UDC 517.11

MATHEMATICS

PHAN DINH DIEU (PHAN ĐÌNH DIỆU)

ON THE DUALS OF CONSTRUCTIVE LOCALLY CONVEX SPACES

(Presented by Academician A. A. Dorodnitsyn on V 6, 1965)

In the present note we use the terms and notation introduced in \(({}^{1-3,7,8})\). In \(({}^{7})\) the concept of a constructive locally convex space was introduced. Let \(\mathfrak M\) be a constructive locally convex space, given by the list:

\[ A_{\mathfrak m},\ \mathfrak P,\ \mathfrak C,\ +,\ \cdot,\ \mathfrak D,\ A_n,\ \mathfrak I,\ \mathfrak D. \tag{1} \]

The notation \(\theta,\iota\) will be understood in the same way as in \(({}^{7})\). As defined in \(({}^{8})\), a set \(\mathfrak A\) of words of type \(\theta\) is called bounded in the space \(\mathfrak M\) if:

\[ \forall l_1 \exists a\,(a>0\ \&\ \forall \theta_1(\theta_1\in\mathfrak A \supset \mathfrak D(l_1,a\cdot\theta_1))). \]

Denote by \(\langle P\rangle\) an algorithm in the alphabet \(A_{\mathfrak m}^{ca}\cup Ч_2\), whose notation is the word \(P\). Introduce the notation:

\[ \mathfrak K \Longleftrightarrow ((\langle c_{0,1}\rangle\in(\iota\to p))\ \&\ \forall \iota_1(\langle c_{0,1}\rangle(\iota_1)>0)), \]

where \(c_{0,1}\) is a variable for words in \(Ч_0\). Since \(\mathfrak I\) is a normal formula, \(\mathfrak K\) is also a normal formula. Denote by \(\mathfrak x\) the subordinate generic letter whose characteristic formula is \(\mathfrak K\).

Let: \(A_r\) be an alphabet; \(\mathfrak X\) a one-parameter formula in a variable of genus \(t_r\); \(\mathfrak R\) a formula in variables \(\varepsilon\) of genus \(t_r\) and \(\eta\) of genus \(\theta\). Denote by \(\mathfrak v\) the subordinate generic letter whose characteristic formula is \(\mathfrak X\). Introduce the notation

\[ \mathfrak R(V,U)\Longleftrightarrow F^{\varepsilon,\eta}_{V,U}\mathfrak R_{\perp}, \]

where \(V\) is any term of genus \(\mathfrak v\); \(U\) is any term of genus \(\theta\).

We shall call the list \((A_r,\mathfrak X,\mathfrak R)\) a fundamental system of bounded sets in the space \(\mathfrak M\), if the conditions

\[ \forall v_1\exists \mathfrak x_1\forall\theta_1(\mathfrak R(v_1,\theta_1)\supset \forall \iota_1 \mathfrak D(\iota_1,\langle \mathfrak x_1\rangle(\iota_1)\cdot\theta_1)); \]

\[ \forall \mathfrak x_1\exists v_1\forall\theta_1(\forall \iota_1 \mathfrak D(\iota_1,\langle \mathfrak x_1\rangle(\iota_1)\cdot\theta_1)\supset \mathfrak R(v_1,\theta_1)). \]

are satisfied.

A fundamental system of bounded sets \((A_r,\mathfrak X,\mathfrak R)\) is called normal if \(\mathfrak X\) is a normal formula.

Theorem 1. For any constructive locally convex space one can construct in it a normal fundamental system of bounded sets.

This is the system \((Ч_0,\mathfrak K,\mathfrak R)\), where the formula \(\mathfrak R\) is defined so that

\[ \mathfrak R(\mathfrak x_1,\theta_1)\equiv \forall \iota_1\mathfrak D(\iota_1,\langle \mathfrak x_1\rangle(\iota_1)\cdot\theta_1). \]

1. Let \(A_\eta \Longleftrightarrow A_{\mathfrak m}\cup Ч_3\). An algorithm \(\lambda\) in \(A_\eta^{ca}\) is called a functional in the space \(\mathfrak M\), if it is an operator of type \((\theta\to \mathbb D)\) (see \(({}^{7})\)).

