# Topological and Functional Properties of SomeF-Algebras of Holomorphic Functions

Authors: Romeo Meštrović
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

LetN
p  (
1
<
p
<
∞
) be the Privalov class of holomorphic functions on the open unit disk D in the complex plane. The space N
p equipped with the topology given by the metric d
p defined by d
p
(
f
,
g
)
=
(
∫
0
2
π
‍
(
l
o
g
(
1
+
|
f
∗
(
e
i
θ
)
-
g
∗
(
e
i
θ
)
|
)
)
p
(
d
θ
/
2
π
)
)
1
/
p, f
,
g
∈
N
p, becomes an F-algebra. For each p
>
1, we also consider the countably normed Fréchet algebra F
p of holomorphic functions on D which is the Fréchet envelope of the space N
p. Notice that the spaces F
p and N
p have the same topological duals. In this paper, we give a characterization of bounded subsets of the spaces F
p and weakly bounded subsets of the spaces N
p with p
>
1. If (
F
p
)
∗ denotes the strong dual space of F
p and N
p
w
∗ denotes the space S
p of complex sequences γ
=
{
γ
n
}
n satisfying the condition γ
n
=
O
e
x
p
-
c
n
1
/
(
p
+
1
), equipped with the topology of uniform convergence on weakly bounded subsets of N
p, then we prove that F
p
∗
=
N
p
w
∗ both set theoretically and topologically. We prove that for each p
>
1  F
p is a Montel space and that both spaces F
p and (
F
p
)
∗ are reflexive.


---

## Body

## 1. Introduction and Preliminaries

LetD denote the open unit disk in the complex plane and let T denote the boundary of D.ThePrivalov class
N
p (1
<
p
<
∞) consists of all holomorphic functions f on D for which (1)
sup
0
<
r
<
1
⁡
∫
0
2
π
log
+
⁡
f
r
e
i
θ
p
d
θ
2
π
<
+
∞
.These classes were firstly considered by Privalov in [1, page 93], where N
p is denoted as A
q.Notice that, forp
=
1, condition (1) defines theNevanlinna class
N of holomorphic functions in D. Recall that theSmirnov class
N
+ is the set of all functions f holomorphic on D such that (2)
lim
r
→
1
⁡
∫
0
2
π
log
+
⁡
f
r
e
i
θ
d
θ
2
π
=
∫
0
2
π
log
+
⁡
f
∗
e
i
θ
d
θ
2
π
<
+
∞
,where f
∗ is the boundary function of f on T; that is, (3)
f
∗
e
i
θ
=
lim
r
→
1
-
⁡
f
r
e
i
θis theradial limit of f which exists for almost every e
i
θ. We denote by H
q
(
0
<
q
≤
∞
) the classicalHardy space on D. It is known (see [2, 3]) that (4)
N
r
⊂
N
p
r
>
p
,
⋃
q
>
0
H
q
⊂
⋂
p
>
1
N
p
,
⋃
p
>
1
N
p
⊂
N
+
⊂
N
,where the above containment relations are proper.Notice that Privalov in [1, page 98] established the inner-outer factorization theorem for the spaces N
p
(
1
<
p
<
∞
) (for another proof see [4]). The study of the spaces N
p
(
1
<
p
<
∞
) on the unit disk was continued in 1977 by Stoll [5] (with the notation (
l
o
g
+
H
)
α in [5]). The topological and functional properties of these spaces were extensively studied in [6]. Different topologies on the spaces N
p were considered and compared in [7] with related applications. Complex-linear isometries of N
p are investigated in [8]. Motivated by some results of Matsugu [9], in [10] the structure of closed weakly dense ideals in Privalov spaces N
p
(
1
<
p
<
∞
) was studied. Notice that the structure of maximal ideals of the algebras N
p and their Fréchet envelopes F
p
(
1
<
p
<
∞
) was investigated in [11]. The interpolation problems for the spaces N
p are treated in [12].Stoll [5, Theorem  4.2] showed that the space N
p (with the notation (
l
o
g
+
H
)
α in [5]) with the topology given by the metric d
p defined by (5)
d
