Reports of the Academy of Sciences of the USSR
1961. Volume 139, No. 5

MATHEMATICS

S. N. SLUGIN

MODULES OVER A \(K\)-SPACE

(Presented by Academician S. L. Sobolev on 31 III 1961)

1. The abstract norming of spaces of type \(B_K\) \((^1)\) makes it possible to study mathematical objects by means of functional analysis in greater detail than in spaces of type \(B\). Here modules over a \(K\)-space with unit \((^1)\) are constructed; one of them is a generalization of a real Banach space and, in particular, has the properties of a space of type \(B_K\); another is a generalization of a real Hilbert space.

2. A \(K\)-space as a generalized ring. A \(K\)-space \(Z\) with unit is transformed in various ways \((^{1-3})\), equivalent to one another (as follows from Theorem 1 \((^3)\)), into a commutative generalized ring (the product \(zu\) exists not for all \(z,u \in Z\)), in which for all elements \(z\) the quantity \(\sqrt[n]{z}\) is defined and for some an inverse \(z^{-1}\) is defined \((zz^{-1}=e_z\), where \(e_z\) is the trace of the element \(z\) \((^{1,2})\)). The maximal extension \(\widetilde Z\) \((^1)\) of the \(K\)-space \(Z\) is an absolute field (with the same unit): the product \(\widetilde z \widetilde u\) and the inverse \(\widetilde z^{-1}\) exist for all \(\widetilde z,\widetilde u \in \widetilde Z\) \((^2)\). For simplicity of notation we shall assume that \(Z \subset \widetilde Z\). Let us note one property of multiplication in \(Z\). For disjoint \(z_i \in Z\), \(z_i \geq 0\), there exists \(\sqrt{z_1}\sqrt{z_2}\), since \((^{2,3})\)

\[ z_i \leq z=z_1 \vee z_2 \]

and \(\sqrt{z}\sqrt{z}\) exists in \(Z\).

3. A quotient in a \(K\)-space.
Definition 1. If the result of the action \(zu^{-1}\) in \(\widetilde Z\) on elements from \(Z\) belongs to \(Z\), we shall call it the quotient \(z/u\).

Let us note some properties of the quotient, immediately following from the properties of the actions \(zu^{-1}\) in \(\widetilde Z\) and the embedding of \(Z\) in \(\widetilde Z\) \((^{1-3})\). The operation of division is normal, i.e., if \(|z| \leq |v|\), \(|u| \geq |w|\), and \(v/w\) exists, then \(z/u\) also exists, and moreover \(|z/u| \leq |v/w|\). In order that \(v=z/u\), it is necessary and sufficient that in \(Z\)

\[ vu=ze_u,\qquad e_v=e_z\wedge e_u . \]

If \(u^{-1}\) and \(zu^{-1}\) exist in \(Z\), then \(zu^{-1}=z/u\). The operation of division is related in the usual way to congruence relations, arithmetic operations (provided multiplication or division of the given elements can be carried out), and extraction of a root of natural degree. If \(udv\) \((^1)\), then

\[ \frac{z}{u+v}=\frac{z}{u}+\frac{z}{v}. \]

Any circumstance will henceforth be called normal (cf. \((^1)\)) if, from the fact that it holds for some \(u \in Z\), it follows that it is preserved for every \(z \in Z\) with modulus \(|z|\leq |u|\).

4. Convergence in a \(K\)-space. In a \(K\)-space \(Z\) one can establish several kinds of convergence. Suppose that in \(Z\) convergence \(z_n\to0\) is defined in some way for certain \(z_n\geq0\). Put \(z_n\to z\) if \(|z_n-z|\to0\).

We shall require that convergence satisfy the conditions: 1) addition is continuous (with respect to a countable sequence); 2) convergence is normal: if \(0\leq z_n\leq u_n\to0\), then \(z_n\to0\); 3) if \(z_n=z\), \(z_n\to0\), then \(z=0\).