Let \(\lambda_1\) be a linear functional in \(\mathfrak M\). \(\lambda_1\) is continuous if

\[ \forall n\exists \iota_1\forall\theta_1(\mathfrak D(\iota_1,\theta_1)\supset M(\lambda_1(\theta_1))<2^{-n}). \]

This means that there is realizable in \(A_\xi^{ca}\) \((A_\xi \Longleftrightarrow A_\eta\cup Ч_0)\) an algorithm \(\lambda_2\) of type

\((\mathbf{n}\to \iota)\) such that

\[ \forall n\theta_1\bigl(\mathfrak{P}(\lambda_2(n),\theta_1)\supset M(\lambda_1(\theta_1)<2^{-n})\bigr). \tag{2} \]

An algorithm \(\lambda_2\) of type \((\mathbf{n}\to \iota)\) satisfying (2) will be called a continuity regulator of the functional \(\lambda_1\).

A linear functional \(\lambda_1\) in \(\mathfrak{M}\) is called quasicontinuous if it cannot be discontinuous.

Theorem 2. There exists a constructive locally convex space for which it is false that every quasicontinuous linear functional is continuous.

Theorem 3. There exists a constructive locally convex space such that no algorithm is possible which, for each continuous linear functional in it, enables one to find a continuity regulator of this functional.

3. We shall say of a word \(t_{\pi,1}\) in the alphabet \(A_\pi(\rightleftarrows Ч_0\cup\{\odot\})\) that it is a complete cipher of a linear discontinuous functional in \(\mathfrak{M}\), and we shall write

\[ \left(t_{\pi,1}\in \frac{\mathrm{compl.\,disc.}}{\mathfrak{m}}\right), \tag{3} \]

if the word \(t_{\pi,1}\) has the form \(P\odot Q\), where \(P\) is a record of a linear discontinuous functional in \(\mathfrak{M}\) and \(Q\) is a record of its continuity regulator. Let

\[ t_{\pi,1}\doteq P\odot Q. \]

We shall denote:

\[ t_{\pi,1}\rightleftarrows \langle P\rangle_{\eta};\qquad t_{\pi,1}\rightleftarrows \langle Q\rangle_{\zeta}. \]

Denote by \(\mathfrak{P}^{\mathfrak{m}}\) formula (3), and by \(\mathfrak{m}\) the generic letter whose characteristic formula is \(\mathfrak{P}^{\mathfrak{m}}\). Introduce the formula

\[ \mathfrak{C}^{\mathfrak{m}}\rightleftarrows \forall\theta_1\bigl(t_{\pi,1}(\theta_1)\overset{\mathrm{B}}{=}t_{\pi,2}(\theta_1)\bigr). \]

Denote by \(\mathfrak{\dot{m}}^{+}\) an algorithm of type \((\mathfrak{m}\mathfrak{m}\to\mathfrak{m})\), and by \(\dot{\mathfrak{m}}\) an algorithm of type \((p\mathfrak{m}\to\mathfrak{m})\), such that

\[ \forall \mathfrak{m}_1\mathfrak{m}_2\theta_1\bigl(\mathfrak{\dot{m}}^{+}(\mathfrak{m}_1\square\mathfrak{m}_2)(\theta_1) \overset{\mathrm{B}}{=}\mathfrak{m}_1(\theta_1)+\mathfrak{m}_2(\theta_1)\bigr); \]

\[ \forall a\mathfrak{m}_1\theta_1\bigl(\dot{\mathfrak{m}}(a\square\mathfrak{m}_1)(\theta_1) \overset{\mathrm{B}}{=}a\cdot\mathfrak{m}_1(\theta_1)\bigr). \]

Denote by \(\mathfrak{D}^{\mathfrak{m}}\) the word \(U\odot V\), where \(U\) is a record of an algorithm transforming each word of type \(\theta\) into the number \(0\); \(V\) is a record of an algorithm transforming each number \(n\) into a fixed word of type \(\iota\).

