Compact Variation, Compact Subdifferetiability and Indefinite Bochner Integral
The notions of compact convex variation and compact convex subdifferential for the mappings from a segment into a locally convex space (LCS) are studied. In the case of an arbitrary complete LCS, each indefinite Bochner integral has compact variation and each strongly absolutely continuous and compa...
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| Цитувати: | Compact variation, compact subdifferentiability and indefinite Bochner integral / I.V. Orlov, F.S. Stonyakin // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 1. — С. 74-90. — Бібліогр.: 24 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859669721344901120 |
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| author | Orlov, I.V. Stonyakin, F.S. |
| author_facet | Orlov, I.V. Stonyakin, F.S. |
| citation_txt | Compact variation, compact subdifferentiability and indefinite Bochner integral / I.V. Orlov, F.S. Stonyakin // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 1. — С. 74-90. — Бібліогр.: 24 назв. — англ. |
| collection | DSpace DC |
| description | The notions of compact convex variation and compact convex subdifferential for the mappings from a segment into a locally convex space (LCS) are studied. In the case of an arbitrary complete LCS, each indefinite Bochner integral has compact variation and each strongly absolutely continuous and compact subdifferentiable a.e. mapping is an indefinite Bochner integral.
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| fulltext |
Methods of Functional Analysis and Topology
Vol. 15 (2009), no. 1, pp. 74–90
COMPACT VARIATION, COMPACT SUBDIFFERENTIABILITY AND
INDEFINITE BOCHNER INTEGRAL
I. V. ORLOV AND F. S. STONYAKIN
Abstract. The notions of compact convex variation and compact convex subdiffer-
ential for the mappings from a segment into a locally convex space (LCS) are studied.
In the case of an arbitrary complete LCS, each indefinite Bochner integral has com-
pact variation and each strongly absolutely continuous and compact subdifferentiable
a.e. mapping is an indefinite Bochner integral.
0. Introduction and preliminaries
As it is well known, the main difference between properties of a Bochner integral
([1]–[6]) on a segment and the classical Lebesgue integral is that the class of indefinite
Bochner integrals (B)
∫ x
a
f(t) dt is essentially smaller than the one of absolutely conti-
nuous mappings. Recall ([2], Theorems 3.7.11, 3.8.5, 3.8.6) that, in the case of Banach-
valued mappings, the class of indefinite Bochner integrals coincides with the one of
absolutely continuous and a.e. differentiable mappings.
This problem was resulted, in particular, in the notion of a Radon-Nikodym property
(RNP). Each absolutely continuous mapping taking values in a space with RNP (or
Radon-Nikodym space) is an indefinite Bochner integral. This class of spaces plays
an important role in the modern theory of Banach spaces and locally convex spaces,
especially in connection with probability theory, harmonic analysis and topology ([7]–
[9]).
In this paper, a rather different way is chosen. Not restricting, as far as possible, the
class of spaces under consideration, a description of the absolute continuous mapping that
are indefinite Bochner integrals is given by means of the nonsmooth analysis. To this end,
the notion of a (set-valued) convex variation of a mapping from a segment into a LCS is
introduced. The use of properties of compact variation (i.1) and compact subdifferential
[10, 12] permitted us to prove the main results of the paper, which include representing
every (strongly) absolutely continuous and compact subdifferentiable a.e. mapping into
an arbitrary LCS as an indefinite Bochner integral (Theorems 3.1), proving that the
class of indefinite Bochner integrals coincides with the class of (strongly) absolutely
continuous and differentiable a.e. mappings in the case of Frechet spaces (Theorem 3.2),
and showing a presence of a compact variation at each indefinite Bochner integral in an
arbitrary complete LCS (Theorem 3.3). Note, in particular, that Theorem 3.2 generalizes
the above-mentioned result from the case of Banach spaces ([2], Theorems 3.7.11, 3.8.5,
3.8.6) to the case of Frechet spaces. Note also that in the case of an arbitrary (not Frechet)
LCS, the indefinite Bochner integral, being absolutely continuous, can nevertheless be
nowhere differentiable (see Example 2.2).