p
f
,
g
=
∫
0
2
π
log
⁡
1
+
f
∗
e
i
θ
-
g
∗
e
i
θ
p
d
θ
2
π
1
/
p
,
f
,
g
∈
N
p
,becomes an F-algebra.Recall that the functiond
1
=
d defined on the Smirnov class N
+ by (5) with p
=
1 induces the metric topology on N
+. Yanagihara [13] showed that, under this topology, N
+ is an F-space.Recently, in [14] the author of this note characterized some topological properties of the spaces N
p
(
1
<
p
<
∞
). For these purposes, the fact that the metric ρ
p defined on N
p as (6)
ρ
p
f
,
g
=
∫
0
2
π
log
p
⁡
1
+
M
f
-
g
θ
d
θ
2
π
1
/
p
,with f
,
g
∈
N
p and (7)
M
f
θ
=
sup
0
⩽
r
<
1
⁡
f
r
e
i
θ
,induces on the space N
p the same topological structure as the initial metric d
p given by (5) (see [14, Theorem  16]) is used.Furthermore, in connection with the spacesN
p
(
1
<
p
<
∞
), Stoll [5] (also see [15] and [10, Section  3]) also studied the spaces F
q
(
0
<
q
<
∞
) (with the notation F
1
/
q in [5]), consisting of those functions f holomorphic on D for which (8)
lim
r
→
1
⁡
1
-
r
1
/
q
log
+
⁡
M
∞
r
,
f
=
0
,where (9)
M
∞
r
,
f
=
max
z
≤
r
⁡
f
z
.Stoll [5, Theorem  3.2] proved that the space F
q with the topology given by the family of seminorms ·
q
,
c
c
>
0 defined for f
∈
F
q as (10)
f
q
,
c
≔
∑
n
=
0
∞
a
n
exp
⁡
-
c
n
1
/
q
+
1
<
∞is a countably normedFréchet algebra.Here, as always in the sequel, we will need some Stoll’s results concerning the spacesF
q only with 1
<
q
<
∞, and, hence, we will assume that q
=
p
>
1 be any fixed number.Theorem 1 (see [5, Theorem  2.2]).
Suppose thatf
(
z
)
=
∑
n
=
0
∞
‍
a
n
z
n is a holomorphic function on D. Then the following statements are equivalent: (a)
f
∈
F
p.
(b)
There exists a sequence{
c
n
}
n of positive real numbers with c
n
→
0 such that (11)
a
n
≤
exp
⁡
c
n
n
1
/
p
+
1
,
n
=
0,1
,
2
,
…
.
(c)
For anyc
>
0, (12)
f
p
,
c
≔
∑
n
=
0
∞
a
n
exp
⁡
-
c
n
1
/
p
+
1
<
∞
.Remark 2.
Notice that, in view of Theorem1 ((a)⇔(c)), by (10) the family of seminorms ·
p
,
c
c
>
0 on F
p is well defined.Recall that a locally convexF-space is called aFréchet space, and aFréchet algebra is a Fréchet space that is an algebra in which multiplication is continuous.Notice that the spaceN
p is not locally convex (see [15, Theorem  4.2] and [16, Corollary]), and, hence, N
p is properly contained in F
p. Moreover, N
p is not locally bounded (see [17, Theorem  1.1]). The most important connection between spaces N
p and F
p is given by the following result.Theorem 3 (see [5, Theorem  4.3]).
For any fixedp
>
1 the following assertions hold: (a)
N
p is a dense subspace of F
p.
(b)
The topology onF
p defined by the family of seminorms (10) is weaker than the topology on N
p given by the metric d
p defined by (5).Remark 4.
Forp
=
1, the space F
1 has been denoted by F
+ and has been studied by Yanagihara in [13, 18]. It was shown in [13, 18] that F
+ is actually the containing Fréchet space for N
+; that is, N
+ with the initial topology embeds densely into F
+, under the natural inclusion, and F
+ and the Smirnov class N
+ have the same topological duals.Observe that the spaceF
p topologised by the family of seminorms ·
p
,
c
c
>
0 given by (10) is metrizable by the metric λ
p defined as (13)
λ
p
f
,
g
=
∑
n
=
1
∞
2
-
n
f
-
g
p
,
1
/
n
p
/
p
+
1
1
+
f
-
g
p
,
1
/
n
p
/
p
+
1
,
f
,
g
∈
F
p
.The following Stoll’s result describes the topological dual of the space F
p.Theorem 5 (see [5, Theorem  3.3]).