Hence the uniqueness of the limit follows. The indicated properties are possessed, for example, by \((o)\)-convergence \((^1)\); by \((bk)\)-convergence, if \(Z\) is a space of type \(B_K\) with a monotone norm: from \(|z|\leq|u|\) it follows that \(\|z\|\leq\|u\|\).

in the \(K\)-space \(W\) of norms; topological convergence, if \(Z\) is a \(KT\)-space \((^{4})\) with the first separation axiom. For such convergence theorem 2 \((^{4})\) remains valid.

5. A generalized module over a \(K\)-space

Elements of \(Z\) and \(\widetilde Z\) shall be called scalars.

Definition 2. Let, for some elements of \(Z\) and some elements of the additive commutative group \(X\), the products \(zx, xz \in X\) be uniquely defined, and let \(1x=x\) and, for \(z,u\in Z,\ x,y\in X\),

\[ zx=xz,\quad z(x+y)=zx+zy,\quad (z+u)x=zx+ux,\quad z(ux)=(zu)x \tag{1} \]

provided that the products exist at least in one part of each equality (1); multiplication by a scalar is normal: if \(ux\) exists in \(X\), \(|z|\leq |u|\), then \(zx\) also exists in \(X\). Then we shall call \(X\) an \(M_z\)-module.

If \(\lambda\) is a real number, then by \(\lambda x\) we mean \(\lambda 1x\).

Definition 3. If \(Z\) is an absolute field (see above) and in an \(M_z\)-module \(X\) the product \(zx\) exists for all \(z\in Z,\ x\in X\), then we shall call \(X\) an absolute \(M_z\)-module.

Definition 4. An absolute \(M_{\widetilde Z}\)-module \(\widetilde X \supset X\) will be called an \(M\)-extension of the \(M_z\)-module \(X\), if the meaning of the product \(zx\) for elements \(z\in Z,\ x\in X\) is the same in \(X\) and in \(\widetilde X\) (provided that \(zx\) exists in \(X\)). Here, instead of an isomorphism, the inclusion \(X\subset \widetilde X\) is adopted also merely for simplicity of notation in what follows.

Theorem 1. For every \(M_z\)-module there exists its \(M\)-extension.

Construct \(\widetilde X\) from all sequences \(\widetilde x=\{\widetilde x_n\}\) \((n=1,\ldots,\infty)\) of all possible “linear” combinations

\[ \widetilde x_n=\sum_{k=1}^{m} z_{kn}x_k, \]

where \(m\) is any non-fixed natural number; \(x_k\in X,\ x_i\ne x_k\) for \(i\ne k\); for each \(k\), \(\{z_{kn}\}\) is a \((o)\)-convergent sequence in \(\widetilde Z\) of finite-valued elements \((^{1})\). We shall consider elements \(\widetilde x\) and \(\widetilde y=\{\widetilde y_n\}\in \widetilde X\) (where \(\widetilde y_n=\sum_{i=1}^{p} u_{in}y_i\)) equal if \(m=p,\ x_k=y_k,\ (o)\!-\!\lim\limits_{n\to\infty} z_{kn}=(o)\!-\!\lim\limits_{n\to\infty} u_{k n}\) \((k=1,\ldots,m)\).

To an element \(x\in X\) we put in correspondence the stationary sequence \(\{x\}\in \widetilde X\). Identifying \(x\) with \(\{x\}\), we obtain the inclusion \(X\subset \widetilde X\). Addition in \(\widetilde X\) is defined “coordinatewise”: \(\widetilde x+\widetilde y=\{\widetilde x_n+\widetilde y_n\}\), by “combining like terms” so that all \(x_k,y_i\) are distinct. Define multiplication by a scalar:

\[ zx=xz=\left\{\sum_{k=1}^{m} z_n z_{kn}x_k\right\}, \]

where the finite-valued \(z_n \xrightarrow{(o)} z\) (the existence of such \(z_n\) is evident from lemma 2.17 of Chapter 4 in \((^{1})\)). Using the \((o)\)-continuity of addition and multiplication \((^{1})\) in \(\widetilde Z\), it is not difficult to verify that all conditions of definition 4 are fulfilled.