2000 Mathematics Subject Classification. Primary 26A45, 28B05, 46J52; Secondary 28C20, 46B22.
Key words and phrases. Compact variation, compact subdifferential, Bochner integral, Frechet space,
locally convex space, strong absolute continuity.
74
COMPACT VARIATION, COMPACT SUBDIFFERENTIABILITY AND BOCHNER INTEGRAL 75
Let us mention a certain similarity of the relevant conditions with the classical condi-
tions of the absolute continuity in measure and bounded variation of a charge in Radon-
Nikodym type theorems for Bochner integral ([5], [13]–[17]).
Let us also introduce, in the conclusion of this section, some auxiliary notions. Thro-
ughout the paper, for arbitrary sets A, B ⊂ E,
A − B = {x − y | x ∈ A, y ∈ B} ,
sup ‖A‖c = sup{‖x‖c | x ∈ A} for every continuous seminorm ‖ · ‖c on E ,
coA denotes the convex hull of A.
Definition 0.1. Given a real vector space (VS) E and a subset A ⊂ E, denote △A =
A − A, w(A) = co(△A). In the case of a LCS E, denote by w(A) the closure of w(A)
and call it the oscillation of A.
The following property of the oscillations is directly verified.
Proposition 0.1. Let E be a real LCS ; A, B ⊂ E. Then
(i) (A
⋂
B �= ∅) ⇒ (w(A
⋃
B) ⊂ w(A) + w(B)) .
(ii) sup ‖w(A)‖c = sup ‖△A‖c for every continuous seminorm ‖ · ‖c on E.
1. Compact variation mappings and their properties
Definition 1.1. For a fixed segment I = [a; b] ⊂ R consider its various partitions
P = {Ik = [xk−1; xk]}n
k=1, a = x0 < x1 < · · · < xn = b, and set P(I) = {P}. Introduce
a partial order in P(I) by
(1.1) (P1 $ P2) :⇔ (P2 is refinement of P1) .
Note that the order (1.1) is obviously inductive.
Definition 1.2. Let E be a real LCS, F : I → E. Associate with every partition
P ∈ P(I) the partial convex variation (or C-variation) of F,
CV (F, P ) =
n∑
k=1
wF (Ik) .
It is obvious that each set CV (F, P ) is closed and absolutely convex.
Proposition 1.1. The set of the all C-variations {CV (F, P ) | P ∈ P(I)} is inductively
ordered by the embedding with respect to P1 $ P2.
Proof. It suffices to consider the case
P1 = {Ik}
n
k=1, P2 = {Ik}k �=m
⋃
{I ′m, I ′′m}, Im = I ′m
⋃
I ′′m.
Then, in view of Proposition 0.1(i),
CV (F, P1) =
n∑
k �=m
wF (Ik) + wF (I ′m
⋃
I ′′m)
⊂
n∑
k �=m
wF (Ik) + wF (I ′m) + wF (I ′′m) = CV (F, P2) . �
Definition 1.3. Let us introduce, using the notation of Definitions 1.1–1.2, the (total)
convex variation (or the (total) C-variation) of F on I by
(1.2) CVI(F ) ≡ CV (F ) =
⋃
P∈P(I)
CV (F, P ) .
76 I. V. ORLOV AND F. S. STONYAKIN
In the case of bounded CV (F ) we call F the bounded C-variation mapping (F ∈
BV (I)), in the case of compact CV (F ) we call F the compact C-variation mapping or
simply compact variation mapping (F ∈ CV (I)).
Note that in any case, CV (F ) is a closed and absolutely convex set. Let us consider
some properties of the C-variation and, first of all, a relationship between the C-variation
and the usual variation of F in the Banach case and in the finite-dimensional case.