Ifγ is a continuous linear functional on F
p, then there exists a sequence {
γ
n
}
n of complex numbers with γ
n
=
O
(
e
x
p
-
c
n
1
/
(
p
+
1
)
), for some c
>
0, such that (14)
γ
f
=
∑
n
=
0
∞
a
n
γ
n
,where f
(
z
)
=
∑
n
=
0
∞
‍
a
n
z
n
∈
F
p, with convergence being absolute. Conversely, if {
γ
n
}
n is a sequence of complex numbers for which (15)
γ
n
=
O
exp
⁡
-
c
n
1
/
p
+
1
,then (14) defines a continuous linear functional on F
p.Let us recall that ifX
=
(
X
,
τ
) is an F-space whose topological dual (the set of all continuous linear functionals on X) X
∗ separates the points of X, then itsFréchet envelope
X
^ is defined to be the completion of the space (
X
,
τ
c
), where τ
c is the strongest locally convex (necessarily metrizable) topology on X that is weaker than τ. In fact, it is known that τ
c is equal to theMackey topology of the dual pair (
X
,
X
∗
), that is, to the unique maximal locally convex topology on X for which X still has dual space X
∗ (see [19, Theorem  1]). For each metrizable locally convex topology τ on X, (
X
,
τ
) is aMackey space; that is, τ coincides with the Mackey topology of the dual pair (
X
,
X
∗
) (see [20, Corollary  22.3, page 210]).Eoff [15, the proof of Theorem  4.2] showed that the topology of F
p, p
>
1 (resp., F
1
=
F
+), is stronger than that of the Fréchet envelope of N
p (resp., N
+). As an immediate consequence of this result, we obtain the following statements.Theorem 6 (see [15, Theorem  4.2, the case p
>
1]).
For eachp
>
1, F
p is the Fréchet envelope of N
p.Theorem 7 (see [21, Theorem  2]; also see [14, Theorem  17]).
The spacesN
p and F
p have the same dual spaces in the sense that every continuous linear functional on F
p (given by (14)) is restricted to one on N
p, and every continuous linear functional on N
p extends continuously to one on F
p.
Hence, the dual spaces ofN
p and F
p can be identified with the collection S
p of complex sequences {
γ
n
} satisfying the growth condition (15).Remark 8.
Theorem7 is proved in [21, Theorem  1] directly, by using the characterization of multipliers from N
q to H
∞. Notice also that the dual space of the Smirnov class N
+ is described by Yanagihara in [13] and applying another method by McCarthy in [22].Remark 9.
Recall that we may introducethe weak topology on N
p in the usual way. The basic weak neighborhoods of zero are defined by V
φ
1
,
…
,
φ
n
;
ɛ
=
f
∈
N
p
:
|
φ
i
(
f
)
|
<
ɛ
,
i
=
1
,
…
,
n, where ɛ
>
0 and n
∈
N are arbitrary, and φ
1
,
…
,
φ
n are arbitrary continuous linear functionals on N
p. The weak topology of N
p is locally convex, and, hence, by [20, Corollary  17.3, page 154], a linear functional on N
p is weakly continuous if and only if it is continuous with respect to the initial metric topology d
p.In Section2, we give a characterization of bounded subsets of the spaces F
p
(
1
<
p
<
∞
) (Theorem 10). As an application, we obtain a characterization of weakly bounded subsets of the spaces N
p (Theorem 11). In Section 3, we prove that F
p
∗
=
N
p
w
∗ both set theoretically and topologically (Theorem 12). Here (
F
p
)
∗ denotes the strong dual space of F
p, and N
p
w
∗ is the space S
p of complex sequences γ
=
{
γ
n
}
n satisfying the growth condition (15), equipped with the topology of uniform convergence on weakly bounded subsets of N
p. Finally, we prove that F
p is a Montel space (Theorem 13) and that both spaces F
p and (
F
p
)
∗ are reflexive (Theorem 14).


## 2. Bounded Subsets of the SpacesF
p

The following result characterizes bounded sets of the spaceF
p.Theorem 10.
Letp
>
1 and let E be a subset of F
p. Then the following assertions are equivalent:(i)
E is a bounded subset of F
p.
(ii)
E is a relatively compact subset of F
p.
(iii)
There exists a constantA
>
0 depending on E and a sequence {
c
n
} of positive real numbers such that c
n
↓
0 and (16)
a
n
≤
A
exp
⁡
c
n
n
1
/
p
+
1
f
o
r
e
a
c
h
f
u
n
c
t
i
o
n
f
z
=
∑
n
=
0
∞
a
n
z
n
∈
E
.Proof.