6. A module over a space of norms

Definition 5. Let to each element \(x\) of an \(M_z\)-module \(X\) there be assigned a norm—an element \(\|x\|\in Z\), with: 1) if \(\|x\|=0\), then \(x=0\); 2) \(\|x+y\|\leq \|x\|+\|y\|\); 3) \(\|zx\|=|z|\|x\|\), if \(zx\) or \(z\|x\|\) exists. Then we shall call \(X\) a \(B_z\)-module.

As usual, we establish that \(\|0\|=0,\ \|x\|\geq 0\). If \(|z|\leq |u|\), \(ux\) exists in \(X\), then \(\|zx\|\leq \|ux\|\).

Definition 6. The trace \(e_x\) of an element \(x\) is the trace \(e_{\|x\|}\) of its norm. If \(e_x d e_y\), i.e. \(\|x\|d\|y\|\), then \(x\) and \(y\) will be called disjoint: \(xdy\). If \(e_z d e_x\), i.e. \(zd\|x\|\), then we shall call \(zdx\).

If \(zdx\), then \(zx=0\), and therefore \(e_x x=x\). Indeed, \(x=(e_x+Ce_x)x\), where \(Ce_x=1-e_x;\ Ce_x dx\). If \(xdy\), then \(\|x\pm y\|=\|x\|+\|y\|\), since \(\|x\pm y\|\geq \|x\|-\|y\|=\|x\|+\|y\|\) when \(\|x\|d\|y\|\).

In \(X\) we define convergence: \(x_n \to x\), if \(\|x_n-x\|\to 0\) in \(Z\) in the sense of item 4 of the present work. It is not difficult to establish the usual properties of convergence in norm: uniqueness of the limit, continuity of the norm, etc.

If \(x_i d x_k\), then for a finite sum or series
\[ \left\|\sum_n x_n\right\|=\sum_n \|x_n\|. \]
This also holds for transfinite series, if convergence in \(Z\) is defined in the corresponding way.

Let us add the requirement of completeness of \(X\) with respect to convergence in norm.

Definition 7. A countably complete \(B_Z\)-module \(X\) will be called a module of type \(B_Z\). If here convergence in \(Z\) is understood as \((o)\)-convergence (or \((bk)\)-convergence, see item 4), then \(X\) will be called a module of type \(B_{\bar K}\) (or, respectively, \(B_{BK}\)). A module of type \(B_K\) is a special case of a module of type \(B_{BK}\) (for \(Z=W\), i.e. \(\|z\|=|z|\), see item 4).*

Theorem 2. A module of type \(B_K\) is a space of type \(B_K\).

Axiom 4 \((^1)\) of a space of type \(B_K\) is fulfilled in any \(B_Z\)-module: let
\[ \|x\|=z_1+z_2,\quad z_i\geqslant 0; \]
take
\[ x_i=\frac{z_i}{z_1+z_2}\,x, \]
then
\[ \|x_i\|=z_i(e_{z_1}\vee e_{z_2})=z_i,\qquad x_1+x_2=e_xx=x. \]

Definition 8. If a \(B_Z\)-module is an absolute \(M_Z\)-module, then we shall call it an absolute \(B_Z\)-module. Such a module of type \(B_Z\) will be called absolute.

Definition 9. An absolute \(B_{\bar Z}\)-module \(\widetilde X\) will be called a \(B\)-extension of the \(B_Z\)-module \(X\), if \(\widetilde X\) is an \(M\)-extension of the \(M_{\bar Z}\)-module \(X\) and the meaning of the norm \(\|x\|\) for \(x\in X\) is the same in \(X\) and \(\widetilde X\).

Theorem 3. For every \(B_Z\)-module there exists its \(B\)-extension.

