gal}$ 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

$$\dim H^i(X, \calO_X(D)) = \dim H^{n-i}(X, \calO_X(K - D)),$$ where $K$ is the canonical divisor on $X$. If $n = 1$, 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 $X$ is a Cohen–Macaulay subvariety (e.g. a locally complete intersection) of codimension $d$ in a non-singular projective variety $Y$, there is duality between the $k$-space $H^i(X, \calF)$ and the space of global Ext's

$$\Ext^{n-i}(X; \calF, \widetilde \omega_X),$$ where $\calF$ is a coherent sheaf on $X$, $\widetilde \omega_X = \Ext_{\calO_Y}^d (\calO_X, \omega_Y)$ (Grothendieck's dualizing sheaf), while $n = \dim X$. Here, the sheaf $\widetilde \omega_X$ is invertible if and only if $X$ is a Gorenstein scheme (cf. Gorenstein ring).

Etale cohomology. Let $X$ be a complete connected non-singular algebraic variety of dimension $d$ over an algebraically closed field $k$; let $n$ be an integer which is relatively prime to the characteristic of the field $k$; let $\calF$ be a locally free (in the étale topology) sheaf of $\ZZ/n\ZZ$-modules on $X$; and let $\mu_n$ be the sheaf of $n$-th power roots of unity. Then there exists a non-degenerate pairing of $\ZZ/n\ZZ$-modules [6]:

$$H^i(X, \calF) \times H^{2d - i}(X, \Hom(\calF, \mu_n^{\tensor d})) \to \ZZ/n\ZZ.$$ A more general duality theorem concerns smooth, but not necessarily complete, varieties [5]. There exists a non-degenerate pairing of $\ZZ/n\ZZ$-modules

$$H_C^i(X, \calF) \times H^{2d-i}(X, \Hom(\calF, \mu_n^{\tensor d})) \to \ZZ/n\ZZ,$$ where cohomology with compact support is found on the left-hand side. If the field $k$ is the algebraic closure of a field $k'$, $X=X' \tensor_{k'} k$ and $\calF = \calF' \tensor_{k'} k$, then the Galois group $\Gal(k/k')$ acts on $H^i(X, \calF)$ and the preceding pairing is a pairing of $\Gal(k/k')$-modules.

Poincaré's duality theorem is an analogue of the first of the theorems given for $l$-adic cohomology: There exists a non-degenerate pairing of $\ZZ_l$-modules

$$H^i(X, \ZZ_l) \times H^{2d - i}(X, \ZZ_l[d]) \to \ZZ_l,$$ where $\ZZ_l[d]$ is the Tate sheaf, which is non-canonically isomorphic to the sheaf $\ZZ_l$ (cf. $l$-adic cohomology). Hence the isomorphism of $\QQ_l$-spaces

$$H^i(X, \QQ_l) \cong \Hom(H^{2d-i}(X, \QQ_l)(d), \ZZ_l),$$ and, in particular, the equality of the Betti numbers

$$b_i(X; l) = b_{2d-i}(X; l).$$ 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 $p > 0$" , Springer (1974) MR0384804 Zbl 0298.14012
[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 $ S $- homotopy and $ S $- cohomotopy groups (see $ S $- duality).

The duality between homology and cohomology consists in the following. Let $ \{ H _ {r} (X,\ A),\ f _ {*} ,\ \partial \} $ 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 $ H _ {r} (X,\ A) $. Then the system $ \{ H ^ {r} (X,\ A),\ f ^ {\ *} ,\ \delta \} $( where $ H ^ {r} (X,\ A) $ is the group of characters of $ H _ {r} (X,\ A) $, and $ f ^ {\ *} $ and $ \delta $ are the homomorphisms, conjugate, respectively, with $ f _ {*} $ and $ \partial $) satisfies the Steenrod–Eilenberg axioms of cohomology theory and represents the cohomology theory over the same category with compact or, respectively, discrete groups $ H ^ {r} (X,\ A) $. 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 $ \{ H _ {r} (X,\ A),\ f _ {*} ,\ \partial \} $, there exists a dual proposition about the system $ \{ H ^ {r} (X,\ A),\ f ^ {*} ,\ \delta \} $, 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 $ K = \{ t ^ {r} \} $ be a (finite) complex. The number

$$ (c _ {r} ,\ c ^ {r} ) \ = \ \sum _ {t ^ {r} \in K} c _ {r} ( t ^ {r} ) c ^ {r} ( t ^ {r} ) \ \mathop{\rm mod} \ 1 $$

