UDC 513.88:513.83

MATHEMATICS

V. D. GOLOVIN

CONTINUOUS LINEAR FORMS

ON INDUCTIVE LIMITS OF DERIVED STRAIGHTS

(Presented by Academician S. N. Bernstein on May 6, 1966)

Let \((A_\iota,\varphi_{\iota\chi})\) be a projective system of sets with respect to a partially ordered set of indices \(I\), filtered to the right. For each \(\iota\in I\) let \(\mathbf{R}^{A_\iota}\) be the vector space of all real-valued functions on the set \(A_\iota\), endowed with the topology of pointwise convergence; for each pair of indices \(\iota\leq\chi\) let \(f_{\chi\iota}:\mathbf{R}^{A_\iota}\to\mathbf{R}^{A_\chi}\) be the mapping assigning to each function \(x_\iota\) on \(A_\iota\) the function \(x_\chi=x_\iota\circ\varphi_{\iota\chi}\) on \(A_\chi\). Then the topological vector spaces \(\mathbf{R}^{A_\iota}\) \((\iota\in I)\) and the continuous linear mappings \(f_{\chi\iota}\) \((\iota\leq\chi)\) form an inductive system; the purpose of the present work is the study of continuous linear forms on the topological vector space
\[ F=\lim_{\longrightarrow}\mathbf{R}^{A_\iota}, \]
which is the (locally convex) inductive limit of the system \((\mathbf{R}^{A_\iota},f_{\chi\iota})\).

Let \(\mathfrak{A}_\iota=\mathfrak{P}(A_\iota)\) for each \(\iota\in I\) be the set of all parts of the set \(A_\iota\), and let \(\psi_{\chi\iota}\), for each pair of indices \(\iota\leq\chi\), be the mapping \(\mathfrak{A}_\iota\to\mathfrak{A}_\chi\) assigning to each part of the set \(A_\iota\) its inverse image under the mapping \(\varphi_{\iota\chi}\); the sets \(\mathfrak{A}_\iota\) \((\iota\in I)\) and the mappings \(\psi_{\chi\iota}\) \((\iota\leq\chi)\) form an inductive system. A subset \(\mathfrak{M}\) of the set
\[ \mathfrak{A}=\lim_{\longrightarrow}\mathfrak{A}_\iota, \]
which is the inductive limit of the system \((\mathfrak{A}_\iota,\psi_{\chi\iota})\), will be called final if, for each \(\iota\in I\), first, there exists only a finite number of elements in \(\mathfrak{M}\) having representatives in \(\mathfrak{A}_\iota\), and, second, the representatives of these elements can be chosen pairwise disjoint in \(A_\iota\).

Let \(m\) be an arbitrary element of the set \(\mathfrak{A}\) having a representative \(M_\iota\subset A_\iota\) in \(\mathfrak{A}_\iota\) for each \(\iota\geq\iota_0=\iota_0(m)\). Then the topological vector spaces \(\mathbf{R}^{M_\iota}\) \((\iota\geq\iota_0)\) and the continuous linear mappings \(\mathbf{R}^{M_\iota}\to\mathbf{R}^{M_\chi}\) (each of which is defined for \(\iota_0\leq\iota\leq\chi\) and carries a function \(x_\iota\) on \(M_\iota\) to the function \(x_\chi=x_\iota\circ\varphi_{\iota\chi}\) on \(M_\chi\)) form an inductive system. For each \(\iota\geq\iota_0\) define a continuous linear mapping
\[ i_{m\iota}:\mathbf{R}^{M_\iota}\to\mathbf{R}^{A_\iota}, \]
which assigns to each function \(x_\iota\) on \(M_\iota\) the function on \(A_\iota\) that coincides with \(x_\iota\) on \(M_\iota\) and is identically equal to zero outside \(M_\iota\). Let
\[ F_m=\lim_{\longrightarrow}\mathbf{R}^{M_\iota} \]
be the inductive limit of the system of spaces \(\mathbf{R}^{M_\iota}\) \((\iota\geq\iota_0)\) and mappings \(\mathbf{R}^{M_\iota}\to\mathbf{R}^{M_\chi}\) \((\iota\leq\chi)\); since, for \(\iota\leq\chi\), the composition \(f_{\chi\iota}\circ i_{m\iota}\) coincides with the composition of the mappings \(\mathbf{R}^{M_\iota}\to\mathbf{R}^{M_\chi}\) and \(i_{m\chi}\), there is defined an (injective) continuous linear mapping
\[ i_m:F_m\to F, \]
which assigns to each element \(x\in F_m\), having representative \(x_\iota\) in \(\mathbf{R}^{M_\iota}\), the element of \(F\) that is the canonical image of the element \(i_{m\iota}(x_\iota)\in\mathbf{R}^{A_\iota}\).