Proposition 1.2. Let E be a real Banach space, F : I → E. Denote by
V (F ) = sup
P∈P(I)
n∑
k=1
‖F (xk) − F (xk−1)‖
the usual total variation of F . Then
(1.3) sup ‖CV (F )‖ � V (F ) .
Proof. Fix ε > 0, P ∈ P(I), and choose x′
k, x′′
k ∈ Ik(k = 1, n) such that
(1.4) ‖F (x′
k) − F (x′′
k)‖ > sup
x′, x′′∈Ik
‖F (x′) − F (x′′)‖ −
ε
2k
= sup ‖wF (Ik)‖ −
ε
2k
.
Summing (1.4) over k = 1, n, we obtain
V (F ) �
n∑
k=1
VIk
(F ) �
n∑
k=1
‖F (x′
k) − F (x′′
k)‖ >
n∑
k=1
sup ‖wF (Ik)‖ −
n∑
k=1
ε
2k
� sup
∥∥∥
n∑
k=1
wF (Ik)
∥∥∥ −
n∑
k=1
ε
2k
> sup ‖CV (F, P )‖ − ε ,
whence V (F ) � sup ‖CV (F, P )‖ for every P ∈ P(I). This immediately implies (1.3) in
view of (1.2). �
Corollary 1.1. If, with the preceding notation, V (F ) < +∞, then F ∈ BV (I).
Proposition 1.3. Let F : I → R. Then
(1.5) supCV (F ) = V (F ) .
Proof. Fix P ∈ P(I). Then Proposition 0.1(i) implies that
sup wF (Ik) = sup
x′, x′′∈Ik
|F (x′) − F (x′′)| (k = 1, n) .
Hence, in view of symmetry of partial C-variations,
supCV (F, P ) = sup
n∑
k=1
wF (Ik) =
n∑
k=1
supwF (Ik)
�
n∑
k=1
|F (xk) − F (xk−1)| = V (F, P ) ,
whence
supCV (F ) � supCV (F, P ) � V (F, P ) ,
and hence
(1.6) supCV (F ) � sup
p∈P(I)
V (F, P ) = V (F ) .
Since, by virtue of (1.3),
(1.7) supCV (F ) = sup |CV (F )| � V (F ) ,
we see that (1.5) follows from (1.6) and (1.7). �
COMPACT VARIATION, COMPACT SUBDIFFERENTIABILITY AND BOCHNER INTEGRAL 77
Corollary 1.2. Let F : I → R
n. Then F ∈ CV (I) if and only if V (F ) < ∞.
Remark 1.1. 1). Below, in Section 2, we’ll show that the case: F ∈ CV (I) but V (F ) =
+∞ is possible for dimE = ∞.
2). The inequality (1.3) can easily be generalized to an arbitrary continuous seminorm
‖ · ‖c in LCS E:
sup ‖CV (F )‖c � Vc(F ) .
In particular, if {‖ · ‖i}i∈Y is a defining system of seminorms in E and Vi(F ) < +∞ (i ∈
Y ), then F ∈ BV (I).
Let us continue with a study of properties of the C-variation.
Theorem 1.1. (Additivity). Let E be a real LCS, F : I → E, I = I ′
⋃
I ′′. Then
CVI(F ) = CVI′(F ) + CVI′′ (F ) .
Proof. Let P ′ = {Ik}m
k=1, P ′′ = {Ik}n
k=m+1 be partitions of I ′ and I ′′, respectively,
P = P ′
⋃
P ′′. Then
(1.8)
CV (F, P ) =
m∑
k=1
wF (Ik) +
n∑
k=m+1
wF (Ik)
= CV (F, P ′) + CV (F, P ′′) ⊂ CVI′ (F ) + CVI′′(F ) .