is taken to be the product of the $ r $- dimensional chain $ c _ {r} $ of $ K $ over a discrete or compact coefficient group $ X $ and the $ r $- dimensional cochain $ c ^ {r} $ of $ K $ over the coefficient group $ X ^ {*} $ dual with $ X $ in the sense of the theory of characters. This product defines the multiplication of a homology class by a cohomology class, and converts $ r $- 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 $ r $- dimensional projective (spectral) homology group $ H _ {r} (R,\ X) $ of a space $ R $ over a discrete or a compact coefficient group $ X $ in any theory (theories of singular homology and cohomology; Aleksandrov–Čech homology and cohomology; Vietoris homology and cohomology; etc.) and an $ r $- dimensional projective (spectral) cohomology group $ H ^ {r} (R,\ X ^ {*} ) $ in the same theory over the group $ X ^ {*} $ dual to $ X $[1], [3], , [9]:

$$ H _ {r} (R,\ X) \mid H ^ {r} (R,\ X ^ {*} ),\ \ \textrm{ for } \ X \mid X ^ {*} . $$

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 $ n $- dimensional orientable manifold, its $ p $- dimensional and $ ( n - p - 1 ) $- dimensional Betti numbers are equal, as are the $ p $- and $ (n-p) $- 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 $ r $- dimensional chain $ c _ {r} $, given on any triangulation $ K $ of an $ n $- dimensional oriented homology manifold $ M ^ {n} $ and taking values in a discrete or compact coefficient group $ X $, with an $ (n - p ) $- dimensional cochain of the cellular complex $ K ^ {*} $ of barycentric stars of $ K $, which assumes, on any star, the value of $ c _ {p} $ on the simplex corresponding to this star. Since the groups of the complexes $ K $ and $ K ^ {*} $ are identical, this correspondence defines an isomorphism of the homology and cohomology groups of complementary dimensions of $ M ^ {n} $:

$$ H _ {r} (M ^ {n} ,\ X) \ \sim \ H ^ {n-r} (M ^ {n} ,\ X) . $$

Here, $ X $ may also be a module, and if the manifold is not orientable, the theorem is true modulo 2. Replacing the group $ H ^ {n-r} (M ^ {n} ,\ X) $ by its dual group $ H _ {n-r} (M ^ {n} ,\ X ^ {*} ) $ yields the duality [1]:

$$ H _ {r} (M ^ {n} , X) \mid H _ {n-r} (M ^ {n} , X ^ {*} ) ,\ \ \textrm{ for } \ X \mid X ^ {*} , $$

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 $ r $- dimensional Betti number modulo 2 of a polyhedron situated in the $ n $- dimensional sphere is equal to the $ ( n - r - 1 ) $- 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]:

$$ \frac{H _ {r} (A,\ X)}{H ^ {r} (A,\ X ^ {*} )} \ \supset \left | \subset \ \frac{H _ {n-r-1} (B,\ X ^ {*} )}{H ^ {n-r-1} (B,\ X)} \right . , $$

where $ X $ is a discrete or compact group of coefficients, $ X ^ {*} \mid X $, $ A $ and $ B $ are mutually complementary sets of an $ n $- dimensional spherical manifold $ M ^ {n} $, $ H _ {r} (A,\ X) $ and $ H ^ {r} (A,\ X ^ {*} ) $ are the $ r $- dimensional Aleksandrov–Čech homology and cohomology groups (with compact support) of the set $ A $ over $ X $ and, respectively, $ X ^ {*} $, and $ H _ {n-r-1} (B,\ X ^ {*} ) $ and $ H ^ {n-r-1} (B,\ X) $ are the $ ( n - r - 1 ) $- dimensional Aleksandrov–Čech spectral homology and cohomology groups of the set $ B $ over $ X ^ {*} $ and $ X $, 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 $ X ^ {*} $, is defined by continuity of the cycle linkage. In the diagram given above, the groups of the first column may be replaced by the $ ( r + 1 ) $- dimensional Steenrod homology and cohomology groups with compact supports, while the groups in the second column can be replaced by the $ ( n - r - 1 ) $- dimensional projective Aleksandrov–Čech homology and cohomology groups. Then, for a compact $ A $, the isomorphism of the main diagonal yields the Steenrod duality theorem in its original form if the cohomology group of the set $ B $ is replaced, in accordance with Poincaré's theorem, by the $ ( r + 1 ) $- dimensional homology group of infinite cycles. If the group $ X $ is compact, the diagrams are isomorphic; if, in addition, the set $ A $ 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 $ r $- dimensional homology (cohomology) group of a closed set $ A $ of a normal locally compact space $ R $ which is acyclic in dimensions $ r $ and $ r + 1 $, over a compact (discrete) group $ X $, is isomorphic to the $ ( r + 1 ) $- dimensional homology (cohomology) group of the complement:

