Abstract

Let    be the Privalov class of holomorphic functions on the open unit disk in the complex plane. The space equipped with the topology given by the metric defined by , , becomes an -algebra. For each , we also consider the countably normed Fréchet algebra of holomorphic functions on which is the Fréchet envelope of the space . Notice that the spaces and have the same topological duals. In this paper, we give a characterization of bounded subsets of the spaces and weakly bounded subsets of the spaces with . If denotes the strong dual space of and denotes the space of complex sequences satisfying the condition , equipped with the topology of uniform convergence on weakly bounded subsets of , then we prove that both set theoretically and topologically. We prove that for each    is a Montel space and that both spaces and are reflexive.

1. Introduction and Preliminaries

Let denote the open unit disk in the complex plane and let denote the boundary of .

The Privalov class () consists of all holomorphic functions on for which These classes were firstly considered by Privalov in [1, page 93], where is denoted as .

Notice that, for , condition (1) defines the Nevanlinna class of holomorphic functions in . Recall that the Smirnov class is the set of all functions holomorphic on such that where is the boundary function of on ; that is, is the radial limit of which exists for almost every . We denote by the classical Hardy space on . It is known (see [2, 3]) that where the above containment relations are proper.

Notice that Privalov in [1, page 98] established the inner-outer factorization theorem for the spaces (for another proof see [4]). The study of the spaces on the unit disk was continued in 1977 by Stoll [5] (with the notation in [5]). The topological and functional properties of these spaces were extensively studied in [6]. Different topologies on the spaces were considered and compared in [7] with related applications. Complex-linear isometries of are investigated in [8]. Motivated by some results of Matsugu [9], in [10] the structure of closed weakly dense ideals in Privalov spaces was studied. Notice that the structure of maximal ideals of the algebras and their Fréchet envelopes was investigated in [11]. The interpolation problems for the spaces are treated in [12].

Stoll [5, Theorem  4.2] showed that the space (with the notation in [5]) with the topology given by the metric defined by becomes an -algebra.

Recall that the function defined on the Smirnov class by (5) with induces the metric topology on . Yanagihara [13] showed that, under this topology, is an -space.

Recently, in [14] the author of this note characterized some topological properties of the spaces . For these purposes, the fact that the metric defined on as with and induces on the space the same topological structure as the initial metric given by (5) (see [14, Theorem  16]) is used.

Furthermore, in connection with the spaces , Stoll [5] (also see [15] and [10, Section  3]) also studied the spaces (with the notation in [5]), consisting of those functions holomorphic on for which where Stoll [5, Theorem  3.2] proved that the space with the topology given by the family of seminorms defined for as is a countably normed Fréchet algebra.

Here, as always in the sequel, we will need some Stoll’s results concerning the spaces only with , and, hence, we will assume that be any fixed number.

Theorem 1 (see [5, Theorem  2.2]). Suppose that is a holomorphic function on . Then the following statements are equivalent: (a).(b)There exists a sequence of positive real numbers with such that (c)For any ,

Remark 2. Notice that, in view of Theorem 1 ((a)(c)), by (10) the family of seminorms on is well defined.

Recall that a locally convex -space is called a Fréchet space, and a Fréchet algebra is a Fréchet space that is an algebra in which multiplication is continuous.

Notice that the space is not locally convex (see [15, Theorem  4.2] and [16, Corollary]), and, hence, is properly contained in . Moreover, is not locally bounded (see [17, Theorem  1.1]). The most important connection between spaces and is given by the following result.

Theorem 3 (see [5, Theorem  4.3]). For any fixed the following assertions hold: (a) is a dense subspace of .(b)The topology on defined by the family of seminorms (10) is weaker than the topology on given by the metric defined by (5).

Remark 4. For , the space has been denoted by and has been studied by Yanagihara in [13, 18]. It was shown in [13, 18] that is actually the containing Fréchet space for ; that is, with the initial topology embeds densely into , under the natural inclusion, and and the Smirnov class have the same topological duals.

Observe that the space topologised by the family of seminorms given by (10) is metrizable by the metric defined as The following Stoll’s result describes the topological dual of the space .

Theorem 5 (see [5, Theorem  3.3]). If is a continuous linear functional on , then there exists a sequence of complex numbers with , for some , such that where , with convergence being absolute. Conversely, if is a sequence of complex numbers for which then (14) defines a continuous linear functional on .

Let us recall that if is an -space whose topological dual (the set of all continuous linear functionals on ) separates the points of , then its Fréchet envelope is defined to be the completion of the space , where is the strongest locally convex (necessarily metrizable) topology on that is weaker than . In fact, it is known that is equal to the Mackey topology of the dual pair , that is, to the unique maximal locally convex topology on for which still has dual space (see [19, Theorem  1]). For each metrizable locally convex topology on , is a Mackey space; that is, coincides with the Mackey topology of the dual pair (see [20, Corollary  22.3, page 210]).

Eoff [15, the proof of Theorem  4.2] showed that the topology of , (resp., ), is stronger than that of the Fréchet envelope of (resp., ). As an immediate consequence of this result, we obtain the following statements.

Theorem 6 (see [15, Theorem  4.2, the case ]). For each , is the Fréchet envelope of .

Theorem 7 (see [21, Theorem  2]; also see [14, Theorem  17]). The spaces and have the same dual spaces in the sense that every continuous linear functional on (given by (14)) is restricted to one on , and every continuous linear functional on extends continuously to one on .
Hence, the dual spaces of and can be identified with the collection of complex sequences satisfying the growth condition (15).