Next, for an arbitrary P1 ∈ P(I) put P2 = P1 ∨ P (i.e., P2 is the mutual refinement
of P1 and P ). Then, applying Proposition 1.1 and (1.8) to P2, we obtain
CV (F, P ) ⊂ CV (F, P2) ⊂ CVI′(F ) + CVI′′ (F ) ,
whence the inclusion
CVI(F ) ⊂ CVI′ (F ) + CVI′′(F )
immediately follows. To check the inverse inclusion, return to (1.8),
CV (F, P ′) + CV (F, P ′′) = CV (F, P ) ⊂ CVI(F ) ,
whence we obtain
CVI′(F ) + CVI′′ (F ) ⊂ CVI(F ) . �
Corollary 1.3. With the preceding notation,
(i) F ∈ BV (I) ⇔ F ∈ BV (I ′) and F ∈ BV (I ′′) ;
(ii) F ∈ CV (I) ⇔ F ∈ CV (I ′) and F ∈ CV (I ′′) .
To formulate the following property, we need to extend the notion of a K-limit, which
was introduced by the first author ([10]) for decreasing the net of closed convex sets (the
projective case), to the case of an increasing net (inductive case).
Definition 1.4. Let {Bt}t∈T be a system of closed convex subsets of a real LCS E,
inductively ordered by inclusion. We say that
B = K − lim
−−−→
t∈T
Bt ,
if for each zero neighborhood U ⊂ E there exists tU ∈ T such that
(1.9) (t & tU ) ⇒ (Bt ⊂ B ⊂ Bt + U) .
Remark 1.2. It obviously follows from (1.9) that
⋃
t∈T
Bt ⊂ B ⊂
⋃
t∈T
(Bt + U) =
⋃
t∈T
Bt + U
for every U ⊂ E, whence ⋃
t∈T
Bt ⊂ B ⊂
⋃
t∈T
Bt .
78 I. V. ORLOV AND F. S. STONYAKIN
In particular, B is convex set.
Theorem 1.2. Let E be a real LCS, F : I → E. If F has compact variation on I, then
CV (F ) = K − lim
−−−−−−→
P∈P(I)
CV (F, P ) .
Proof. It follows from Definition 1.3 and ([18], I.1.1) that for each fixed neighborhood
Ũ ⊂ E of zero,
CV (F ) =
⋃
P∈P
CV (F, P ) =
⋂
U⊂E
[ ⋃
P∈P
CV (F, P ) + U
]
⊂
⋃
P∈P
(CV (F, P ) + Ũ) .
Thus, the sets {CV (F, P ) + Ũ | P ∈ P(I)} forms an open covering of the compact set
CV (F ). Choose a finite subcovering
{CV (F, Pj) + Ũ}n
j=1
and put P�U = P1∨ ...∨Pn. Then, by Proposition 1.1, CV (F, Pj) ⊂ CV (F, P�U ), j = 1, n,
whence
CV (F, Pj) + Ũ ⊂ CV (F, P�U ) + Ũ . (j = 1, n)
Hence,
CV (F, P�U ) ⊂ CV (F ) ⊂
n⋃
j=1
(CV (F, Pj) + Ũ) ⊂ CV (F, P�U ) + Ũ .
Replacing P�U with an arbitrary refinement P & P�U , the last inclusion remains valid:
CV (F, P ) ⊂ CV (F ) ⊂ CV (F, P ) + Ũ ,
which corresponds to (1.9). �
Remark 1.3. As it is well known, even in the scalar case for discontinuous F , in general,
V (F, P ) � V (F ) (and hence CV (F, P ) � CV (F )) as the step of the partition, λ(P ),
tends to 0. Let us check now that CV (F, P ) → CV (F )) as λ(P ) → 0 for a continuous
mapping F : I → E.
Theorem 1.3. Let E be a real LCS, F : I → E be a continuous mapping. If F ∈ CV (I)
then for every neighborhood U ⊂ E of zero there exists δU > 0 such that for each
P ∈ P(I),
(λ(P ) < δU ) ⇒ (CV (F, P ) ⊂ CV (F ) ⊂ CV (F ) + U) .
