UDC 513.88+513.83

MATHEMATICS

V. D. GOLOVIN

DUALITY IN SPACES OF HOLOMORPHIC FUNCTIONS WITH SINGULARITIES

(Presented by Academician S. N. Bernstein on 9 VIII 1965)

1. Let \(O\) be a nonempty open subset of the complex plane, and let \(\mathfrak D\) be the set of all closed discrete subsets of \(O\). For each \(D \in \mathfrak D\), let \(H(O,D)\) be the vector space of all holomorphic functions on the set \(O\) with singularities in \(D\), endowed with the topology of uniform convergence on every compact subset in \(O \setminus D\). The set \(\mathfrak D\) is filtered by inclusion \(\subset\), and \(H(O,D) \subset H(O,D')\) if \(D \subset D'\), while the topology in \(H(O,D)\) coincides with that induced from \(H(O,D')\). The vector space

\[ H(O,\mathfrak D)=\bigcup_{D\in\mathfrak D} H(O,D) \]

of all holomorphic functions on \(O\) with isolated singularities is endowed with the strongest of the locally convex topologies for which the canonical mappings
\[ \varphi_D:H(O,D)\to H(O,\mathfrak D)\quad (D\in\mathfrak D) \]
are continuous. The study of duality for the space \(H(O,\mathfrak D)\) is the subject of the present paper.

Let \(M\) be an arbitrary open subset in \(O\), and let \(\mathfrak F\) be the set of all finite subsets of \(M\). The vector space \(H(M,\mathfrak F)\) of all holomorphic functions on \(M\) with a finite number of singularities, which is the union of the subspaces \(H(M,F)\) \((F\in\mathfrak F)\), is endowed with the strongest of the locally convex topologies for which the canonical mappings
\[ \psi_F:H(M,F)\to H(M,\mathfrak F)\quad (F\in\mathfrak F) \]
are continuous. Further, let \(P(U)\), for every nonempty open subset \(U\) of the Riemann sphere \(S_2\), denote the vector space of all holomorphic functions on \(U\) with zero at the point at infinity (if the latter belongs to \(U\)), endowed with the topology of uniform convergence on every compact subset of \(U\) (see \((^1)\)). Finally, let \(P_M(S_2\setminus\{a\})\), for each \(a\in M\), be the subspace of the topological vector space \(H(M,\{a\})\) consisting of the restrictions to \(M\) of all possible functions belonging to \(P(S_2\setminus\{a\})\).

Theorem 1. For every finite subset \(F=\{a_1,\ldots,a_n\}\) in \(M\), the space \(H(M,F)\) is the topological direct sum of the subspaces \(H(M)(=H(M,\varnothing))\) and \(P_M(S_2\setminus\{a_k\})\) \((k=1,\ldots,n)\).

Corollary. The space \(H(M,\mathfrak F)\) is the topological direct sum of its subspaces \(H(M)\) and \(P_M(S_2\setminus\{a\})\) \((a\in M)\).

Let \(M\) be a nonempty relatively compact open set in \(O\). Then, for each \(D\in\mathfrak D\), the intersection \(D\cap M\) is finite; consequently, the restriction mappings
\[ \pi_M:H(O,\mathfrak D)\to H(M,\mathfrak F) \]
and
\[ \rho_M:H(O,D)\to H(M,D\cap M) \]
are defined, and for them the diagram

\[ \begin{array}{ccc} H(O,D) & \xrightarrow{\ \varphi_D\ } & H(O,\mathfrak D)\\ {\scriptstyle \rho_M}\downarrow & & \downarrow{\scriptstyle \pi_M}\\ H(M,D\cap M) & \xrightarrow{\ \psi_{D\cap M}\ } & H(M,\mathfrak F) \end{array} \]

commutes.

obviously, commutative. A direct verification shows that the topology of the space \(H(O,D)\) is the weakest among all topologies for which the maps \(\rho_M\) are continuous. Hence it follows, in particular, that the maps \(\pi_M\) are continuous, since the maps \(\pi_M\circ \varphi_D(=\psi_{D\cap M}\circ \rho_M)\) are continuous ((2), Ch. II, § 2, corollary of proposition 1).