$$ H _ {r} (A,\ X) \ \sim \ H _ {r+1} (R \setminus A,\ X) $$

and

$$ H ^ {r} (A,\ X) \ \sim \ H ^ {r+1} (R \setminus A,\ X) . $$

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 $ H _ \Phi ^ {p} (X,\ {\mathcal F} ) $ of a complex space $ X $ with values in a coherent analytic sheaf $ {\mathcal F} $ and supports in the family $ \Phi $ 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 $ X $ be a complex manifold of dimension $ n $ with a countable base, let $ \Omega $ be the sheaf of holomorphic differential forms of degree $ n $ and let $ {\mathcal F} $ be a locally free analytic sheaf on $ X $. For each integer $ p $, $ 0\leq p\leq n $, one defines the bilinear mapping

$$ \tag{*} H ^ {p} (X,\ {\mathcal F} ) \times H _ {c} ^ {n-p} (X,\ \mathop{\rm Hom} ( {\mathcal F},\ \Omega )) \ \rightarrow \ \mathbf C , $$

which may be written as the composition of a $ \cup $- multiplication

$$ H ^ {p} (X,\ {\mathcal F} ) \times H _ {c} ^ {n-p} (X,\ \mathop{\rm Hom} ( {\mathcal F},\ \Omega )) \ \rightarrow \ H _ {c} ^ {n} (X,\ \Omega ) $$

( $ c $ denotes the family of compact supports) and linear forms $ s $ on $ H _ {c} ^ {n} (X ,\ \Omega ) $, known as traces, of the form

$$ s ( \widehat \omega ) \ = \ (-1) ^ {n} \int\limits _ { X } \omega , $$

where $ \omega $ is the form of type $ (n,\ n) $ with compact support which corresponds to the class $ \widehat \omega $ 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 $ H ^ {p+1} (X,\ {\mathcal F} ) $ is separable, it defines an isomorphism of vector spaces

$$ ( H ^ {p} (X,\ {\mathcal F} )) ^ \prime \ \cong \ H _ {c} ^ {n-p} (X,\ \mathop{\rm Hom} ( {\mathcal F} , \Omega )) . $$

The roles of the sheaves $ {\mathcal F} $ and $ \mathop{\rm Hom} ( {\mathcal F} ,\ \Omega ) $ may be interchanged, since the operation $ \mathop{\rm Hom} ( \cdot ,\ \Omega ) $ on locally free sheaves is involutory.

In particular, if the manifold $ X $ is compact, $ K $ is the canonical and $ D $ is an arbitrary divisor on $ X $, Serre's theorem implies the equality of the dimensions of the spaces $ H ^ {p} (X,\ {\mathcal O} _ {X} (D)) $ and $ H ^ {n-p} ( X ,\ {\mathcal O} _ {X} ( K-D )) $, 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 $ {\mathcal F} $ is an arbitrary coherent analytic sheaf on the manifold $ X $, there exists a real topological duality between the individual spaces associated with the topological vector spaces $ H _ \Phi ^ {p} (X,\ {\mathcal F} ) $ and $ \mathop{\rm Ext} _ \Psi ^ {n-p} (X; \ {\mathcal F},\ \Omega ) $, where $ \Phi $ is the family of closed supports, $ \Psi $ is the family of compact supports, or vice versa, while the $ \mathop{\rm Ext} _ \Psi ^ {n-p} (X; \ {\mathcal F} ) $ denote the derived functor (cf. Derived functor) of the functor $ \mathop{\rm Hom} _ \Psi (X; \ {\mathcal F} ,\ \cdot ) $. The space $ H _ \Phi ^ {p} (X,\ {\mathcal F} ) $ is separable if $ \mathop{\rm Ext} _ \Psi ^ {n-p+1} (X; \ {\mathcal F} ,\ \Omega ) $ is separable, and vice versa [2], [3]. This implies, for a compact manifold $ X $, an isomorphism of finite-dimensional spaces

$$ ( H ^ {p} (X,\ {\mathcal F} )) ^ \prime \ \cong \ \mathop{\rm Ext} ^ {n-p} (X; \ {\mathcal F} ,\ \Omega ). $$

If $ X $ is a Stein manifold, one obtains topological duality between $ H ^ {0} (X,\ {\mathcal F} ) $ and $ \mathop{\rm Ext} _ {c} ^ {n} (X; \ {\mathcal F} ,\ \Omega ) $, and also between $ H _ {c} ^ {p} (X,\ {\mathcal F} ) $ and $ \mathop{\rm Ext} ^ {n-p} (X; \ {\mathcal F} ,\ \Omega ) $.