We will write the last condition in the form
CV (F ) = Kλ − lim
λ→0
CV (F, P ) .
Proof. Fix U ⊂ E and choose W ⊂ E such that W + W ⊂ U . Then, using Theorem 1.2,
choose PW = {Ik}N
k=1 ∈ P(I) such that
CV (F, PW ) ⊂ CV (F ) ⊂ CV (F, PW ) + W .
Next, using uniform continuity of F , choose δ > 0 and W ′ ⊂ E such that
(1.10) (|x′ − x′′| < δ) ⇒ (F (x′) − F (x′′) ∈ W ′) .
Finally, choose a partition Pδ = {Ij}n
j=1 with λ(Pδ) < δ and set P̃ = PW ∨ Pδ.
Then, by Proposition 1.1,
CV (F, PW ) ⊂ CV (F, P̃ ) ⊂ CV (F, Pδ) +
N∑
k=1
wF (I
j(k)
± ) ,
where the intervals I
j(k)
± contain inside right or left ends of the interval Ik, respectively,
k = 1, N .
COMPACT VARIATION, COMPACT SUBDIFFERENTIABILITY AND BOCHNER INTEGRAL 79
Since, by virtue of (1.10), wF (I
j(k)
± ) ⊂ W ′, we finally have
CV (F, Pδ) ⊂ CV (F ) ⊂ CV (F, PW ) + W ⊂ [CV (F, Pδ) +
2N︷ ︸︸ ︷
W ′ + · · · + W ′ +W
⊂ CV (F, Pδ) + (W + W ) ⊂ CV (F, Pδ) + U . �
At the end of this section, let us introduce the notion of a compact-Lipschitz (or
K-Lipschitz) mapping and explain its relation with that of a compact variation.
Definition 1.5. Let E be a real LCS, F : I → E. We say that a mapping F is compact-
Lipschitz on I (or F ∈ LipK(I)) if, for some compact absolutely convex set C ⊂ E, the
inclusion
F (x2) − F (x1) ∈ C · (x2 − x1)
holds for arbitrary x1, x2 ∈ I.
It is obvious that, in the case of a Banach space E, each compact-Lipschitz mapping
is also Lipschitz. It is also easy to see that each compact-Lipschitz mapping is absolutely
continuous. Let us check now that each K-Lipschitz mapping is also of compact variation.
Proposition 1.4. Let E be a real LCS, F : I → E. If F ∈ LipK(I) then F ∈ CV (I).
Proof. Obviously, the inclusion F (x2) − F (x1) ∈ C · (x2 − x1) implies the inclusion
wF ([x; x+h]) ⊂ C ·h for each [x; x+h] ⊂ I. Whence the inclusion CV (F, P ) ⊂ C ·(b−a)
for each P ∈ P(I) and, hence, the inclusion CV (F ) ⊂ C · (b − a) easily follow. �
2. K-subdifferentiability. Examples of K-subdifferentiable and nowhere
K-subdifferentiable mappings
First, let us introduce the notion of a K-limit for a decreasing system of closed convex
sets, which has been already mentioned earlier in connection with Definition 1.4.
Definition 2.1. Let {Bt}t∈T be a system of the closed convex subsets of a real LCS E,
inductively ordered oppositely to inclusion. We say that
B = K − lim
←−−
t∈T
Bt ,
if B is closed and for each neighborhood U ⊂ E of zero there exists tU ∈ T such that
(2.1) (t & tU ) ⇒ (B ⊂ Bt ⊂ B + U) .
Remark 2.1. It obviously follows from (2.1) that
B ⊂
⋂
t∈T
Bt ⊂
⋂
U⊂E
(B + U) = B ,
whence, since
⋂
t∈T Bt is closed,
B =
⋂
t∈T
Bt .
In particular, B is a closed convex set.
Let us pass to the notion of a K-subdifferential (or compact subdifferential). The main
properties of K-subdifferential, including various forms of the mean value theorem, were
investigated by the authors in [11, 12].