Remark 8. Theorem 7 is proved in [21, Theorem  1] directly, by using the characterization of multipliers from to . Notice also that the dual space of the Smirnov class is described by Yanagihara in [13] and applying another method by McCarthy in [22].

Remark 9. Recall that we may introduce the weak topology on in the usual way. The basic weak neighborhoods of zero are defined by , where and are arbitrary, and are arbitrary continuous linear functionals on . The weak topology of is locally convex, and, hence, by [20, Corollary  17.3, page 154], a linear functional on is weakly continuous if and only if it is continuous with respect to the initial metric topology .

In Section 2, we give a characterization of bounded subsets of the spaces (Theorem 10). As an application, we obtain a characterization of weakly bounded subsets of the spaces (Theorem 11). In Section 3, we prove that both set theoretically and topologically (Theorem 12). Here denotes the strong dual space of , and is the space of complex sequences satisfying the growth condition (15), equipped with the topology of uniform convergence on weakly bounded subsets of . Finally, we prove that is a Montel space (Theorem 13) and that both spaces and are reflexive (Theorem 14).

2. Bounded Subsets of the Spaces

The following result characterizes bounded sets of the space .

Theorem 10. Let and let be a subset of . Then the following assertions are equivalent:(i) is a bounded subset of .(ii) is a relatively compact subset of .(iii)There exists a constant depending on and a sequence of positive real numbers such that and

Proof. (ii)⇒(i): it follows immediately from the fact that every relatively compact set in a topological vector space is bounded.
(iii)⇒(i): for given choose a positive integer such that for each . Then by condition (iii) for every function we have where is a constant depending only on . Hence, is a bounded set in every normed space , , and, thus, is a bounded subset of .
(i)⇒(iii): suppose that is a bounded set in . For arbitrary and the set defined as is a neighbourhood of 0 in the space . Since, by the assumption, is a bounded set in , there exists such that . This yields From (19) we find that Then putting into (20), we find that Since is arbitrary, from (22) we immediately have which immediately yields (16).
(iii)⇒(ii): since in a metric space compactness and sequential compactness are equivalent, it is necessary to show that a set satisfying condition (16) is a relatively compact subset of ; that is, for every sequence there exists a subsequence of which is convergent in . We will inductively construct such a subsequence of a sequence in the following way. Take Since, by assumption (16), it follows that there is a subsequence of ( denotes the th term of this subsequence) such that the appropriate subsequence of a sequence is convergent; assume that Take ; that is, denote by the first term of the obtained subsequence .
Since, by assumption (16), we have it follows that there exists a subsequence of ( denotes the th term of this subsequence) such that the corresponding subsequence of a sequence is convergent; assume that Put ; that is, denote by the first term of the obtained subsequence which is different from .
By continuing the above diagonal procedure and taking into account that in view of (16) and (24) after steps we obtain a subsequence of ( denotes the th term of this subsequence) such that the corresponding subsequence of converges; assume that Take ; that is, denote by the first term of the obtained subsequence which is different from for all .
In this way we have obviously constructed a subsequence of a sequence such that Furthermore, by the above construction, is a convergent sequence for any fixed ; assume that Let be a function defined as where the coefficients are defined by (26) and (32). Since by (29) and (30) we have it follows from Theorem 1 that belongs to the space .
It remains to show that in as . Let be any fixed positive real number. Choose a nonnegative integer such that and also choose a nonnegative integer for which Take . For every choose such that Take . Then from (32), (33), (35), (36), and (37) it follows that for each there holds Therefore, in the space . From this and the fact that is arbitrary we conclude that in . Hence, is a relatively compact subset of . This completes the proof.

Theorem 11. Let be a subset of . is a weakly bounded subset of if and only if there is a constant depending on and a sequence of positive real numbers with such that

Proof. The proof follows immediately from Theorem 7, the equivalence (i)(iii) of Theorem 10, and the well known fact that in every locally convex topological vector space a set is bounded if and only if it is weakly bounded (see, e.g., [23, page 68]).

3. The Dual Spaces of the Spaces and

Let be a topological vector space over the field . Consider its strong dual space which consists of all continuous linear functionals and is equipped with the usual strong topology . Recall that in the case when is a locally convex space, the strong topology on coincides with the topology of uniform convergence on bounded subsets in . In this case the topology coincides with the topology of uniform convergence on bounded sets in , that is, with the topology on generated by the seminorms of the form where runs over the family of all bounded sets in (see, e.g., [24, 21, page 21]).

The space is a locally convex topological vector space, so one can consider its strong dual space which is called strong bidual space for . More precisely, the strong bidual space for consists of all continuous linear functionals and is equipped with the strong topology . Furthermore, is called reflexive if the evaluation map defined as (, ) is surjective and continuous (in this case is an isomorphism of topological vector spaces) (see [20, page 189]). Accordingly, is a reflexive space if and only if coincides with the continuous dual of its continuous dual space , both as linear space and as topological space.

In particular, in the case of locally convex space , the strong topology coincides with the topology on (see Theorem 7) generated by the family of seminorms of the form where is an arbitrary bounded subset of .

Furthermore, denote by the space of complex sequences satisfying the growth condition (15), equipped with the topology of uniform convergence on weakly bounded subsets of . (Let us recall that a subset is weakly bounded if it is bounded with respect to the weak topology on described in Remark 9.) More precisely, this topology on the space is defined by the family of seminorms of the form where is an arbitrary weakly bounded subset of .