Theorem 2. The topology of the space \(H(O,\mathfrak D)\) is the weakest of all topologies for which the maps \(\pi_M\) are continuous (where \(M\) is an arbitrary relatively compact open set in \(O\)).

Corollary 1. For every \(D\in\mathfrak D\) the topology of the space \(H(O,D)\) coincides with that induced from \(H(O,\mathfrak D)\).

Corollary 2. The space \(H(O,\mathfrak D)\) is separated and complete.

Theorem 3. In order that a subset of the space \(H(O,\mathfrak D)\) be bounded, it is necessary and sufficient that it be contained in one of the spaces \(H(O,D)\) \((D\in\mathfrak D)\) and be bounded in it.

Corollary 1. A sequence of elements of the space \(H(O,\mathfrak D)\) is convergent if and only if it is contained in one of the spaces \(H(O,D)\) \((D\in\mathfrak D)\) and converges in it to the same limit.

Corollary 2. \(H(O,\mathfrak D)\) is a Montel space and, in particular, reflexive.

Remark. One can show that \(H(O,\mathfrak D)\) is not a space of countable type and, consequently, is not metrizable (cf. \((^3)\)).

1. Speaking of neighborhoods of a nonempty set \(A\) on the Riemann sphere, we shall assume that they are open and contain no connected components not meeting \(A\). The spaces \(P(U)\) (where \(U\) is an arbitrary neighborhood of the set \(A\subset S_2\)) and the restriction maps \(P(U')\to P(U)\) \((U\subset U')\) form an inductive system. The inductive limit of this system \(R(A)\) is canonically identified with the vector space of holomorphic functions, each of which is defined in some neighborhood of the set \(A\); two such functions are regarded as equivalent (and represent one and the same element of the space \(R(A)\)) if they coincide in some neighborhood of the set \(A\) (cf. \((^4)\)).

The vector space \(H'(M)\), dual to \(H(M)\) (where \(M\) is an arbitrary open subset of \(O\)), is canonically identified with the space \(R(S_2\setminus M)\) (see \((^1,^5,^6)\)). In this case, for every \(D\in\mathfrak D\), the vector space \(H'(O,D)\) is identified with \(R((S_2\setminus O)\cup D)\), since there is a canonical isomorphism of the space \(H(O,D)\) onto \(H(O\setminus D)\). The vector spaces \(R((S_2\setminus O)\cup D)\) \((D\in\mathfrak D)\) and the maps
\[ R((S_2\setminus O)\cup D')\to R((S_2\setminus O)\cup D)\quad (D\subset D'), \]
dual to the embeddings \(H(O,D)\to H(O,D')\), form a projective system. The projective limit \(R(S_2\setminus O,\mathfrak D)\) of this system is canonically identified with the vector space \(H'(O,\mathfrak D)\), dual to \(H(O,\mathfrak D)\). Every continuous linear form on \(H(O,\mathfrak D)\) can be represented in the form
\[ f\mapsto \int_\Gamma f(\zeta)\,g(\zeta)\,d\zeta, \]
where \(f\in H(O,\mathfrak D)\) (more precisely, \(f\in H(O,D)\) for some \(D\in\mathfrak D\)), \(g\) is a holomorphic function in some neighborhood \(U\) of the set \((S_2\setminus O)\cup D\) with zero at the point at infinity; \(\Gamma\) is the oriented boundary of some compact set in \(O\setminus D\), consisting of paths in \(U\). The map \({}^t\varphi_D\), dual to \(\varphi_D\), is for every \(D\) the canonical projection assigning to an element of the space \(R(S_2\setminus O,\mathfrak D)\) its representative in \(R((S_2\setminus O)\cup D)\).