Definition 2.2. Let E be a real LCS, F : I → E, x ∈ I, δ > 0. The set
∂KF (x, δ) = co
{
F (x + h) − F (x)
h
∣∣∣ 0 < |h| < δ
}
is said to be a partial K-subdifferential of F at x, corresponding to δ. A set ∂KF (x) ⊂ E
is said to be a K-subdifferential of F at x, if
80 I. V. ORLOV AND F. S. STONYAKIN
(i) ∂KF (x) = K − lim←−−
δ→0
∂KF (x, δ) ;
(ii) ∂KF (x) is a compact set.
In an analogous way, the right and left K-subdifferentials, ∂+
KF (x) and ∂−
KF (x), respec-
tively, can be defined.
Remark 2.2. It is shown in [12] that if F is differentiable at x then F is K-subdifferentiable
at x and ∂KF (x) = {F ′(x)}, but the inverse is not true.
Proposition 2.1. Each K-Lipschitz mapping is everywhere K-subdifferentiable.
Proof. Obviously, the inclusion F (x2) − F (x1) ∈ C(x2 − x1) implies the inclusion
∂KF (x, δ) ⊂ C
for each x ∈ I, δ > 0, whence ∂KF (x, δ) is compact. Because convergence of a decreasing
sequence of compact sets to its intersection is necessarily topological, both conditions of
Definition 2.2 are fulfilled. �
Let us pass to examples. First, consider an example of differentiable (and, hence,
K-subdifferentiable) a.e. compact variation mapping, not having the usual (strong)
bounded variation.
Example 2.1. Let E be a real infinite-dimensional separable Hilbert space, {en}∞n=1 be
an orthonormal basis in E. Define the mapping F : [0; 1] → E by
{
F (0) = 0 ; F ( n
n+1 ) =
∑n
k=1
ek
k
(n ∈ N) ; F (1) =
∑∞
k=1
ek
k
;
F is linear on the segments
[
n−1
n
; n
n+1
]
(n ∈ N) .
Thus, F is continuous and countably piecewise linear on [0; 1]. For each partition
Pn : 0 < 1
2 < · · · < n
n+1 < 1 we obtain
V (F, Pn) =
( n∑
k=1
∥∥∥F
( k
k + 1
)
− F
(k − 1
k
)∥∥∥
)
+
∥∥∥F (1) − F
( n
n + 1
)∥∥∥ =
n∑
k=1
∥∥∥
ek
k
∥∥∥
+
∥∥∥
∞∑
k=n+1
ek
k
∥∥∥ =
n∑
k=1
1
k
+
∞∑
k=n+1
1
k2
→ +∞ as n → ∞ ,
whence V (F ) = + ∞ follows.
However, let us show now that F ∈ CV (I). Using piecewise linearity of F and
additivity of C-variation (Theorem 1.1), we obtain
(2.2) CVI(F ) ⊃ CV[0; n
n+1 ]
(F ) =
n∑
k=1
CV[ k−1
k
; k
k+1 ]
(F ) =
n∑
k=1
[
−
ek
k
;
ek
k
]
.
Let P = {Ij}m
j=1 be an arbitrary partition of I. Choose n such that xm−1 < n
n+1 < xm
and set
P ′ = P ∨
{[k − 1
k
;
k
k + 1
]}n
k=1
.
Then, taking into account Proposition 1.1, (2.2) implies
(2.3)
CV (F, P ) ⊂ CV (F, P ′) ⊂ CV[0; n
n+1 ]
(F ) + wF
([ n
n + 1
; 1
])
=
n∑
k=1
[
−
ek
k
;
ek
k
]
+ w
{ n+p∑
k=1
ek
k
(p ∈ N) ;
∞∑
k=1
ek
k
}
,
and it remains to estimate the last term in the right-hand side of (2.3).