Define in the \(M\)-extension \(\widetilde X\) the norm
\[ \|\widetilde x\|=(o)\text{-}\lim_{n\to\infty}\|\widetilde x_n\|. \]
Then \(\widetilde X\) satisfies all the requirements of Definition 9.

7. Module over a space of scalar products

Definition 10. Let to each pair \(x,y\) of elements of the \(M_Z\)-module \(X\) there correspond a scalar product \((x,y)\in z\), with: 1) \((x,x)\geqslant 0\); if \((x,x)=0\), then \(x=0\); 2) \((x,y)=(y,x)\); 3) \((x+y,y')=(x,y')+(y,y')\); 4) \((zx,y)=z(x,y)\), if \(zx\) or \(z(x,y)\) exists. We shall call \(X\) an \(H_Z\)-module.

Definition 11. If an \(H_Z\)-module is an absolute \(M_Z\)-module, then we shall call it an absolute \(H_Z\)-module. Such a module of type \(H_Z\) (see Definition 13) will be called absolute.

Definition 12. An absolute \(H_{\bar Z}\)-module \(\widetilde X\) will be called an \(H\)-extension of the \(H_Z\)-module \(X\), if \(\widetilde X\) is an \(M\)-extension of the \(M_Z\)-module \(X\) and the meaning of \((x,y)\) for \(x,y\in X\) is the same in \(X\) and \(\widetilde X\).

Theorem 4. For every \(H_Z\)-module there exists its \(H\)-extension.

Define in the \(M\)-extension \(\widetilde X\) the scalar product
\[ (\widetilde x,\widetilde y)=(o)\text{-}\lim_{n\to\infty}(\widetilde x_n,\widetilde y_n). \]
All the conditions of Definition 12 are fulfilled.

Define in \(X\), as usual, the norm
\[ \|x\|=\sqrt{(x,x)}. \]
Then \(e_x=e_{(x,x)}\). As before it is proved that \(e_xx=x\). Therefore
\[ (x,y)e_y=(x,e_yy)=(x,y). \]

Theorem 5. For an \(H_Z\)-module \(X\) relation (2) is fulfilled; \(X\) is a \(B_Z\)-module.

Let \(x,y\in X\). In \(\widetilde Z\) there exists
\[ \widetilde z=(x,y)/(y,y). \]
Construct in the \(H\)-extension \(\widetilde X\) the element \(x-\widetilde z y\). For the \(H_{\widetilde Z}\)-module \(\widetilde X\), in \(\widetilde Z\) the relations
\[ 0\leqslant (x-\widetilde z y,\ x-\widetilde z y)=(x,x)-(x,y)^2/(y,y) \]
are fulfilled. Multiply by \((y,y)\). Then
\[ (x,x)(y,y)-(x,y)^2e_y\geqslant 0,\qquad (x,x)(y,y)\geqslant (x,y)^2\geqslant 0. \]
Since extraction of the root is an increasing operation \((^1,^2)\), it follows from this the comparability in \(Z\)
\[ |(x,y)|\leqslant \|x\|\|y\|. \tag{2} \]

Verification of the axioms of a \(B_Z\)-module is carried out in the usual way. It is clear that an absolute \(H_Z\)-module is an absolute \(B_Z\)-module and that an \(H\)-extension is simultaneously also a \(B\)-extension, if in \(\widetilde X\) one sets \(\|\tilde x\| = \sqrt{(\tilde x,\tilde x)}\) (taking into account the \((o)\)-continuity of the root \(({}^1)\)).

From comparability (2) it follows that \(e_{(x,y)} \le e_x \wedge e_y\); if \(xdy\), then \(x \perp y\).

Definition 13. A countably complete \(H_Z\)-module \(X\) will be called a module of type \(H_Z\). If here convergence in \(Z\) is understood as \((o)\)-convergence (or \((bk)\)-convergence, see item 4), then \(X\) will be called a module of type \(H_K\) (or, respectively, \(H_{B_K}\)).