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 $ X $ be a complex manifold of dimension $ n $ with a countable base; let $ K $ be a Stein compactum in $ X $. For any coherent analytic sheaf $ {\mathcal F} $ on $ X $ and any integer $ p\geq 0 $ the space $ \mathop{\rm Ext} ^ {n-p} (K; \ {\mathcal F} ,\ \Omega ) $ has a topology of type DFS (is strongly dual to a Fréchet–Schwartz space), and its dual space is algebraically isomorphic to $ H _ {K} ^ {p} (X,\ {\mathcal F} ) $. According to another theorem of this type [6], under the same assumptions, if $ Y \subset X $ is open, the space $ H _ {Y} ^ {p} (X,\ {\mathcal F} ) $ has a topology of type QFS (is a Fréchet–Schwartz quotient space), $ \mathop{\rm Ext} _ {c} ^ {n-p} (Y; \ {\mathcal F} ,\ \Omega ) $ has a topology of type QDFS (is a quotient space of type DFS), while the associated separable spaces are in topological duality. The space $ H _ {Y} ^ {p} (X,\ {\mathcal F} ) $ is separable if and only if $ \mathop{\rm Ext} _ {c} ^ {n-p+1} (Y; \ {\mathcal F} ,\ \Omega ) $ is.

The third type of duality theorem is represented by the following duality theorem [8]: For any open subset $ Y \subset X = \mathbf C P ^ {1} $, the strong dual to the space $ \Gamma (Y,\ {\mathcal O} _ {X} / \Gamma (X,\ {\mathcal O} _ {X} )) $ is isomorphic to $ \Gamma (X \setminus Y,\ {\mathcal O} _ {X} ) $. This theorem may be generalized as follows [7]: Let $ X $ be an $ n $- dimensional complex manifold, countable at infinity, let $ Y\subset X $ be open, let $ {\mathcal F} $ be a coherent analytic sheaf on $ X $, and let $ p \geq 0 $ be an integer. Consider canonical mappings of topological vector spaces

$$ \alpha : \ H ^ {p} (X,\ {\mathcal F} ) \ \rightarrow \ H ^ {p} (Y,\ {\mathcal F} ) , $$

$$ \gamma : \ H _ {X\setminus Y} ^ {p+1} (X,\ {\mathcal F} ) \ \rightarrow \ H ^ {p+1} (X,\ {\mathcal F} ) , $$

$$ \beta : \ \mathop{\rm Ext} _ {c} ^ {n-p-1} (X; \ {\mathcal F} ,\ \Omega ) \ \rightarrow \ \mathop{\rm Ext} _ {c} ^ {n-p-1} (X\setminus Y; \ {\mathcal F} ,\ \Omega ) . $$

For the separable space associated with $ \mathop{\rm Coker} \ \beta $ to be isomorphic to the strong dual of $ \mathop{\rm Coker} \ \alpha $, it is necessary and sufficient for $ \mathop{\rm Ker} \ \gamma $ to be closed. (An example of a non-closed $ \mathop{\rm Ker} \ \gamma $ is known.) In particular, if the sheaf $ {\mathcal F} $ is locally free and if

$$ H ^ {p} (X,\ {\mathcal F} ) \ = \ H ^ {p+1} (X,\ {\mathcal F} ) \ = \ 0 , $$

then the separable spaces associated with $ H ^ {p} (Y,\ {\mathcal F} ) $ and $ H _ {c} ^ {n-p-1} (X \setminus Y ,\ \mathop{\rm Hom} ( {\mathcal F} ,\ \Omega )) $ 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

$$ a(z) \ = \ \sum _ { n=0 } ^ \infty \frac{a _ n}{z ^ {n+1}} $$

into the series

$$ A(z) \ = \ \sum _ { n=0 } ^ \infty \frac{a _ n}{n!} z ^ {n} , $$

and conversely, under the condition that

$$ \overline{\lim\limits}\; _ {n \rightarrow \infty} \ | a _ {n} | ^ {1 / n} \ = \ \sigma \ < \ + \infty . $$

This is a duality relation between functions which are analytic in a neighbourhood of infinity $ | z | > \sigma $ and entire functions of exponential type $ \sigma $. For instance, Pólya's theorem is obtained in this manner: Let $ k( \phi ) $ be the supporting function of the convex envelope of the set of singularities of a function $ a(z) $ under analytic continuation to a half-plane of the form $ \mathop{\rm Re} (ze ^ {-i \phi} ) > c $, and let

$$ h( \phi ) \ = \ \overline{\lim\limits}\; _ {r \rightarrow \infty} \ \frac{ \mathop{\rm ln} \ | A (re ^ { i \phi} \ ) |}{r} $$

be the growth indicator of the entire function $ A(z) $; then

$$ h( \phi ) \ = \ k(- \phi ),\ \ \sigma \leq \phi \leq 2 \pi . $$

By virtue of this duality relation the problem of analytic continuation of the function $ a(z) $ to the disc $ | z | < \sigma $ is equivalent to the study of the growth of the corresponding entire function $ A(z) $ in different directions.