COMPACT VARIATION, COMPACT SUBDIFFERENTIABILITY AND BOCHNER INTEGRAL 81
Denote it by w(A). Since
△A =
{
±
n+p+q∑
k=n+p+1
ek
k
; 0 ; ±
∞∑
k=n+p+1
ek
k
(p ∈ N0, q ∈ N)
}
,
we have
(2.4) sup ‖w(A)‖ = sup ‖△A‖ �
∞∑
k=n+1
1
k2
=: rn → 0 as n → ∞ .
Thus, w(A) is contained in the closed ball Brn
(0). From here and (2.3) it follows that
CV (F, P ) ⊂
n∑
k=1
[
−
ek
k
;
ek
k
]
+ Brn
(0) ⊂
∞∑
k=1
[
−
ek
k
;
ek
k
]
+ Brn
(0) ,
whence for each n ∈ N,
CV (F ) ⊂
∞∑
k=1
[
−
ek
k
;
ek
k
]
+ Brn
(0) ,
and, finally,
(2.5) CV (F ) ⊂
∞∑
k=1
[
−
ek
k
;
ek
k
]
.
(It is easy to check the exact equality in (2.5)). Since the set in the right-hand side
of (2.5) is compact (the Hilbert cube), CV (F ) is compact, too. So, F ∈ CV (I). Note
that, in view of countable piecewise linearity, F is differentiable a.e.
Secondly, consider an example of a mapping that is nowhere differentiable K-Lipschitz
(and, hence, of compact variation) and everywhere K-subdifferentiable.
Example 2.2. Let EI be the space of the all real functions ξ = ξ(θ) on I = (0; 1) with
the defining system of seminorms {‖·‖t}t∈I that corresponds to the topology of pointwise
convergence. Then EI is a separable and complete LCS. Define a mapping y(·) : I → EI
by
y(s)(θ) = s for 0 < s � θ < 1 , y(s)(θ) = θ for 0 < θ � s < 1 .
1). For each s ∈ I and corresponding △s > 0 small enough (such that both cases
s � θ � s + △s and s −△s � θ � s are impossible) it follows that
y(s + △s) − y(s)
△s
(θ) =
1, for 0 < s < s + △s � θ < 1 ,
0, for 0 < θ � s < s + △s < 1 ,
y(s −△s) − y(s)
−△s
(θ) =
1, for 0 < s −△s < s � θ < 1 ,
0, for 0 < θ � s −△s < s < 1 ,
and, therefore, y possess the left and the right derivatives everywhere on I, y′
+(s) = y1
s(·) ,
y′
−(s) = y2
s(·), where
y1
s(θ) = 1 for 0 < s < θ < 1 , y1
s(θ) = 0 for 0 < θ � s < 1 ,
and
y2
s(θ) = 1 for 0 < s � θ < 1 , y2
s(θ) = 0 for 0 < θ < s < 1 .
It is obvious that y1
s(·) �= y2
s(·) for each s ∈ I and, therefore, the mapping y is nowhere
differentiable on I. However, in fact, y(s) is K-subdifferentiable at each s ∈ I and
(2.6) ∂Ky(s) = [y′
−(s); y′
+(s)] = [y1
s(·); y2
s(·)] .
(Here we consider that interval [y; y] as the one-point set {y}).
82 I. V. ORLOV AND F. S. STONYAKIN
Indeed, let ∂̃Ky(s) be a formal K-subdifferential (without the requirement of com-
pactness). Taking into account separability of EI , the equalities
(ℓ ◦ y)′±(s) = ℓ(y′
±(s)) (∀ℓ ∈ E∗)
and
[(ℓ ◦ y)′−(s); (ℓ ◦ y)′+(s)] = ℓ([y′
−(s); y′
+(s)]) = ℓ(∂̃Ky(s))
imply (2.6) (here ℓ(A) = {ℓ(x) | x ∈ A} for an arbitrary A ⊂ E). Obviously, the vector
segment ∂̃Ky(s) is compact and, therefore, ∂Ky(s) = ∂̃Ky(s) exists everywhere on I.