Let \((A_r,\mathfrak{X},\mathfrak{R})\) be a normal fundamental system of bounded sets in \(\mathfrak{M}\). Construct a formula \(\mathfrak{D}^{\mathfrak{m}}\) such that, whatever the word \(\nu_1\) of type \(\nu\) and the word \(\mathfrak{m}_1\) of type \(\mathfrak{m}\), one has

\[ \mathfrak{D}^{\mathfrak{m}}(\nu_1,\mathfrak{m}_1)\equiv \forall\theta_1\bigl(\mathfrak{R}(\nu_1,\theta_1)\supset M(\mathfrak{m}_1(\theta_1))\leq 1\bigr). \]

It is not hard to prove that the list

\[ A_\pi,\ \mathfrak{P}^{\mathfrak{m}},\ \mathfrak{C}^{\mathfrak{m}},\ \mathfrak{m}^{+},\ \dot{\mathfrak{m}},\ \mathfrak{D}^{\mathfrak{m}},\ A_r,\ \mathfrak{X},\ \mathfrak{D}^{\mathfrak{m}} \]

is a constructive locally convex space, which we shall call conjugate to the space \(\mathfrak{M}\).

4. Let \(\mathfrak{M}\) be a normed space. It may be regarded as a locally convex space. Consequently, one can construct the space conjugate to the normed space \(\mathfrak{M}\). This conjugate is a locally convex space.

Theorem 4. Let \(\mathfrak{M}\) be a normed space with multiplication by real duplets, and let \(N\) be its norm. If the set of points \(\theta_1\) such that \(N(\theta_1)=1\) is compact then the space conjugate to \(\mathfrak{M}\) is normable.

* See the definition in \((^3)\), § 11.
** See the definition in \((^8)\).

Theorem 5. There exists a constructive normed space such that its conjugate is not normable (not even seminormable) (see the definition in \((8)'\)).

Theorem 6. There exists a constructive multinormed space such that its conjugate is not multinormable (see \((8)\)).

Theorem 7. There exists a constructive complete Hilbert space such that it is not isomorphic to its conjugate.

For the proof of Theorems 5–7 it suffices to consider the space \(l_2\), constructed in \((3)\), § 13.

Theorem 8. For natural numbers \(m,n\), \(0 \leqslant m \leqslant n\), there exists a \(T'\)-separable, \(n\)-dimensional in the strong sense locally convex space such that its conjugate is an \(m\)-dimensional, normable space.

Theorem 9. For any natural number \(m\) there exists a \(T'\)-separable, infinite-dimensional locally convex space such that its conjugate is an \(m\)-dimensional, normable space.

Theorems 8 and 9 are proved with the aid of the space \(\mathfrak S\), constructed in \((7)\).

1. We shall consider bounded sets in the inductive limit of a sequence of locally convex spaces.

Let \(A_m\) and \(A_n\) be alphabets; \(\mathfrak P\) a two-parameter formula in the variables \(k\) and \(\alpha\), where \(k\) is a variable for natural numbers and \(\alpha\) is a variable of type \(t_m\); \(\mathfrak E\) a three-parameter formula in the variables \(k\) and \(\beta,\gamma\) of type \(t_m\); \(+\) an algorithm of type \((t_m t_m \dot{\to} t_m)\); \(\cdot\) an algorithm of type \((pt_m \dot{\to} t_m)\); \(\mathfrak D\) a fixed word of type \(t_m\); \(\mathfrak I\) a normal two-parameter formula in the variables \(k\) and \(\delta\) of type \(t_n\); \(\mathfrak L\) a three-parameter formula in the variables \(k,\xi\) of type \(t_n\) and \(\eta\) of type \(t_m\).