Duality in spaces of analytic functions

Let $ G $ be an open set in the extended complex plane $ \overline{\mathbf C}\; $ and let $ A(G) $ be the space of analytic functions in $ G $ with topology defined by the system of norms

$$ p _ {n} (f \ ) \ = \ \max _ {z \in K _ n} \ | f(z) | ,\ \ f \in A (G) , $$

where $ \{ K _ {n} \} $ is an increasing system of compact sets contained in $ G $ and exhausting $ G $; thus, the convergence $ f _ {n} \rightarrow f $ in $ A(G) $ means uniform convergence $ f _ {n} (z) \rightarrow f(z) $ on all compact subsets of $ G $. Let $ \infty \in G $, let $ A _ {0} (G) $ be the subspace of $ A(G) $ of functions for which $ f( \infty ) = 0 $ and let $ F $ be a compact subset of $ \overline{\mathbf C}\; $. Consider the system $ {\mathcal O} ^ {F} $ of all open sets $ G \supset F $ and the set of functions $ \cup _ {G \in {\mathcal O} ^ {F} } A (G) $. Two functions $ f _ {1} (z) $ and $ f _ {2} (z) $ in this set are considered to be equivalent if their restrictions to some set $ G \in {\mathcal O} ^ {F} $ coincide. The equivalence relation introduced subdivides the entire set under consideration into classes $ \overline{f}\; $. Each class is said to be a local analytic function on $ F $, and the set of such functions is denoted by $ A (F \ ) $. The class $ A(F \ ) $ is naturally converted into a linear space, with the topology of the inductive limit of sequences of normed spaces $ B _ {n} $ introduced on it. This space is constructed as follows. Let $ \{ G _ {n} \} $ be a decreasing sequence in $ {\mathcal O} ^ {F} $ such that $ G _ {n+1} \subset G _ {n} $ and $ \forall G \in {\mathcal O} ^ {F} $,

$$ \exists n _ {0} : \ n \geq n _ {0} \ \Rightarrow \ G _ {n} \subset G . $$

Now $ B _ {n} $ is the space of bounded analytic functions in $ G _ {n} $ with norm

$$ \| f \| \ = \ \max _ {z \in G _ n} \ | f(z) | . $$

The simplest fact about the duality of spaces of analytic functions is the following. Let $ G $ be an open set, let (for the sake of being specific) $ \infty \in G $, and $ F= \overline{\mathbf C}\; \setminus G $. The space $ A(F \ ) $ is dual (conjugate) to the space $ A _ {0} (G) $ in the sense of the theory of linear topological spaces. This duality is established as follows: If $ \Lambda (f \ ) $ is a continuous linear functional on $ A _ {0} (G) $, then there exists a unique element $ \overline{g}\; \in A(F \ ) $ such that

$$ A(f \ ) \ = \ \int\limits _ \gamma f (z) g (z) \ dz , $$

where $ \gamma $ is some (composite) contour lying in $ G $ and including $ F $, while $ g \in \overline{g}\; $, and $ \Lambda (f \ ) $ does not depend on $ g \in \overline{g}\; $. The spaces $ A(E) $ may be defined for arbitrary sets $ E \subseteq \mathbf C $, and not only for the cases considered here, when $ E=G $ is an open set and $ E=F $ 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 $ \{ f _ {n} \} $ of a locally convex space $ X $ is complete if and only if, for an arbitrary linear functional $ \Lambda $ continuous on $ X $, it follows from $ \Lambda (f _ {n} )= 0 $, $ n=1,\ 2 \dots $ that $ \Lambda \equiv 0 $. This fact forms the connection between completeness problems in spaces of analytic functions and various uniqueness theorems for analytic functions. The functional $ \Lambda $ is connected (cf. b) above) with some analytic function $ F(z) $. The condition $ \Lambda (f _ {n} ) =0 $, $ n=1,\ 2 \dots $ renders $ F(z) $ equal to zero at certain points or else renders the coefficients of $ F(z) $ equal to zero. The uniqueness theorems lead to the conclusion that $ F(z) \equiv 0 $, so that $ \Lambda \equiv 0 $ as well. The following duality principle of uniqueness and completeness problems has been formulated for spaces of analytic functions in a disc. Let $ A _ {R} $ and $ A _ {P} $ be, respectively, the spaces of functions which are analytic in the discs $ | z | < R $ and $ | \zeta | < P $, where $ 0<R,\ P \leq \infty $, and let $ F (z,\ \zeta ) $ be a function which is analytic in the bicylinder $ | z | < R $, $ | \zeta | < P $. Let $ L _ {z} $ and $ \Lambda _ \zeta $ be linear functionals defined on $ A _ {R} $ and $ A _ {P} $, and let $ {\mathcal O} \subset A _ {R} $ and $ \Omega \subset A _ {P} $ be subsets of functions which can be represented, respectively, as $ \Lambda _ \zeta F(z,\ \zeta ) $ and $ L _ {z} F(z,\ \zeta ) $. A sequence of functions $ \Lambda _ {\zeta , n} F(z,\ \zeta ) $ is complete in $ {\mathcal O} $ if and only if for each $ \phi \in \Omega $ it follows from $ \Lambda _ {\zeta , n} \phi =0 $, $ n = 1,\ 2 \dots $ that $ \phi ( \zeta ) \equiv 0 $. In particular, if $ R=P= \infty $ and if $ F(z,\ \zeta ) = e ^ {z \zeta} $, both sets $ {\mathcal O} $ and $ \Omega $ 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 $ E $ is a subspace in a normed space $ X $ and $ \omega $ is an arbitrary element of $ X $, one has