Let
\[ A=\lim_{\longleftarrow}A_\iota \]
be the projective limit of the system \((A_\iota,\varphi_{\iota\chi})\), and let \(\varphi_\iota:A\to A_\iota\) \((\iota\in I)\) be the canonical mappings. For each \(\iota\in I\) define a continuous linear mapping \(h_\iota:\mathbf{R}^{A_\iota}\to\mathbf{R}^{A}\), assigning to each function \(x_\iota\) on \(A_\iota\) the function \(x=x_\iota\circ\varphi_\iota\) on \(A\). Since \(h_\iota(x_\iota)=h_\chi(x_\chi)\), if the canonical images of the elements \(x_\iota\in\mathbf{R}^{A_\iota}\) and \(x_\chi\in\mathbf{R}^{A_\chi}\)

coincide in \(F\), and hence a continuous linear mapping has been defined,

\[ h:F\to \mathbf{R}^{A}, \]

which assigns to each element \(x\in F\) the function \(x_{\iota}\circ\varphi_{\iota}\) on \(A\), where \(x_{\iota}\) is a representative of the element \(x\) in \(\mathbf{R}^{A_{\iota}}\). The main result of the paper is the following

Theorem. A continuous linear form \(s\) on \(F\) is representable in the form

\[ s=t\circ h, \]

where \(t\) is some continuous linear form on \(\mathbf{R}^{A}\) (uniquely determined by the form \(s\)), if and only if, whatever finite subset \(\mathfrak{M}\) of \(\mathfrak{A}\) is taken,

\[ s\circ i_{\mathfrak{m}}=0 \]

for all but, possibly, finitely many elements \(\mathfrak{m}\in\mathfrak{M}\).

Proof. Let \(s\) be an arbitrary continuous linear form on \(F\), and let \(f_{\iota}\), for each \(\iota\in I\), be the canonical continuous linear mapping \(\mathbf{R}^{A_{\iota}}\to F\). Then, for each \(\iota\in I\), the composition \(s_{\iota}=s\circ f_{\iota}\) is uniquely representable in the form

\[ x_{\iota}\to \sum_{\alpha\in A_{\iota}} x_{\iota}(\alpha)s_{\iota}(\alpha), \]

where \(\alpha\to s_{\iota}(\alpha)\) is a numerical function on \(A_{\iota}\), for which the set \(\operatorname{supp}(s_{\iota})\) of those \(\alpha\in A_{\iota}\) such that \(s_{\iota}(\alpha)\ne 0\) is finite; moreover,

\[ s_{\iota}(\alpha)=\sum_{\varphi_{\iota\chi}(\beta)=\alpha} s_{\chi}(\beta) \]

for any pair of indices \(\iota\leq \chi\) and every \(\alpha\in A_{\iota}\). In particular,

\[ \operatorname{supp}(s_{\iota})\subset \varphi_{\iota\chi}(\operatorname{supp}(s_{\chi}))\qquad(\iota\leq \chi). \]

