Abstract
Full Text
UDC 517.11
MATHEMATICS
PHAN DINH DIEU (PHAN ĐÌNH DIỆU)
ON SPACES OF CONSTRUCTIVE INFINITELY DIFFERENTIABLE FUNCTIONS AND ON FUNCTIONALS IN THEM
(Presented by Academician P. S. Novikov, 3 I 1968)
1. In the present note the terms and notation introduced in the papers \((^{1,2})\) are used. By a word of type \(\mathrm{ш}\) we shall mean a complete code of an almost uniformly continuous function \((^9)\). In \((^9)\) an algorithm \(J\) was introduced with the following property: whatever the natural number \(n\) and the word \(\mathrm{ш}_1\) of type \(\mathrm{ш}\), \(J(n \square \mathrm{ш}_1)\) is a primitive of order \(n\) of the complete code \(\mathrm{ш}_1\). Let \(\mathrm{ш}_1\) and \(\mathrm{ш}_2\) be words of type \(\mathrm{ш}\). If the condition
\[ Д(\mathrm{ш}_1,\mathrm{ш}_2)\ \rightleftarrows\ \forall x\bigl(\mathrm{ш}_2(x)-\mathrm{ш}_2(0)=J(1\square \mathrm{ш}_1)(x)\bigr) \]
is fulfilled, the function \(\mathrm{ш}_1\) is the derivative of the function \(\mathrm{ш}_2\). Then we shall also say that \(\mathrm{ш}_1\) is a derivative of \(\mathrm{ш}_2\). Introduce the formula
\[ \mathfrak{P}_f\ \rightleftarrows\ \bigl(\langle\gamma\rangle\in(\mathrm{H}\to\mathrm{ш})\bigr)\ \&\ \forall n\, Д\bigl(\langle\gamma\rangle(n+1),\langle\gamma\rangle(n)\bigr) \]
where \(\gamma\) is a variable for words in the alphabet \(\mathcal{Ч}_0\). The elements of the set \(\mathfrak{P}_f\) will be called complete records of constructive infinitely differentiable functions, or words of type \(f\).
One can construct a formula \(\mathfrak{E}_f\), algorithms \(+_f\) of type \((ff\to f)\) and an algorithm \(\dot f\) of type \((\mathbb{D}f\to f)\) such that, whatever the words \(f_1,f_2\) of type \(f\) and the real duplex \(\alpha\),
\[ \mathfrak{E}_f(f_1,f_2)\ \equiv\ \bigl(\langle f_1\rangle(0)\equiv_{\mathrm{ш}}\langle f_1\rangle(n)\bigr). \]
\[ \forall n\bigl(\langle +_f(f_1\square f_2)\rangle(n)\simeq \langle f_1\rangle(n)+_{\mathrm{ш}}\langle f_2\rangle(n)\bigr); \]
\[ \forall n\bigl(\langle \dot f(\alpha\square f_1)\rangle(n)\simeq \alpha\cdot_{\mathrm{ш}}\langle f_1\rangle(n)\bigr). \]
By means of \(\mathfrak{D}_f\) denote the record of an algorithm that transforms any natural number into the null element of type \(\mathrm{ш}\). By virtue of the results from \((^2)\), § 11.2, one can construct an algorithm \(N_1\) of type \((\mathrm{нн}f\to Д)\) such that, whatever the natural numbers \(m,n\) and the word \(f_1\) of type \(f\), \(N_1(m\square n\square f_1)\) is an exact upper bound of the function \(\langle f_1\rangle(n)\) on the segment \(-m\Delta m\).
After this construct an algorithm \(N\) such that, for any \(m,n\) and \(f_1\),
\[ N(m\square n\square f_1)\simeq \max\bigl(N_1(m\square 0\square f_1)\square\ldots\square N_1(m\square n\square f_1)\bigr). \]
Let \(m\) be some fixed positive natural number. Introduce the formula
\[ \mathfrak{P}_m\ \rightleftarrows\ \mathfrak{P}_f\ \&\ \forall x\bigl(|x|\ge m\supset \langle\gamma\rangle(0)(x)=0\bigr). \]
It can be verified that the list*
\[ \mathcal{Ч}_0,\ \mathfrak{P}_m,\ \mathfrak{E}_f,\ +_f,\ \dot f,\ \mathfrak{D}_f,\ \widetilde{N}_m \square \tag{1} \]
defines a constructive countably normed space (see the definition in \((^7)\)). This space will be denoted by \(K_m\). It is pre-
* The meaning of the notation \(N_m\square\) is given in \((^3)\).
is the space of complete records of all constructive infinitely differentiable functions that vanish outside the segment \(-m\Delta m\). The points of the space \(K_m\) will be called words of type \(\phi^m\), the variables for which are denoted by \(\phi_1^m,\phi_2^m,\ldots\). Being a computably normed space, \(K_m\) is a metrizable locally convex space \((^{6,7})\). The notions of completeness and separability introduced for metric spaces in \((^{2,3})\) are easily defined for the space \(K_m\). The following holds.