We agree to denote:

\[ \mathfrak P(N,R) \Leftrightarrow F^{k,\alpha}_{N,R}\,\mathfrak P_{\perp}; \qquad \mathfrak E(N,R,S) \Leftrightarrow F^{k,\beta,\gamma}_{N,R,S}\,\mathfrak E_{\perp}; \qquad (R+S) \Leftrightarrow +(R \square S); \]

\[ (a\cdot R) \Leftrightarrow \cdot(a\square R); \qquad \mathfrak I(N,T) \Leftrightarrow F^{k,\delta}_{N,T}\,\mathfrak I_{\perp}; \qquad \mathfrak L(N,T,R) \Leftrightarrow F^{k,\xi,\eta}_{N,T,R}\,\mathfrak L_{\perp}, \]

where \(N\) is any natural number, \(R\) and \(S\) are any terms of type \(t_m\), \(T\) is any term of type \(t_n\), and \(a\) is any rational number. Suppose that

\[ \forall k\alpha\bigl(\mathfrak P(k,\alpha)\supset \mathfrak P(k+1,\alpha)\bigr); \qquad \forall k\,\mathfrak P(k,\mathfrak D); \]

\[ \forall k\beta\gamma\bigl(\mathfrak P(k,\beta)\&\mathfrak P(k,\gamma)\supset (\mathfrak E(k,\beta,\gamma)\equiv \mathfrak E(k+1,\beta,\gamma))\bigr); \]

\[ \forall k\beta\gamma\bigl(\mathfrak P(k+1,\beta)\&\mathfrak P(k,\gamma)\&\mathfrak E(k+1,\beta,\gamma)\supset \mathfrak P(k,\beta)\bigr); \]

\[ \forall k\beta\gamma\bigl(\mathfrak P(k,\beta)\&\mathfrak P(k,\gamma)\supset!(\beta+\gamma)\&\mathfrak P(k,\beta+\gamma)\bigr); \]

\[ \forall k a\alpha\bigl(\mathfrak P(k,\alpha)\supset!(a\cdot\alpha)\&\mathfrak P(k,a\cdot\alpha)\bigr). \]

Under these assumptions we shall call the list

\[ A_m,\ \mathfrak P,\ \mathfrak E,\ +,\ \cdot,\ \mathfrak D,\ A_n,\ \mathfrak I,\ \mathfrak L \tag{4} \]

an expanding sequence of constructive locally convex spaces, if for each fixed natural number \(k\) this list represents a constructive locally convex space (we denote it by \(\mathfrak M_k\)) and if the identity operator from each \(\mathfrak M_k\) into \(\mathfrak M_{k+1}\) is continuous (see the definition of a continuous operator in \((7)\)).

Let the list \((4)\) be an expanding sequence of locally convex spaces. Introduce the notation:

\[ \mathfrak P^{*} \Leftrightarrow \exists k\,\mathfrak P(k,\alpha); \]

\[ \mathfrak E^{*} \Leftrightarrow \exists k\bigl(\mathfrak P(k,\beta)\&\mathfrak P(k,\gamma)\&\mathfrak E(k,\beta,\gamma)\bigr); \]

\[ \mathfrak I^{*} \Leftrightarrow \bigl(\langle c_{0,1}\rangle_s \in (\mathrm H\to t_n)\bigr)\& \forall k\,\mathfrak I\bigl(k,\langle c_{0,1}\rangle_s(k)\bigr); \]

\[ \mathfrak L^{*} \Leftrightarrow \exists c_{0,2}m\Bigl( \bigl(\langle c_{0,2}\rangle_r \in (\mathrm H\to t_m)\bigr)\& \forall k\bigl(\mathfrak P(k,\langle c_{0,2}\rangle_r(k))\& \]

\[ \&\mathfrak L\bigl(k,\langle c_{0,1}\rangle_s(k),\langle c_{0,2}\rangle_r(k)\bigr)\bigr)\& F^{\beta,\gamma}_{\alpha,\Sigma(m\square c_{0,2})}\,\mathfrak E^{*}\Bigr), \]

where \(s\) is the number of the alphabet \(A_n \cup \mathcal U_0\), \(r\) is the number of the alphabet \(A_m \cup \mathcal U_0\), and \(\Sigma\) is an algorithm in \(A_r^{\alpha}\) having the following property:

\[ \Sigma(m \square c_{0,2}) \simeq \langle c_{0,2}\rangle_r(0)+\cdots+\langle c_{0,2}\rangle_r(m). \]

It is not difficult to prove that the list

\[ A_m,\ \mathfrak P^*,\ \mathfrak C^*,\ +,\ \cdot,\ \mathcal O,\ \mathcal U_0,\ \mathfrak I^*,\ \mathcal O^* \tag{5} \]

is a locally convex space. We shall call it the inductive limit of the sequence of spaces \(\mathfrak M_k\).