$$ \tag{1} \sup _ {\begin{array}{c} l \in E ^ \perp \\ \| l \| \leq 1 \end{array} } \ | l ( \omega ) | \ = \ \inf _ {x \in E} \ \| \omega - x \| , $$

where $ E ^ \perp $ is the annihilator of $ E $, i.e. the totality of linear functionals $ l $ which vanish on the elements of $ E $. 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 $ G $ be an $ n $- connected domain whose boundary $ \partial G $ consists of rectifiable contours, let $ B ^ {1} $ be the class of analytic functions $ f(z) $ in $ G $, $ | f(z) | \leq 1 $, let $ E ^ {1} $ be the class of analytic functions in $ G $ which may be represented by the Cauchy integral over their boundary values, and let $ \omega ( \zeta ) $ be some integrable function on $ \partial G $. Then:

$$ \tag{2} \sup _ {f \in B ^ {1}} \ \left | \int\limits _ {\partial G} f( \zeta ) \omega ( \zeta ) \ d \zeta \right | \ = \ \inf _ {\phi \in E ^ {1}} \ \int\limits _ {\partial G} | \omega ( \zeta ) - \phi ( \zeta ) | \ | d \zeta | . $$

The left-hand side of this equation is a linear extremal problem for bounded functions (e.g. if $ \omega (z) = 1 / \{ 2 \pi i ( \zeta - z _ {0} ) ^ {2} \} $, the resulting problem is $ \sup _ {f \in B ^ {1} } \ | f ^ {\ \prime} (z _ {0} ) | $, 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 $ \omega ( \zeta ) $ on $ \partial G $ 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 $ f ^ {*} (z) \in B ^ {1} $ and $ \phi ^ {*} (z) \in E ^ {1} $, the problem of their uniqueness, etc. The function $ F ^ {*} (z) $ proves to have important geometric properties: In the problem of the Schwarz lemma it maps $ g $ onto an $ n $- sheeted disc; in other problems with an $ \omega ( \zeta ) $ which is analytic on $ \partial G $, the function $ f ^ {*} (z) $ maps $ G $ into an $ m\geq n $- 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 $ \{ F,\ G,\ f \} $ in which $ F,\ G $ are vector spaces over a field $ k $, and $ f $ is a bilinear functional (form) on $ F \times G $ which has the property of being non-degenerate (or separating): If $ f(x,\ y) = 0 $ for each $ y $, $ y \in G $, then $ x= {\mathcal O} $; if $ f(x,\ y)=0 $ for each $ x $, $ x \in F $, then $ y= {\mathcal O} $. One also says that $ f $ realizes the duality and that $ F ,\ G $ form a dual pair; if $ f $ is fixed, one writes $ f (x,\ y) = (x ,\ y) $. The most important example is the natural duality: $ F= (F,\ \tau ) $ is a locally convex topological vector space with the topology $ \tau $, $ G= ( (F,\ \tau ) ) ^ \prime $ is the dual space (cf. Adjoint space) of all linear $ \tau $- continuous functionals on $ F $ and $ (x,\ x ^ \prime ) = x ^ \prime (x) $ if $ x \in F $, $ x ^ \prime \in G $; the fact that this form is non-degenerate is a consequence of, for example, the local convexity of the topology $ \tau $( a corollary of the Hahn–Banach theorem). The main subject of duality theory are methods for constructing objects in $ F $ or $ G $ which are dual to given ones with respect to the form $ ( \cdot ,\ \cdot ) $; 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 $ k = \mathbf R $ or $ \mathbf C $, the polar of a set $ A $, $ A \subset F $, is the set

$$ A ^ \circ \ = \ \{ {y \in G} : { \mathop{\rm Re} (x,\ y ) \leq 1,\ \forall x \in A} \} . $$