Theorem 1. The space \(K_m\) is complete and separable.
Let \(\mathfrak A\) and \(\mathfrak B\) be sets of points of the space \(K_m\). We shall say that \(\mathfrak A\) is a convex set in \(K_m\) if
\[ \forall \phi_1^m \phi_2^m xy\,(x+y=1\ \&\ \phi_1^m\in\mathfrak A\ \&\ \phi_2^m\in\mathfrak A \supset x\cdot\phi_1^m+y\cdot\phi_2^m\in\mathfrak A), \]
and that the set \(\mathfrak A\) absorbs the set \(\mathfrak B\) if
\[ \exists x\,(x>0\ \&\ \forall \phi_1^m\,(\phi_1^m\in\mathfrak B \supset x\cdot\phi_1^m\in\mathfrak A)). \]
Theorem 2. In the space \(K_m\) one can construct a convex set \(\mathfrak A\) possessing the properties: 1) \(\mathfrak A\) absorbs every bounded set in the space \(K_m\); 2) \(\mathfrak A\) contains no neighborhood of zero in \(K_m\).
2. Let \(\mathfrak M\) be some constructive locally convex space and let \(\lambda\) be a linear operator mapping \(K_m\) into \(\mathfrak M\). Let \(\mathfrak A\) be a set of points of the space \(K_m\). By \(\lambda(\mathfrak A)\) we denote the set of points of the space \(\mathfrak M\) defined by the formula \(\exists\phi_1^m(\phi_1^m\in\mathfrak A \& \lambda(\phi_1^m)=\theta_1)\), where \(\theta_1\) is a variable for points of \(\mathfrak M\). The operator \(\lambda\) is called bounded if, for any bounded set \(\mathfrak A\) in \(K_m\), the set \(\lambda(\mathfrak A)\) is bounded in \(\mathfrak M\); the operator \(\lambda\) is called continuous if, for any neighborhood \(\mathfrak B\) of zero in \(\mathfrak M\), there is a neighborhood \(\mathfrak A\) of zero in \(K_m\) such that \(\lambda(\mathfrak A)\) is contained in \(\mathfrak B\); the operator \(\lambda\) is called p-continuous if, for any sequence \(\varphi\) of points of the space \(K_m\) converging to zero, the sequence \((\lambda\circ\varphi)\) converges to zero in \(\mathfrak M\). The notion of p-continuity was introduced for normed spaces in \((^{5})\).
Theorem 3. There exist a constructive locally convex space \(\mathfrak M\) and a linear operator \(\lambda\), mapping \(K_m\) into \(\mathfrak M\), such that \(\lambda\) is bounded and p-continuous, but not continuous.
On the other hand, from the main theorem of \((^{3})\) and Theorem 1 it follows:
Theorem 4. If \(\mathfrak M\) is a metrizable locally convex space, then every linear operator mapping \(K_m\) into \(\mathfrak M\) is continuous.
3. Let us now consider functionals in the space \(K_m\). By Theorem 4, every linear functional in \(K_m\) is continuous.
Denote by \(K_m'\) the locally convex space conjugate to the space \(K_m\) (see the definition of the conjugate space in \((^{8})\)).
Recall that, according to the definition of the conjugate space introduced in \((^{8})\), a point of the space \(K_m'\) is a word of the form \(P\odot Q\), where \(P\) is the record of a linear continuous functional in \(K_m\), and \(Q\) is the record of its regulator of continuity. Let \(T\,(T \doteq P\odot Q)\) be a point of the space \(K_m'\). Then \(T\) denotes the functional whose record is the word \(P\).
The variables for points of the space \(K_m'\) will be denoted by \(F_1^m,F_2^m,\ldots\).
Let \(\mathfrak A\) be a set of points of the space \(K_m\) and \(\mathfrak B\) a set of points of the space \(K_m'\). We shall say that the set \(\mathfrak B\) is bounded on the set \(\mathfrak A\) if there exists a positive real duplex \(a\) such that
\[ \forall F_1^m\phi_1^m\,(F_1^m\in\mathfrak B\ \&\ \phi_1^m\in\mathfrak A \supset |F_1^m(\phi_1^m)|\leq a). \]
Theorem 5. There exists a bounded set \(\mathfrak B\) of the space \(K_m'\) such that it is not bounded in any neighborhood of zero of the space \(K_m\).