We shall say that:

1) The inductive limit (5) is \(\alpha\)-regular if, for every bounded set \(\mathfrak A\) in (5), there is a realizable number \(k\) such that \(\mathfrak A\) is contained in \(\mathfrak M_k\).

2) The inductive limit (5) is \(\beta\)-regular if, for every set \(\mathfrak A\) bounded in (5) and contained in \(\mathfrak M_k\), there is a realizable number \(j\) such that \(\mathfrak A\) is bounded in \(\mathfrak M_j\).

3) The inductive limit (5) is regular if it is \(\alpha\)-regular and \(\beta\)-regular (cf. the classical definition in \((^5)\)).

We shall say that in the sequence (4) each \(\mathfrak M_k\) is a subspace of the space \(\mathfrak M_{k+1}\) if the following conditions are satisfied:

\[ \forall k j \delta\bigl(\mathfrak I(k,\delta)\equiv \mathfrak I(j,\delta)\bigr); \]

\[ \forall k\delta\alpha\bigl(\mathfrak I(k,\delta)\ \&\ \mathfrak P(k,\alpha)\supset(\mathcal O(k,\delta,\alpha)\equiv \mathcal O(k+1,\delta,\alpha))\bigr). \]

The inductive limit (5) is called strict if, in the sequence (4), each \(\mathfrak M_k\) is a \(Z\)-closed subspace of \(\mathfrak M_j\) for \(j>k\).

The theorem on the regularity of strict inductive limits in classical mathematics \((^6)\) carries over to constructive mathematics in the following form:

Theorem 10. Let \(\mathfrak M\) be a strict inductive limit of a sequence of constructive locally convex spaces \(\mathfrak M_k\). Then:

a) if \(\mathfrak A\) is a bounded set in \(\mathfrak M\), then

\[ \neg \forall k\,\mathfrak N_1\bigl((\theta_1\in\mathfrak A)\ \&\ \neg(\theta_1\in\mathfrak M_k)\bigr). \]

b) \(\mathfrak M\) is \(\beta\)-regular.

On the other hand, one can prove the following theorem:

Theorem 11. There exists a strict inductive limit \(\mathfrak M\) of a sequence of constructive locally convex spaces \(\mathfrak M_k\) such that: a) there is no algorithm which, for each bounded set \(\mathfrak A\) in \(\mathfrak M\), makes it possible to find a number \(k\) such that \(\mathfrak A\) is contained in \(\mathfrak M_k\); b) it is false that for every bounded set \(\mathfrak A\) in \(\mathfrak M\) there is a least number \(k\) such that \(\mathfrak A\) is contained in \(\mathfrak M_k\).

Part a) is proved with the aid of the theorem stating that there exists an algorithm of type \((\mathbb N\to\mathbb N)\) whose applicability problem is not decidable \((^1,^3)\); part b) is proved with the aid of Corollary 1 of Theorem 1 in \((^4)\).

In conclusion the author expresses his deep gratitude to A. A. Markov for a number of valuable comments and suggestions.

Moscow State University
named after M. V. Lomonosov

Received
23 IV 1965

CITED LITERATURE

\(^1\) A. A. Markov, Tr. Mat. Inst. im. V. A. Steklov AN SSSR, 42 (1954).
\(^2\) N. A. Shanin, ibid., 52, 226 (1958).
\(^3\) N. A. Shanin, ibid., 67, 15 (1962).
\(^4\) A. O. Slisenko, DAN, 152, No. 2, 292 (1963).
\(^5\) B. M. Makarov, UMN, 18, 3 (111), 171 (1963).
\(^6\) J. Dieudonné, L. Schwartz, Ann. Inst. Fourier, 1, 61 (1950).
\(^7\) Fanding Zieu, DAN, 162, No. 4 (1965).
\(^8\) Fanding Zieu, DAN, 162, No. 5 (1965).