The duality generates various locally convex topologies on $ F $( and also on $ G $); for instance, the weak topology $ \sigma (F,\ G ) $( generated by a given duality), specified by the family of semi-norms $ | ( \cdot ,\ y ) | $, $ y \in G $, is the weakest topology for which all the mappings $ ( \cdot ,\ y ) $ are continuous; the Mackey topology $ \mu (F,\ G) $, with a neighbourhood base of zero formed by the polars $ A ^ \circ $ of the absolutely convex $ \sigma (G,\ F \ ) $- compact subsets $ A $ in $ G $; and the strong topology $ \beta (F,\ G) $, a base of which is formed by the polars of the bounded subsets $ A $ in $ (G,\ \sigma (G,\ F \ )) $. For any $ A $, $ A \subset F $, the set $ A ^ {\circ \circ} $ is the convex $ \sigma (F,\ G) $- closed hull of the set $ A \cup \{ {\mathcal O} \} $( the bipolar theorem). The space $ G $ is identical with $ (F,\ \sigma (F,\ G) ) ^ \prime $( the basic theorem of duality theory which proves that any duality may be interpreted as natural). The space $ (F ^ {\ \prime} ,\ \sigma (F ^ {\ \prime} ,\ F \ )) $ is said to be the weak conjugate (or dual) of $ F $.

Let $ F $ be a locally convex space over $ \mathbf R $ or $ \mathbf C $. Each one of the following conditions is necessary and sufficient for a set $ A $, $ A \subset F $, to be bounded: a) $ A $ is bounded in the weak topology; and b) $ A ^ \circ $ is an absorbing set. If $ A $ is a neighbourhood of zero, $ A ^ \circ $ is $ \sigma (F ^ {\ \prime} ,\ F \ ) $- compact. A metric space $ F $ is complete if and only if a set $ A $, $ A \subset F ^ {\ \prime} $, is closed in the topology $ \sigma (F ^ {\ \prime} ,\ F \ ) $ whenever all intersections $ A \cap U ^ \circ $ are closed in the same topology, where $ U $ runs through the set of neighbourhoods of zero in $ F $( Krein–Shmul'yan theorem). If $ F $ is a complete separable space and $ f $ is a linear functional on $ F ^ {\ \prime} $, then $ f \in {( F ^ {\ \prime} ,\ \sigma ( F ^ {\ \prime} ,\ \widetilde{F} ))} {} ^ \prime $ if and only if $ \lim\limits _ {n} \ x _ {n} = {\mathcal O} $ in the topology $ \sigma ( F ^ {\ \prime} ,\ F \ ) $ implies that $ \lim\limits _ {n} \ f(x _ {n} ) = 0 $( Grothendieck's theorem). A subset $ A $ of a complete space $ F $ is relatively $ \sigma (F,\ F ^ {\ \prime} ) $- compact if it is relatively $ \sigma (F,\ F ^ {\ \prime} ) $- sequentially compact (Eberlein's theorem). A convex subset $ A $ of a Fréchet space over $ \mathbf R $ is $ \sigma (F,\ F ^ {\ \prime} ) $- compact if and only if for any $ f \in F ^ {\ \prime} $ there exists an $ a \in A $ such that $ \sup _ {A} \ f = f (a) $( James' theorem). $ \mu ( F ,\ G ) $ is the finest and $ \sigma ( F ,\ G ) $ is the coarsest among the topologies $ \tau $ for which $ ( F ,\ \tau ) ^ \prime = G $( 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 $ (F,\ \tau ) $ suffices for $ \tau $ to coincide with the Mackey topology: a) $ F $ is a barrelled space; and b) $ F $ is a bornological space (in particular, a metric space). The strong topology $ \beta (F,\ G) $, generally speaking, does not preserve the duality; if $ X = G $ is locally convex and $ X ^ \prime = F $, the space $ X ^ {*} = (X ^ \prime ,\ \beta (X ^ \prime ,\ X )) $ is said to be the strong dual of $ X $, and if, in addition, $ \beta (X ^ \prime ,\ X) $ preserves the duality (i.e. if $ {X ^ {*}} ^ \prime = X $), the space $ X $ is said to be semi-reflexive ( $ X $ is a reflexive space if $ X ^ {**} = X $).

Let $ H $ be a subspace of $ F $; $ \{ H,\ G / H ^ {0} \} $ and $ \{ F / H,\ H ^ {0} \} $ will then be dual pairs with respect to natural factorizations of the form $ ( \cdot ,\ \cdot ) $. If a family of dualities $ \{ F _ \alpha ,\ G _ \alpha ,\ ( \cdot ,\ \cdot ) _ \alpha \} $ is given, the duality of the product space $ F = \prod _ \alpha F _ \alpha $ and the subspace $ \{ \alpha : {g _ \alpha \neq 0 \ \textrm{ is finite }} \} $ is realized by the form

$$ (f,\ g) \ = \ \sum _ \alpha (f _ \alpha ,\ g _ \alpha ) _ \alpha , $$

where

$$ f \ = \ \{ f _ \alpha \} \ \in \ \prod _ \alpha F _ \alpha ,\ \ g \ = \ \{ g _ \alpha \} \ \in \oplus _ \alpha G _ \alpha . $$