(ii)⇒(i): it follows immediately from the fact that every relatively compact set in a topological vector space is bounded.
(iii)⇒(i): for givenc
>
0 choose a positive integer m such that c
n
<
c
/
2 for each n
>
m. Then by condition (iii) for every function f
(
z
)
=
∑
n
=
0
∞
‍
a
n
z
n
∈
E we have (17)
f
p
,
c
=
∑
n
=
0
∞
a
n
exp
⁡
-
c
n
1
/
p
+
1
≤
A
∑
n
=
0
∞
exp
⁡
c
n
-
c
n
1
/
p
+
1
≤
A
∑
n
=
0
m
exp
⁡
c
n
-
c
n
1
/
p
+
1
+
∑
n
=
m
+
1
∞
exp
⁡
-
c
2
n
1
/
p
+
1
=
K
c
=
K
,where K is a constant depending only on c. Hence, E is a bounded set in every normed space (
F
p
,
·
p
,
c
), c
>
0, and, thus, E is a bounded subset of F
p.
(i)⇒(iii): suppose thatE is a bounded set in F
p. For arbitrary c
>
0 and η
>
0 the set V defined as (18)
V
=
g
∈
F
p
:
g
p
,
c
<
ηis a neighbourhood of 0 in the space F
p. Since, by the assumption, E is a bounded set in F
p, there exists α
>
0 such that α
E
⊂
V. This yields (19)
∑
n
=
0
∞
α
a
n
exp
⁡
-
c
n
1
/
p
+
1
<
η
f
o
r
e
a
c
h
f
z
=
∑
n
=
0
∞
a
n
z
n
∈
E
.From (19) we find that (20)
a
n
≤
η
α
exp
⁡
c
n
1
/
p
+
1
f
o
r
e
a
c
h
n
∈
N
.Then putting (21)
a
n
∗
=
sup
⁡
a
n
f
:
f
z
=
∑
n
=
0
∞
a
n
f
z
n
∈
Einto (20), we find that (22)
a
n
∗
≤
η
α
exp
⁡
c
n
1
/
p
+
1
f
o
r
e
a
c
h
n
∈
N
.Since c
>
0 is arbitrary, from (22) we immediately have (23)
a
n
∗
=
O
exp
⁡
o
n
1
/
p
+
1
,which immediately yields (16).
(iii)⇒(ii): since in a metric space compactness and sequential compactness are equivalent, it is necessary to show that a setE satisfying condition (16) is a relatively compact subset of F
p; that is, for every sequence {
f
n
}
⊂
E there exists a subsequence of {
f
n
} which is convergent in F
p. We will inductively construct such a subsequence of a sequence {
f
n
} in the following way. Take (24)
f
n
z
=
∑
k
=
0
∞
a
k
n
z
k
f
o
r
e
a
c
h
n
∈
N
.Since, by assumption (16), (25)
a
0
n
≤
A
exp
⁡
c
0
f
o
r
e
a
c
h
n
∈
N
,it follows that there is a subsequence {
f
n
(
0
)
} of {
f
n
} (f
n
(
0
) denotes the nth term of this subsequence) such that the appropriate subsequence {
a
0
n
(
0
)
} of a sequence {
a
0
n
} is convergent; assume that (26)
lim
n
→
∞
⁡
a
0
n
0
=
a
0
.Take f
i
0
=
f
0
(
0
); that is, denote by f
i
0 the first term of the obtained subsequence {
f
n
(
0
)
}.
Since, by assumption (16), we have (27)
a
1
n
≤
A
exp
⁡
c
1
f
o
r
e
a
c
h
n
∈
N
,it follows that there exists a subsequence {
f
n
(
1
)
} of {
f
n
(
0
)
} (f
n
(
1
) denotes the nth term of this subsequence) such that the corresponding subsequence {
a
1
n
(
1
)
} of a sequence {
a
0
n
(
0
)
} is convergent; assume that (28)
lim
n
→
∞
⁡
a
1
n
1
=
a
1
.Put f
i
1
=
f
1
(
1
); that is, denote by f
i
1 the first term of the obtained subsequence {
f
n
(
1
)
} which is different from f
i
0.