Theorem 6. A module of type \(H_Z\) (or respectively \(H_K\), \(H_{B_K}\)) is a module of type \(B_Z\) (respectively \(B_K\), \(B_{B_K}\)). A module of type \(H_K\) is a space of type \(B_K\).

A module of type \(H_K\) is also a special case of a module of type \(H_{B_K}\).

Definition 14. A countable system of elements \(e_n\) of an \(H_Z\)-module will be called orthonormal if \(e_i \perp e_k\), \(\|e_n\| = e_n\), where \(e_n\) are unit elements from the base \((|)\) of the \(K\)-space \(Z\).

As usual, the system \(\{e_n\}\) is called complete in \(X\) if from \(x \perp e_n\) for all \(n\) it follows that \(x=0\).

Theorem 7. If in a module \(X\) of type \(H_K\) there is a complete orthonormal system \(\{e_n\}\) \((n=1,\ldots,\infty)\), then for every element \(x \in X\) the expansion

\[ x=\sum_{n=1}^{\infty}(x,e_n)e_n,\qquad \|x\|^2=\sum_{n=1}^{\infty}(x,e_n)^2 \]

holds.

Let us note that here axiom 5 \(({}^1)\) of the \(K\)-space \(Z\) is used only for countable sets.

If the set of all norms \(\|x\|\) coincides with \(Z\), then from the completeness of \(\{e_n\}\) in \(X\) follows the completeness of \(\{e_n\}\) in \(Z\), i.e. \(\sup\{e_n\}=1\). In particular, disjoint elements may be taken as the \(\{e_n\}\), but the completeness condition of \(\{e_n\}\) is still expressed through orthogonality.

8. Examples. Let \(X\) and \(Z\) consist of real bounded almost everywhere functions \(x(s,t)\) and \(z(s)\), square-summable (\(x\) on \(E\), \(z\) on the projection \(S\) onto the \(s\)-axis of the set \(E\)). We naturally make \(X\) into a module over \(Z\) in the sense of Definition 2. Let \(m\) disjoint measurable sets \(S_i\) have union \(S\); the sets \(T(s)\) are all measurable sections for \(s=\mathrm{const}\) of the set \(E\). Put

\[ (x,y)=\int_{T(s)} x(s,t)y(s,t)\,dt,\qquad \|z\|=\left\{\operatorname*{vrai\,sup}_{S_i}|z(s)|\right\}\quad (i=1,\ldots,m). \]

Then \(X\) becomes a module of type \(H_{B_K}\), but not an absolute one. The proof of the countable completeness of \(X\) is similar to the proof of the countable completeness of the space \(L^2\).

If, however, \(X\) consists of functions \(x(t)\) square-summable on the set \(E_1\), and \(Z\) consists of vectors \(z=\{\zeta_1,\ldots,\zeta_m\}\) (\(m\) fixed), \(e_i\) are measurable, \(e_i \cap e_k=\Lambda\),

\[ \bigcup_{i=1}^{m} e_i=E_1, \]

and the products \(zx\) and \((x,y)\) are defined in the following way:

\[ (zx)(t)=\zeta_k x(t)\quad \text{for } t\in e_k,\qquad (x,y)=\left\{\int_{e_k} xy\,dt\right\}_{k=1,\ldots,m}, \]

then \(X\) becomes an absolute module of type \(H_K\) (convergence in \(Z\) is coordinatewise).

Gorky State University
named after N. I. Lobachevsky

Received
31 III 1961

CITED LITERATURE

1. L. V. Kantorovich, B. Z. Vulikh, A. G. Pinsker, Functional analysis in semiordered spaces, 1950.
2. B. Z. Vulikh, Matem. sborn., 22 (64), 1, 27 (1948).
3. B. Z. Vulikh, Matem. sborn., 33 (75), 2, 343 (1953).
4. S. N. Lugin, DAN, 131, No. 6, 126 (1960).