Let \(\mathfrak{M}\) be an arbitrary finite subset of \(\mathfrak{A}\); suppose that \(\mathfrak{M}\) is infinite and that \(s\circ i_{\mathfrak{m}}\ne 0\) for every \(\mathfrak{m}\in\mathfrak{M}\). Then one can choose indices \(\iota_{1}<\iota_{2}<\cdots\) so that, for each integer \(n>0\), the set \(\mathfrak{M}\) contains at least \(n\) elements having pairwise disjoint representatives \(M_{n,1},\ldots,M_{n,n}\) in \(A_{\iota_n}\), for which

\[ \operatorname{supp}(s_{\iota_n})\cap M_{n,k}\ne \varnothing \qquad (n=1,2,\ldots;\ k=1,\ldots,n). \]

Let \(n(s_{\iota})\), for each \(\iota\in I\), be the number of elements of the set \(\operatorname{supp}(s_{\iota})\); then \(n(s_{\iota_k})\geq k\) for every \(k=1,2,\ldots\), and, consequently, the family of numbers \(n(s_{\iota})\) \((\iota\in I)\) is unbounded. On the other hand, if \(s=t\circ h\), where \(t\) is some continuous linear form on \(\mathbf{R}^{A}\), then, for each \(x\in F\),

\[ s(x)=\sum_{\alpha\in A} y(\alpha)t(\alpha), \]

where \(y=h(x)\), and \(\alpha\to t(\alpha)\) is a numerical function on \(A\), nonzero only on some finite set \(\operatorname{supp}(t)\). In this case

\[ \operatorname{supp}(s_{\iota})\subset \varphi_{\iota}(\operatorname{supp}(t)) \]

for every \(\iota\in I\); consequently, the numbers \(n(s_{\iota})\) \((\iota\in I)\) are bounded above by the number of elements of the set \(\operatorname{supp}(t)\).

Conversely, suppose that \(s\) is a continuous linear form on \(F\) not representable in the form \(s=t\circ h\). Then the numbers \(n(s_{\iota})\) \((\iota\in I)\) cannot be bounded in the aggregate. Indeed, in the contrary case the family of these numbers stabilizes, since \(n(s_{\iota})\leq n(s_{\chi})\) for \(\iota\leq \chi\). Replacing, if necessary, the set \(I\) by a cofinal subset, one may assume that \(\varphi_{\iota\chi}\) maps \(\operatorname{supp}(s_{\chi})\) bijectively onto \(\operatorname{supp}(s_{\iota})\), whatever \(\iota\leq\chi\). Then there exists in \(A\) a finite set \(M\) for which

\[ \varphi_{\iota}(M)=\operatorname{supp}(s_{\iota}) \]

for every \(\iota\in I\); since, moreover, \(s_{\iota}(\alpha)=s_{\chi}(\beta)\) if \(\alpha=\varphi_{\iota\chi}(\beta)\) and \(\beta\in\operatorname{supp}(s_{\chi})\), the continuous linear form on \(\mathbf{R}^{A}\)

\[ x\to t(x)=\sum_{\alpha\in A} x(\alpha)t(\alpha), \]

where \(t(\alpha)=s_{\iota}(\varphi_{\iota}(\alpha))\) for \(\alpha\in M\) and \(t(\alpha)=0\) for \(\alpha\notin M\), is such that \(s=t\circ h\), and the latter representation is unique, since the image of the mapping \(h\) is everywhere dense in \(\mathbf R^{A}\).

