View source for Duality in complex analysis

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202801.png" /> be a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202802.png" /> and denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202803.png" /> the space of all functions holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202804.png" /> with the topology of uniform convergence on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202805.png" /> (the projective limit topology). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202806.png" /> be a compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202807.png" />. Similarly, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202808.png" /> be the space of all functions holomorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202809.png" /> endowed with the following topology: A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028010.png" /> converges to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028012.png" /> if there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028013.png" /> such that all the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028015.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028017.png" /> (the inductive limit topology).

The description of the dual spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028019.png" /> is one of the main problems in the concrete functional analysis of spaces of holomorphic functions.

==Grothendieck–Köthe–Sebastião e Silva duality.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028020.png" /> be a domain in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028021.png" /> and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028022.png" /></td> </tr></table>

Then one has the isomorphism (see [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028023.png" /></td> </tr></table>

defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028025.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028027.png" /> for some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028028.png" />; and the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028029.png" /> separates the singularities of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028031.png" />. The integral in (a1) does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028032.png" />.

==Duality and linear convexity.==
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028033.png" />, the complement of a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028034.png" /> is not useful for function theory. Indeed, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028035.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028036.png" />. However, a generalized notion of "exterior" does exist for linearly convex domains and compacta.

A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028037.png" /> is called linearly convex if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028038.png" /> there exists a complex hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028039.png" /> through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028040.png" /> that does not intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028041.png" />. A compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028042.png" /> is called linearly convex if it can be approximated from the outside by linearly convex domains. Observe that the topological dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028043.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028044.png" />.

Some examples:

1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028045.png" /> be convex; then for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028046.png" /> of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028047.png" /> there exists a hyperplane of support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028048.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028049.png" /> that contains the complex hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028050.png" />.

2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028051.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028053.png" />, are arbitrary plane domains.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028054.png" /> be approximated from within by the sequence of linearly convex domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028055.png" /> with smooth boundaries: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028056.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028058.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028059.png" />. Such an approximation does not always exist, unlike the case of usual convexity. For instance, this approximation is impossible in Example 2) if at least one of the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028060.png" /> is non-convex.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028061.png" />, one has the isomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028062.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028063.png" /> is the adjoint set (the generalized complement) defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028064.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028066.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028067.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028068.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028069.png" /></td> </tr></table>

The index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028070.png" /> depends on the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028071.png" />, which is holomorphic on the larger compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028072.png" />. The integral in (a4) does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028073.png" />.

A similar duality is valid for the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028074.png" /> as well (see [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]]).

A. Martineau has defined a strongly linearly convex domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028075.png" /> to be a linearly convex domain for which the above-mentioned isomorphism holds. It is proved in [[#References|[a7]]] that a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028076.png" /> is strongly linearly convex if and only if the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028077.png" /> with any complex line is connected and simply connected (see also [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]]).

==Duality based on regularized integration over the boundary.==
L. Stout obtained the following result for bounded domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028078.png" /> with the property that, for a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028079.png" />, the Szegö kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028080.png" /> is real-analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028081.png" />. Apparently, this is true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028082.png" /> is a strictly pseudo-convex domain with real-analytic boundary. Then the isomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028083.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028084.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028087.png" /> (see [[#References|[a12]]], [[#References|[a13]]]).

==Nacinovich–Shlapunov–Tarkhanov theorem.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028088.png" /> be a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028089.png" /> with real-analytic boundary and with the property that any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028090.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028091.png" /> contains a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028092.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028093.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028094.png" />. This is always the case if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028095.png" /> is a strictly pseudo-convex domain with real-analytic boundary.

For any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028096.png" /> there exists a unique solution of the [[Dirichlet problem|Dirichlet problem]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028097.png" /></td> </tr></table>

Here, the isomorphism (a3) can be defined by the formula (see [[#References|[a14]]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028098.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

The integral is well-defined for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028099.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280100.png" /> is a sequence of domains with smooth boundaries which approximate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280101.png" /> from within) since the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280102.png" />, which is harmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280103.png" />, can be harmonically continued into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280104.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280105.png" /> because of the real analyticity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280107.png" />. The integral in (a4) does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280108.png" />.

==Duality and cohomology.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280109.png" /> be the Dolbeault cohomology space

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280110.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280111.png" /> is the space of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280112.png" />-closed forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280113.png" /> that are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280114.png" /> on some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280115.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280116.png" /> (the neighbourhood depends on the cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280117.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280118.png" /> is the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280119.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280120.png" />-exact forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280121.png" /> (coboundaries).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280122.png" /> is a bounded pseudo-convex domain, then one has [[#References|[a15]]], [[#References|[a16]]] an isomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280123.png" /></td> </tr></table>

defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280124.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280125.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280126.png" />. Here, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280127.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280128.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280129.png" /> is a sequence of domains with smooth boundaries approximating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280130.png" /> from within. Although <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280131.png" /> depends on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280132.png" />, the integral in (a5) does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280133.png" /> (given (a5), the same formula is valid for larger <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280134.png" /> as well).

A new result [[#References|[a17]]] consists of the following: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280135.png" /> be a bounded pseudo-convex domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280136.png" /> that can be approximated from within by a sequence of strictly pseudo-convex domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280137.png" />; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280138.png" /> be the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280139.png" /> consisting of the differential forms of type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280140.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280141.png" /> is a function that is harmonic in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280142.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280143.png" /> (which depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280144.png" />) such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280145.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280146.png" /> be the space of all forms of type (a6) such that the harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280147.png" /> is representable for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280148.png" /> by the Bochner–Martinelli-type integral (cf. also [[Bochner–Martinelli representation formula|Bochner–Martinelli representation formula]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280149.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280150.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280151.png" />. Then one has an isomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280152.png" /></td> </tr></table>

it is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280153.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280154.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280155.png" />. Note that (a7) gives a more concrete description of the duality than does (a5). The integral in (a7) is also independent of the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280156.png" />.