Let \(M\) be a nonempty open subset of \(O\) and \(F\) a finite subset of \(M\) \((F\in\mathfrak F)\). Then, by Theorem 1, the space
\[ R(S_2\setminus M, F)=R(S_2\setminus M)\times \prod_{a\in F} R(\{a\}) \]

canonically identifiable with the space dual to \(H(M,F)\); similarly, by virtue of the corollary to Theorem 1, the space

\[ R(S_2\setminus M,\mathfrak F)=R(S_2\setminus M)\times \prod_{a\in M} R(\{a\}) \]

is canonically identifiable with the vector space \(H'(M,\mathfrak F)\), dual to \(H(M,\mathfrak F)\). Here the mapping \({}^{t}\psi_F\), dual to \(\psi_F\), for each \(F\in\mathfrak F\) sends a family of functions \(g_0\in R(S_2\setminus M)\), \(g_a\in R(\{a\})\) \((a\in M)\), which is an element of the space \(R(S_2\setminus M,\mathfrak F)\), to the family \(g_0,g_a\) \((a\in F)\) as an element of the space \(R(S_2\setminus M,F)\).

The spaces \(R(S_2\setminus M,\mathfrak F)\) (where \(M\) is an arbitrary relatively compact open set in \(O\)) and the mappings dual to the restriction mappings \(H(M',\mathfrak F)\to H(M,\mathfrak F)\) \((M\subset M')\), form an inductive system whose limit is canonically identifiable with the vector space \(R(S_2\setminus O,\mathfrak D)\). The mapping \({}^{t}\pi_M\), dual to \(\pi_M\), sends each family of functions \(g_0\in R(S_2\setminus M)\), \(g_a\in R(\{a\})\) \((a\in M)\) to a family of functions \(h_D\in R((S_2\setminus O)\cup D)\) \((D\in\mathfrak D)\) in such a way that, for every \(D\in\mathfrak D\), the function \(h_D\) coincides with \(g_0\) in some neighborhood of the set \(S_2\setminus O\), and in some neighborhood of the point \(a\in D\) this same function \(h_D\) coincides with \(g_a\), if \(a\in M\), and with \(g_0\), if \(a\in O\setminus M\).

Thus, every continuous linear form on \(H(O,\mathfrak D)\) is representable in the form

\[ f\mapsto \int_{\Gamma_0} f_0(\xi)g_0(\xi)\,d\xi +\sum_{a\in M}\int_{\Gamma_a} f_a(\xi)g_a(\xi)\,d\xi, \]

where \(M\) is a relatively compact open set in \(O\); \(f_a\) is the principal part of the Laurent expansion of the function \(f\in H(O,\mathfrak D)\) in a neighborhood of the point \(a\in M\), and \(f_0=f-\sum_{a\in M} f_a\); \(g_0\) and \(g_a\) \((a\in M)\) are functions holomorphic in some neighborhoods \(U_0\) and \(U_a\), respectively, of the sets \(S_2\setminus M\) and \(\{a\}\) \((a\in M)\), with the point at infinity being a zero of \(g_0\); finally, \(\Gamma_0\) and \(\Gamma_a\) \((a\in M)\) are oriented boundaries of compact sets containing, respectively, \(M\) and \(S_2\setminus\{a\}\) \((a\in M)\), which consist of paths in \(U_0\) and \(U_a\) \((a\in M)\), respectively.

1. The vector space \(R(A)\), where \(A\) is an arbitrary nonempty subset of the Riemann sphere, shall be endowed with the strongest of the locally convex topologies for which the canonical mappings \(P(U)\to R(A)\) are continuous (where \(U\) is an arbitrary neighborhood of the set \(A\)). Then the topological vector space \(R((S_2\setminus O)\cup D)\) is identifiable with the strong dual of the space \(H(O,D)\) ((1), Proposition 16). The vector space \(R(S_2\setminus M,D\cap M)\), endowed with the product of the topologies of the spaces \(R(S_2\setminus M)\) and \(R(\{a\})\) \((a\in D\cap M)\), is identifiable with the strong dual of the space \(H(M,D\cap M)\). Further, the topology in \(R(S_2\setminus M,\mathfrak F)\), being the weakest of the locally convex topologies for which the mappings \(\psi_{D\cap M}\) \((D\in\mathfrak D)\) are continuous and obviously coinciding with the product of the topologies of the spaces \(R(S_2\setminus M)\) and \(R(\{a\})\) \((a\in M)\), is identifiable with the strong topology in the dual of \(H(M,\mathfrak F)\). Finally, by virtue of Mackey’s theorem:

Theorem 4. The strongest of the locally convex topologies in \(R(S_2\setminus O,\mathfrak D)\), for which the mappings \({}^{t}\pi_M\) are continuous (where \(M\) is an arbitrary relatively compact open set in \(O\)), is canonically identifiable with the strong topology in the space dual to \(H(O,\mathfrak D)\).