Thus, one may suppose that the numbers \(n(s_{\iota})\) \((\iota\in I)\) are not bounded in the aggregate. We construct by induction a sequence of indices \(\iota_1<\iota_2<\cdots\) and a sequence of elements \(\alpha_n\in A_{\iota_n}\) \((n=1,2,\ldots)\) in the following way. Choose the index \(\iota_1\) and the element \(\alpha_1\in \operatorname{supp}(s_{\iota_1})\) so that \(n(s_{\iota_1})\ge 2\) and the number of elements of the set \(\operatorname{supp}(s_{\iota})\setminus \varphi_{\iota_1\iota}^{-1}(\alpha_1)\) is not bounded for \(\iota>\iota_1\). If the indices \(\iota_1<\cdots<\iota_{n-1}\) and the elements \(\alpha_1,\ldots,\alpha_{n-1}\) have already been chosen, then we determine the index \(\iota_n\) and the element \(\alpha_n\in \operatorname{supp}(s_{\iota_n})\) so that the number of elements of the set
\[ \operatorname{supp}(s_{\iota_n})\setminus \bigcup_{m<n}\varphi_{\iota_m\iota_n}^{-1}(\alpha_m) \]
is \(\ge 2\), \(\varphi_{\iota_m\iota_n}(\alpha_n)\ne \alpha_m\) for \(m<n\), and so that the number of elements of the set
\[ \operatorname{supp}(s_{\iota})\setminus \bigcup_{m\le n}\varphi_{\iota_m\iota}^{-1}(\alpha_m) \]
is not bounded for \(\iota>\iota_n\). Let \(\mathfrak m_n\), for each \(n=1,2,\ldots\), be the canonical image in \(\mathfrak A\) of the one-point set \(\{\alpha_n\}\in\mathfrak A_{\iota_n}\); since \(\alpha_n\in \operatorname{supp}(s_{\iota_n})\), the form \(s\) is such that \(s\circ i_{\mathfrak m_n}\ne 0\) for every \(n=1,2,\ldots\). According to the assumptions made, the set \(\mathfrak M=\{\mathfrak m_1,\ldots\}\) cannot be final; therefore there exists \(\iota\in I\) such that all elements of some infinite part of the set \(\mathfrak M\) have representatives in \(\mathfrak A_{\iota}\). We shall suppose that, for every \(n=1,2,\ldots\), the element \(\mathfrak m_n\in\mathfrak A\) has a representative \(M_n\) in \(\mathfrak A_{\iota}\). Since \(\varphi_{\iota_m\iota_n}(\alpha_n)\ne\alpha_m\) for \(m<n\), one may suppose that the sets \(M_n\) \((n=1,2,\ldots)\) are pairwise disjoint. For every \(n=1,2,\ldots\) define a function \(x_n\) on \(A_{\iota_n}\) by putting \(x_n(\alpha)=0\) (for \(\alpha\in A_{\iota_n}\setminus \alpha_n\)) and \(x_n(\alpha_n)=n/s_{\iota_n}(\alpha_n)\). Then \(s(f_{\iota_n}(x_n))=n\), for every \(n=1,2,\ldots\); consequently, the form \(s\) is not bounded on the sequence of elements \(f_{\iota_n}(x_n)\in F\) \((n=1,2,\ldots)\). On the other hand, \(f_{\iota_n}(x_n)=f_{\iota}(y_n)\), where \(y_n\) is a function on \(A_{\iota}\) which takes the value \(n/s_{\iota_n}(\alpha_n)\) on \(M_n\) and is equal to zero on \(A_{\iota}\setminus M_n\). Since the sets \(M_n\) \((n=1,2,\ldots)\) are pairwise disjoint, the functions \(y_n\) \((n=1,2,\ldots)\) form a bounded sequence in \(R^{A_{\iota}}\); hence the sequence of elements \(f_{\iota}(y_n)\) is bounded in \(F\). Since the form \(s\) is not bounded on the elements \(f_{\iota}(y_n)\) \((n=1,2,\ldots)\), it cannot be continuous on \(F\); thus the theorem is proved.

Corollary. Every continuous linear form \(s\) on \(F\) is uniquely representable in the form \(s=t\circ h\), where \(t\) is some continuous linear form on \(\mathbf R^{A}\), if every final subset in \(\mathfrak A\) is finite.

Let \(\mathbf R^{(A_{\iota})}\), for every \(\iota\in I\), be the vector space of all real numerical functions \(s_{\iota}\), each of which is defined on \(A_{\iota}\) and is equal to zero everywhere outside some (depending on \(s_{\iota}\)) finite set \(\operatorname{supp}(s_{\iota})\). For each pair of indices \(\iota\le \varkappa\) let \(g_{\iota\varkappa}:\mathbf R^{(A_{\varkappa})}\to \mathbf R^{(A_{\iota})}\) be the mapping which assigns to each function \(s_{\varkappa}\in\mathbf R^{(A_{\varkappa})}\) the function
\[ \alpha\to s_{\iota}(\alpha)= \sum_{\varphi_{\iota\varkappa}(\beta)=\alpha} s_{\varkappa}(\beta), \]
belonging to the space \(\mathbf R^{(A_{\iota})}\). The vector spaces \(\mathbf R^{(A_{\iota})}\) \((\iota\in I)\) and the linear mappings \(g_{\iota\varkappa}\) \((\iota\le\varkappa)\) form a projective system. The projective limit
\[ G=\varprojlim \mathbf R^{(A_{\iota})} \]
of the system \((\mathbf R^{(A_{\iota})}, g_{\iota\varkappa})\) will be identified with the vector space \(F'\), conjugate to \(F\).