Other descriptions of the spaces dual to spaces of holomorphic functions for special classes of domains can be found in [[#References|[a18]]], [[#References|[a19]]], [[#References|[a20]]], [[#References|[a21]]], [[#References|[a22]]], [[#References|[a23]]], [[#References|[a24]]], [[#References|[a10]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Grothendieck, "Sur certain espaces de fonctions holomorphes" ''J. Reine Angew. Math.'' , '''192''' (1953) pp. 35–64; 77–95 {{MR|0062335}} {{MR|0058865}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Köthe, "Dualität in der Funktionentheorie" ''J. Reine Angew. Math.'' , '''191''' (1953) pp. 30–39 {{MR|0056824}} {{ZBL|0050.33502}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Sebastião e Silva, "Analytic functions in functional analysis" ''Portug. Math.'' , '''9''' (1950) pp. 1–130</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A. Martineau, "Sur la topologies des espaces de fonctions holomorphes" ''Math. Ann.'' , '''163''' (1966) pp. 62–88 {{MR|190697}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> L. Aizenberg, "The general form of a linear continuous functional in spaces of functions holomorphic in convex domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280157.png" />" ''Soviet Math. Dokl.'' , '''7''' (1966) pp. 198–202</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> L. Aizenberg, "Linear convexity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280158.png" /> and the distributions of the singularities of holomorphic functions" ''Bull. Acad. Polon. Sci. Ser. Math. Astr. Phiz.'' , '''15''' (1967) pp. 487–495 (In Russian) {{MR|0222346}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> S.V. Zhamenskij, "A geometric criterion of strong linear convexity" ''Funct. Anal. Appl.'' , '''13''' (1979) pp. 224–225</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> M. Andersson, "Cauchy–Fantappié–Leray formulas with local sections and the inverse Fantappié transform" ''Bull. Soc. Math. France'' , '''120''' (1992) pp. 113–128 {{MR|}} {{ZBL|0757.32008}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> S.G. Gindikin, G.M. Henkin, "Integral geometry for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280159.png" />-cohomologies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280160.png" />-linearly concave domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280161.png" />" ''Funct. Anal. Appl.'' , '''12''' (1978) pp. 6–23</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> S.V. Znamenskij, "Strong linear convexity. I: Duality of spaces of holomorphic functions" ''Sib. Math. J.'' , '''26''' (1985) pp. 331–341 {{MR|}} {{ZBL|0596.32017}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> S.V. Znamenskij, "Strong linear convexity. II: Existence of holomorphic solutions of linear systems of equations" ''Sib. Math. J.'' , '''29''' (1988) pp. 911–925 {{MR|}} {{ZBL|0689.47004}} {{ZBL|0672.47009}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> L. Aizenberg, S.G. Gindikin, "The general form of a linear continuous functional in spaces of holomorphic functions" ''Moskov. Oblast. Ped. Just. Uchen. Zap.'' , '''87''' (1964) pp. 7–15 (In Russian) {{MR|0180699}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> E.L. Stout, "Harmonic duality, hyperfunctions and removable singularities" ''Izv. Akad. Nauk Ser. Mat.'' , '''59''' (1995) pp. 133–170 {{MR|1481618}} {{ZBL|0876.32003}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> M. Nacinovich, A. Shlapunov, N. Tarkhanov, "Duality in the spaces of solutions of elliptic systems" ''Ann. Scuola Norm. Sup. Pisa'' , '''26''' (1998) pp. 207–232 {{MR|1631573}} {{ZBL|0919.35040}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> J.P. Serre, "Une théorème de dualité" ''Comment. Math. Helvetici'' , '''29''' (1955) pp. 9–26</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> A. Martineau, "Sur les fonctionelles analytiques et la transformation de Fourier–Borel" ''J. Anal. Math.'' , '''9''' (1963) pp. 1–164</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> L. Aizenberg, "Duality in complex analysis" , ''Israel Math. Conf. Proc.'' , '''11''' (1997) pp. 27–35 {{MR|1476701}} {{ZBL|0907.46018}} </TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> H.G. Tillman, "Randverteilungen analytischer funktionen und distributionen" ''Math. Z.'' , '''59''' (1953) pp. 61–83 {{MR|}} {{ZBL|0051.08901}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> S. Rolewicz, "On spaces of holomorphic function" ''Studia Math.'' , '''21''' (1962) pp. 135–160 {{MR|0154146}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> L. Aizenberg, B.S. Mityagin, "The spaces of functions analytic in multi-circular domains" ''Sib. Mat. Zh.'' , '''1''' (1960) pp. 153–170 (In Russian) {{MR|124526}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> L. Aizenberg, "The spaces of functions analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280162.png" />-circular domains" ''Soviet Math. Dokl.'' , '''2''' (1961) pp. 75–82</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> L.J. Ronkin, "On general form of functionals in space of functions, analytic in semicircular domain" ''Soviet Math. Dokl.'' , '''2''' (1961) pp. 673–686 {{MR|131577}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top"> S.G. Gindikin, "Analytic functions in tubular domains" ''Soviet Math. Dokl.'' , '''3''' (1962)</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top"> S.D. Simonzhenkov, "Description of the conjugate space of functions that are holomorphic in the domain of a special type" ''Sib. Math. J.'' , '''22''' (1981) {{MR|610784}} {{ZBL|}} </TD></TR></table>