- In the space \(K_m\) the locally convex topology corresponding to the multinorm \(N_{m\square}\) will be called the initial topology. In addition to the initial topology, we define in \(K_m\) another topology in the following way.
Let \((A_r,\mathfrak X,\mathfrak R)\) be a normal fundamental system of bounded sets in the space \(K_m'\). Such a system can be constructed by the method indicated in (8). After this, construct a formula \(\mathfrak D\) such that, whatever the element \(\mathfrak x_1\) of the set \(\mathfrak X\) and the point \(\phi_1^m\) of the space \(K_m\) may be,
\[ \mathfrak D(\mathfrak x_1,\phi_1^m)\equiv \forall F_1^m\bigl(\mathfrak R(\mathfrak x_1,F_1^m)\supset |F_1^m(\phi_1^m)|<1\bigr). \]
It is not difficult to verify that the list \((A_r,\mathfrak X,\mathfrak D)\) defines a locally convex topology on the linear space
\[ \mathcal Ch_0,\ \mathfrak P_m,\ \mathfrak E_f,\ +,\ \cdot,\ \mathfrak D_f . \]
We shall call this topology the strong topology in \(K_m\). It can be shown that in \(K_m\) the strong topology majorizes the initial topology. But the converse is not true. Namely, we have:
Theorem 6. In the space \(K_m\) the initial topology does not majorize the strong topology.
Thus, in \(K_m\) the initial and strong topologies are not equivalent to one another. On the other hand, despite the nonequivalence of these topologies, it can be proved that in \(K_m\) every set bounded in the initial topology is also bounded in the strong topology, and conversely.
- If in the list (1) \(m\) denotes a variable ranging over the positive natural numbers, then this list defines an expanding sequence of countably normed spaces, and hence an expanding sequence of locally convex spaces (8). It is easy to verify that in this sequence each space \(K_m\) is a \(\beta\)-closed (6) subspace of the space \(K_l\) \((l>m)\). By definition in (8), we can construct the strict inductive limit of the sequence of spaces \(K_m\). This strict inductive limit is a locally convex space; we shall denote it by \(K^0\). Let us note that the points of the space \(K^0\) are words \(\gamma\) of type \(f\) satisfying the condition:
\[ \exists m\forall x\bigl(|x|\ge m\supset \langle\gamma\rangle(0)(x)=0\bigr). \]
In addition to the space \(K^0\), for the sequence (1) we construct the so-called \(FR\)-strict inductive limit \(K\) as follows: as the points of the space \(K\) we take words of the form \(m\tau\gamma\), where \(m\) is a positive natural number and \(\gamma\) is a word of type \(f\) satisfying the condition:
\[ \forall x\bigl(|x|\ge m\supset \langle\gamma\rangle(0)(x)=0\bigr). \]
The equality relation, linear operations and topology in \(K\) are defined in the analogous way as for the space \(K^0\).
One may regard \(K\) as a constructive analogue of the space of all finite infinitely differentiable functions in classical mathematics (see (10)).
Theorem 7. There is no algorithm which transforms each point \(X\) of the space \(K^0\) into a natural number \(m\) such that \(m\tau X\) is a point of the space \(K\).
Theorem 8. The space \(K\) is multinormable.
Let \(\mathfrak A\) be a set of points of the space \(K\). We shall say that \(\mathfrak A\) is contained in \(K_m\) if from \(n_t X \in \mathfrak A\) it follows that \(X \in K_m\).
Theorem 9. One can construct in the space \(K\) a bounded set \(\mathfrak A\) such that there is no natural number \(m\) for which \(\mathfrak A\) would be contained in \(K_m\).
This means that there exists a strict inductive limit of a sequence of locally convex spaces that is not regular (see \((^8)\)).
Let us note that linear continuous functionals in the space \(K\) may be regarded as constructive analogues of generalized functions defined by Sobolev–Schwartz in classical mathematics.
Theorems 2, 5, 9 are proved with the aid of the universal arithmetical algorithm (see, for example, \((^4)\)), and on the basis of these theorems Theorems 3, 6 are proved. Theorems 2–6, 9 show that certain assertions of classical functional analysis (see, for example, \((^{10},{}^{11})\)) do not carry over to constructive mathematics.
Institute of Mathematics
Hanoi, DRV
Received
21 XII 1967
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