For every \(\iota\in I\) a linear mapping \(k_{\iota}:\mathbf R^{(A)}\to \mathbf R^{(A_{\iota})}\) is defined, assigning to each element \(s\in\mathbf R^{(A)}\) the function
\[ \alpha\to s_{\iota}(\alpha)= \sum_{\varphi_{\iota}(\beta)=\alpha} s(\beta), \]

belonging to the space \(\mathbf{R}^{(A_\iota)}\). Since \(k_\iota = g_{\iota\kappa}\circ k_\kappa\) for all \(\iota \le \kappa\), a linear mapping is defined,

\[ k:\mathbf{R}^{(A)}\to G, \]

which assigns to each element \(s\in \mathbf{R}^{(A)}\) the element of \(G\) whose canonical projection in \(\mathbf{R}^{(A_\iota)}\) is equal to \(k_\iota(s)\), for each \(\iota\in I\). Since the image of the mapping \(h\) is everywhere dense in \(\mathbf{R}^A\), the mapping \(k\) is injective.

Let an element \(\mathfrak{m}\in \mathfrak{A}\) have, for each \(\iota \ge \iota_0\), a representative \(M_\iota\) in \(\mathfrak{A}_\iota\). Then the vector spaces \(\mathbf{R}^{(M_\iota)}\) \((\iota\ge \iota_0)\) and the naturally defined linear mappings \(\mathbf{R}^{(M_\kappa)}\to \mathbf{R}^{(M_\iota)}\) \((\iota\le \kappa)\) form a projective system (a projective spectrum), whose limit we denote by \(G_{\mathfrak{m}}\). For each \(\iota\ge \iota_0\) define the restriction mapping \(j_{m_\iota}:\mathbf{R}^{(A_\iota)}\to \mathbf{R}^{(M_\iota)}\); since for every pair of indices \(\iota \le \kappa\) the composition \(j_{m_\iota}\circ g_{\iota\kappa}\) coincides with the composition of the mappings \(j_{m_\kappa}\) and \(\mathbf{R}^{(M_\kappa)}\to \mathbf{R}^{(M_\iota)}\), a (contractive) linear mapping is thereby defined,

\[ j_{\mathfrak{m}}:G\to G_{\mathfrak{m}}, \]

which assigns to each element \(s\in G\) the element \(j_{\mathfrak{m}}(s)\in G_{\mathfrak{m}}\) whose canonical image in \(\mathbf{R}^{(M_\iota)}\) is the restriction to \(M_\iota\) of the canonical image of the element \(s\) in \(\mathbf{R}^{(A_\iota)}\), for each \(\iota\ge \iota_0\).

The theorem proved in the present paper is equivalent to the following assertion:

The image of the mapping \(k\) consists of those and only those elements \(s\in G\) for which, whatever final set \(\mathfrak{M}\subset \mathfrak{A}\) may be, \(j_{\mathfrak{m}}(s)=0\) for every \(\mathfrak{m}\in \mathfrak{M}\), except possibly for a finite number of them. In particular, the mapping \(k\) is an isomorphism of the space \(\mathbf{R}^{(A)}\) onto \(G\), if every final subset in \(\mathfrak{A}\) is finite.

Kharkov State University
named after A. M. Gorky

Received
17 IV 1966

CITED LITERATURE

1. N. Bourbaki, Set Theory, Moscow, 1965.
2. A. Douady, C. R., 259, 2946 (1964).