By continuing the above diagonal procedure and taking into account that in view of (16) and (24) (29)
a
k
n
≤
A
exp
⁡
c
k
k
1
/
p
+
1
f
o
r
e
a
c
h
n
∈
N
,after s
+
1 steps we obtain a subsequence {
f
n
(
s
)
} of {
f
n
(
s
-
1
)
} (f
n
(
s
) denotes the nth term of this subsequence) such that the corresponding subsequence {
a
s
n
(
s
)
} of {
a
(
s
-
1
)
n
(
s
-
1
)
} converges; assume that (30)
lim
n
→
∞
⁡
a
s
n
s
=
a
s
.Take f
i
s
=
f
s
(
s
); that is, denote by f
i
s the first term of the obtained subsequence {
f
n
(
s
)
} which is different from {
f
n
(
j
)
} for all j
=
0,1
,
…
,
s
-
1.
In this way we have obviously constructed a subsequence{
f
i
n
} of a sequence {
f
n
} such that (31)
f
i
n
z
=
∑
k
=
0
∞
a
k
n
n
z
k
f
o
r
e
a
c
h
n
∈
N
.Furthermore, by the above construction, {
a
k
n
(
k
)
}
n is a convergent sequence for any fixed k
∈
N; assume that (32)
lim
n
→
∞
⁡
a
k
n
k
=
a
k
f
o
r
e
a
c
h
f
i
x
e
d
k
∈
N
.Let f be a function defined as (33)
f
z
=
∑
k
=
0
∞
a
k
z
k
,where the coefficients a
k are defined by (26) and (32). Since by (29) and (30) we have (34)
a
k
≤
A
exp
⁡
c
k
k
1
/
p
+
1
f
o
r
e
a
c
h
k
∈
N
,it follows from Theorem 1 that f belongs to the space F
p.
It remains to show thatf
i
n
→
f in F
p as n
→
∞. Let c
>
0 be any fixed positive real number. Choose a nonnegative integer k
1
=
k
1
(
c
) such that (35)
c
k
<
c
2
f
o
r
e
a
c
h
k
>
k
1
,and also choose a nonnegative integer k
2
=
k
2
(
c
) for which (36)
∑
k
=
k
2
+
1
∞
exp
⁡
-
c
2
k
1
/
p
+
1
<
ɛ
4
A
.Take k
0
=
m
a
x
{
k
1
,
k
2
}. For every k
∈
{
0,1
,
…
,
k
0
} choose n
k
∈
N such that (37)
a
k
n
n
-
a
k
<
ɛ
2
k
0
+
1
exp
⁡
c
k
1
/
p
+
1
f
o
r
e
a
c
h
n
>
n
k
.Take m
=
m
a
x
{
n
k
:
k
=
0,1
,
…
,
k
0
}. Then from (32), (33), (35), (36), and (37) it follows that for each n
>
m there holds (38)
f
i
n
-
f
p
,
c
=
∑
k
=
0
∞
a
k
n
n
-
a
k
exp
⁡
-
c
k
1
/
p
+
1
=
∑
k
=
0
k
0
a
k
n
n
-
a
k
exp
⁡
-
c
k
1
/
p
+
1
+
∑
k
=
k
0
+
1
∞
a
k
n
n
-
a
k
exp
⁡
-
c
k
1
/
p
+
1
≤
ɛ
2
k
0
+
1
k
0
+
1
+
2
A
∑
k
=
k
0
+
1
∞
exp
⁡
c
k
-
c
k
1
/
p
+
1
≤
ɛ
2
+
2
A
∑
k
=
k
2
+
1
∞
exp
⁡
-
c
2
k
1
/
p
+
1
≤
ɛ
2
+
ɛ
2
=
ɛ
.Therefore, f
i
n
→
f in the space (
F
p
,
·
p
,
c
). From this and the fact that c
>
0 is arbitrary we conclude that f
i
n
→
f in F
p. Hence, E is a relatively compact subset of F
p. This completes the proof.Theorem 11.
LetE be a subset of N
p. E is a weakly bounded subset of N
p if and only if there is a constant A
>
0 depending on E and a sequence {
c
n
} of positive real numbers with c
n
↓
0 such that (39)
a
n
≤
A
exp
⁡
c
n
n
1
/
p
+
1
f
o
r
e
a
c
h
f
z
=
∑
n
=
0
∞
a
n
z
n
∈
E
.Proof.