Similarly, the topology of the space \(R((S_2\setminus O)\cup D)\) is the strongest of the locally convex topologies for which all mappings \({}^{t}\rho_M\) are continuous. From the commutative diagram

\[ \begin{array}{ccc} R(S_2\setminus M,\mathfrak F) & \xrightarrow{\ t_{\pi_M}\ } & R(S_2\setminus O,\mathfrak D)\\ {\scriptstyle t_{\psi_{D\cap M}}}\downarrow & & \downarrow{\scriptstyle t_{\varphi_D}}\\ R(S_2\setminus M,D\cap M) & \xrightarrow{\ t_{\rho_M}\ } & R((S_2\setminus O)\cup D) \end{array} \]

it follows that the mappings \(t_{\varphi_D}\) \((D\in\mathfrak D)\) are continuous.

Theorem 5. The topology of the space \(R(S_2\setminus O,\mathfrak D)\) is the weakest among all topologies for which the mappings \(t_{\varphi_D}\) \((D\in\mathfrak D)\) are continuous.

Theorem 6. In order that a subset of the space \(R(S_2\setminus O,\mathfrak D)\) be bounded, it is necessary and sufficient that it be the image under the mapping \(t_{\pi_M}\), for some relatively compact open \(M\subset O\), of a bounded set from \(R(S_2\setminus M,\mathfrak F)\).

An analogous proposition also holds for the space \(R((S_2\setminus O)\cup D)\). Moreover, for any \(D\in\mathfrak D\) the mapping \(t_{\varphi_D}\) is an epimorphism, and every bounded set in \(R((S_2\setminus O)\cup D)\) is the image under \(t_{\varphi_D}\) of some bounded set from \(R(S_2\setminus O,\mathfrak D)\).

1. In conclusion we shall explain why the space \(H(O,\mathfrak D)\) cannot be embedded in the space \(H(O,\mathfrak S)\) of holomorphic functions on \(O\) with a finite number of non-isolated singularities.

Let \(\mathfrak S\) be the set of all closed subsets in \(O\), each of which has only a finite number of non-isolated points, and let \(H(O,S)\), for each \(S\in\mathfrak S\), be the vector space of all holomorphic functions on \(O\) with singularities in \(S\), endowed with the topology of uniform convergence on each compact subset of \(O\setminus S\). The set \(\mathfrak S\) is filtered by inclusion, \(H(O,S)\subset H(O,S')\) for \(S\subset S'\), and the topology in \(H(O,S)\) coincides with that induced from \(H(O,S')\). The vector space \(H(O,\mathfrak S)\), which is the union of the subspaces \(H(O,S)\) \((S\in\mathfrak S)\), is naturally endowed with the strongest of the locally convex topologies for which the canonical mappings \(H(O,S)\to H(O,\mathfrak S)\) \((S\in\mathfrak S)\) are continuous. It turns out, however, that the topology in \(H(O,\mathfrak S)\) so defined is trivial: its only neighborhood of zero coincides with the whole space.

Kharkov State University
named after A. M. Gorky

Received
12 VI 1965

CITED LITERATURE

1. G. Köthe, J. reine u. angew. Math., 191, 29 (1953).
2. N. Bourbaki, Topological Vector Spaces, Moscow, 1959.
3. J. Dieudonné, C. R., 238, No. 2, 194 [[unclear: final digit]] (1954).
4. J. Sebastião e Silva, Port. Math., 9, 1 (1950).
5. C. L. da Silva Dias, Bol. Soc. Mat. São Paulo, 5, 1 (1950).
6. A. Grothendieck, J. reine u. angew. Math., 192, 35 (1953).