The dualities of the inductive and the projective limits $ \lim\limits _ \alpha \ \mathop{\rm ind} \ F _ \alpha $, $ \lim\limits _ \alpha \ \mathop{\rm pr} \ F _ \alpha $ are described in a similar manner. The presence of duality-preserving topologies in the spaces $ F $, $ F _ \alpha $ makes it possible to interpret these statements as the description of natural dualities for $ \prod _ \alpha F _ \alpha $( the Tikhonov topology), $ F/H $( the quotient topology), $ H $( the induced topology), $ \lim\limits _ \alpha \ \mathop{\rm ind} \ F _ \alpha $ and $ \lim\limits _ \alpha \ \mathop{\rm pr} \ F _ \alpha $, respectively. In the case of a normed space $ F $ the natural isomorphism of $ H ^ {*} $ and $ F ^ {*} / H ^ {0} $ is an isometry

$$ \| f \mid H \| \ = \ \mathop{\rm dist} (f,\ H ^ {0} ),\ \ f \in F ^ {*} . $$

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 $ E ^ {*} $ of a topological vector space $ E $: the weak- $ * $- topology. It is the weakest topology on $ E ^ {*} $ in which all mappings $ x : \ f \rightarrow f (x) $, $ x \in E $, 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 $ A $ in a vector space $ X $ to be connected with a dual object in the conjugate space — its polar $ A ^ \circ = \{ {x ^ {*} \in X ^ {*}} : {\langle x ^ {*} ,\ x \rangle \leq 1,\ x \in A} \} $. 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 $ f: \ X \rightarrow \overline{\mathbf R}\; $ and the dual object — the conjugate function given on the conjugate space $ X ^ {*} $ and defined by the formula

$$ f ^ {*} ( x ^ {*} ) \ = \ \sup _ {x \in X} ( \langle x ^ {*} ,\ x \rangle - f (x) ) . $$

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 $ A ^ {\circ \circ} = A $ and the conjugation operator $ f ^ {**} =f $, which exists for convex closed sets containing zero and convex closed functions which are everywhere larger than $ - \infty $. 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:

$$ \tag{1} \begin{array}{r} I. \ (c,\ x) \ \rightarrow \ \inf \\ Ax \geq b,\ x \geq 0 \end{array} \ \ \begin{array}{r} II. \ (b,\ y) \ \rightarrow \ \sup \\ A ^ {*} y \leq c,\ y \geq 0 . \end{array} $$

Here

$$ x,\ c \ \in \ \mathbf R ^ {n} ,\ \ (c,\ x) \ = \ \sum _ {i= 1} ^ { n } c _ {i} x _ {i} ,\ \ y,\ b \ \in \ \mathbf R ^ {m} , $$

$$ (b,\ y) \ = \ \sum _ {i = 1} ^ { m } b _ {i} y _ {i} ,\ \ A: \ \mathbf R ^ {n} \rightarrow \mathbf R ^ {m} . $$

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:

$$ \tag{2} f (x) \ \rightarrow \ \inf ,\ \ x \in X , $$

where $ X $ is a linear space, $ f: \ X \rightarrow \overline{\mathbf R}\; $, is included in a class of similar problems which depend on a parameter:

$$ F (x,\ y) \ \rightarrow \ \inf ,\ \ x \in X , $$

where $ Y $ is some other linear space, $ F: \ X \times U \rightarrow \overline{\mathbf R}\; $, $ F (x,\ 0) = f(x) $( the function $ F $ is known as a perturbation of $ f \ $). As a rule, $ F $ is assumed to be convex. The problem which is dual to the problem in relation to a given perturbation is the problem

$$ \tag{2*} -F ^ {*} (0,\ y ^ {*} ) \ \rightarrow \ \sup ,\ \ y ^ {*} \in Y ^ {*} , $$

where $ F ^ {*} $ is the function dual to (conjugate with) $ F $ in the sense of Legendre–Young–Fenchel (cf. Dual functions). For the simplest problems in convex programming, of the type

$$ \tag{3} f _ {0} (x) \ \rightarrow \ \inf ,\ \ f _ {i} (x) \ \leq \ 0,\ \ i= 1 \dots m ,\ \ x \in B , $$

where $ X $ is a linear space, $ f _ {i} : \ X \rightarrow \overline{\mathbf R}\; $ are convex functions on $ X $ and $ B $ is a convex set in $ X $( linear programming problems are special cases of (3)), the following standard perturbations, which depend on the parameters $ y \in \overline{\mathbf R}\; {} ^ {m} $, $ y= (y _ {1} \dots y _ {m} ) $, $ f _ {0} (x) \rightarrow \inf $, $ f _ {i} (x) \leq y _ {i} $, $ i= 1 \dots m $, $ x \in B $, are usually employed. The duality theorems for the general classes of linear problems state that, if certain assumptions regarding the perturbation $ F $ 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 $\hat{\hat{A}} = A$.