The proof follows immediately from Theorem7, 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 SpacesF
p and N
p

LetX be a topological vector space over the field C. Consider itsstrong dual space
X
∗ which consists of all continuous linear functionals f
:
X
→
C and is equipped with the usual strong topology β
(
X
∗
,
X
). Recall that in the case when X is a locally convex space, the strong topology β
(
X
∗
,
X
) on X
∗ coincides with the topology of uniform convergence on bounded subsets in X. In this case the topology β
(
X
∗
,
X
) coincides with the topology of uniform convergence on bounded sets in X, that is, with the topology on X
∗ generated by the seminorms of the form (40)
φ
B
=
sup
x
∈
B
⁡
φ
x
,
φ
∈
X
∗
,where B runs over the family of all bounded sets in X (see, e.g., [24, § 21, page 21]).The spaceX
∗ is a locally convex topological vector space, so one can consider its strong dual space X
∗
∗
≔
(
X
∗
)
∗ which is calledstrong bidual space for X. More precisely, the strong bidual space X
∗
∗ for X consists of all continuous linear functionals h
:
X
∗
→
C and is equipped with the strong topology β
(
X
∗
∗
,
X
∗
). Furthermore, X is calledreflexive if the evaluation map J
:
X
→
(
X
)
∗
∗ defined as J
(
x
)
(
f
)
=
f
(
x
) (x
∈
X, f
∈
X
∗) is surjective and continuous (in this case J is an isomorphism of topological vector spaces) (see [20, page 189]). Accordingly, X is a reflexive space if and only if X coincides with the continuous dual of its continuous dual space X
∗, both as linear space and as topological space.In particular, in the case of locally convex spaceF
p, the strong topology β
(
(
F
p
)
∗
,
F
p
) coincides with the topology on (
F
p
)
∗
=
S
p (see Theorem 7) generated by the family of seminorms of the form (41)
γ
B
=
sup
f
∈
B
⁡
γ
f
,
γ
∈
F
p
∗
=
S
p
,where B is an arbitrary bounded subset of F
p.Furthermore, denote byN
p
w
∗ the space S
p of complex sequences γ
=
{
γ
n
}
n satisfying the growth condition (15), equipped with the topology of uniform convergence on weakly bounded subsets of N
p. (Let us recall that a subset E
⊂
N
p isweakly bounded if it is bounded with respect to the weak topology on N
p described in Remark 9.) More precisely, this topology on the space S
p is defined by the family of seminorms of the form (42)
γ
E
′
=
sup
f
∈
E
⁡
γ
f
,
γ
∈
N
p
∗
=
S
p
,where E is an arbitrary weakly bounded subset of N
p.Notice that (see Theorems5 and 7) for γ
=
{
γ
n
}
n
=
0
∞
∈
S
p and f
(
z
)
=
∑
n
=
0
∞
‍
a
n
z
n
∈
F
p (or f
∈
N
p) (43)
γ
f
=
∑
n
=
0
∞
a
n
γ
n
.Then we have the following result which is analogous to Yanagihara’s result [25, Theorem  3] concerning the spaces F
+ and N
+.Theorem 12.
For eachp
>
1
F
p
∗
=
N
p
w
∗ both set theoretically and topologically.Proof.
Proof follows immediately from Theorem7, the equivalence (i)⇔(iii) of Theorems 10 and 11 in view of (41)–(43).A locally convex topological vector space is called barrelled if every closed, absolutely convex, absorbing set is a neighborhood of zero. It is known that everyF-space is barreled (see [24, page 263 § 21, the assertion (
3
)]). The most important fact about barrelled spaces is that every pointwise bounded family of continuous linear functionals is equicontinuous. Furthermore, each barrelled space is a Mackey space (see, e.g., [20, pages 171–173]). Recall that aMontel space is a barrelled topological vector space in which every closed bounded set is compact. It is known that every Montel space is reflexive (e.g., see [24, page 369]).Theorem 13.
F
p is a Montel space for each p
>
1.Proof.
As noticed above, everyF-space is barreled (see [24, page 263 § 21, 5.(
3
)]). Furthermore, by Theorem 10, every closed bounded subset of F
p is compact. Therefore, F
p is a Montel space.Theorem 14.
The spaceF
p is reflexive. Hence, (
F
p
)
∗ is reflexive.Proof.
The first assertion follows from Theorem13 and the fact that every Montel space is reflexive (see [24, page 369, the assertion (
1
)]). The second assertion follows from the first assertion and the fact that the strong dual of a reflexive F-space is also reflexive (see [20, page 303, the assertion (
5
)]).Theorem 15.
For eachp
>
1
F
p
∗
∗
=
F
p.Proof.
The assertion follows immediately from Theorem14.



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*Source: 850709-2015-09-17.xml*