2). Let us show that y ∈ LipK(I). First, we prove compactness of the set D :=
{yi
s(·) | i = 1, 2 ; s ∈ I} in EI . Consider an arbitrary sequence {y1
sk
(·)}∞k=1 of func-
tions from D. Denote by s0 some partial limit of the sequence {sk}∞k=1 and choose a
subsequence {skℓ
}∞ℓ=1 that converges to s0 from the right, for definiteness. Then
y1
sk
ℓ
(θ) − y1
s0
(θ) =
−1, for θ ∈ (s0; skℓ
) ,
0, for θ /∈ (s0; skℓ
) .
This means that
‖y1
sk
ℓ
(θ) − y1
s0
(θ)‖t = |(y1
sk
ℓ
− y1
s0
)(t)| → 0
as ℓ → ∞, i.e., the sequence {y1
sk
ℓ
(·)}∞ℓ=1 converges in the space EI to the function y1
s0
(·).
Analogously, for the case skℓ
→ s0 − 0, y1
sk
ℓ
(·) converges to y2
s0
(·) ∈ D.
Thus, an arbitrary sequence of functions from D has a partial limit contained in D.
Therefore D is sequentially compact and moreover compact. Hence, the set U := co D is
convex compact in E, because E is a separable and complete space. Note that for each
s ∈ I,
∂Ky(s) = [y1
s(·); y2
s(·)] ⊂ U ,
whence ∂Ky(I) ⊂ U . Now, using the mean value theorem ([12]) for K-subdifferentials
on an arbitrary segment [α; β] ⊂ (0; 1), we obtain
y(β) − y(α) ∈ U · (β − α) ,
whence y ∈ LipK(I).
Let us give, for completeness, a sketch of the proof of the “K-mean value theorem” [12]
that was used above in Example 2.2.
Proposition 2.2. Let E be a real LCS, a mapping F : [a; b] → E be continuous on [a; b]
and K-subdifferentiable on (a; b). Then
(2.7)
F (b) − F (a)
b − a
∈ co ∂KF ((a; b)) .
Proof. We use the following result for scalar-valued functions (see [11], Theorem 9).
If a function f : [a; b] → R is continuous on [a; b] and K-subdifferentiable on (a; b),
then
(2.8)
f(b) − f(a)
b − a
� sup∂Kf((a; b)) .
Assume that inclusion (2.7) is not true. Then, in view of a known corollary from
Hahn-Banach theorem ([21], Corollary 2.1.4), there is a functional ℓ ∈ E∗ such that
ℓ
(
F (b) − F (a)
b − a
)
> sup ℓ (co ∂KF ((a; b))) .
Denoting by f = ℓ ◦ F , the last inequality takes the form
f(b) − f(a)
b − a
> sup∂Kf((a; b)) ,
COMPACT VARIATION, COMPACT SUBDIFFERENTIABILITY AND BOCHNER INTEGRAL 83
which contradicts (2.8). �
Next, in the monographs [2, 19], examples of Lipschitz mappings acting from an
interval into Banach spaces, which are nowhere differentiable, are considered. Let us
show now that the corresponding mappings are, in fact, nowhere K-subdifferentiable.
Example 2.3. Let L1(I) be the space of the all Lebesgue integrable real-valued functions
x = ξ(t) on I = [0; 1] with the norm ‖x‖ =
∫ 1
0
|ξ(t)| dt. Define a mapping y(·) : I → L1(I)
by
y(s)(t) = 1 for 0 � t � s , y(s)(t) = 0 for s � t � 1 .
Suppose that y(·) is K-subdifferentiable at some point s0 ∈ I. Then, for some sequence
hk ց 0, the corresponding sequence
ỹk =
y(s0 + hk) − y(s0)
hk
(k ∈ N)
converges in L1(I). Hence,
(2.9) ∀ε > 0 ∃k0 ∈ N ∀k � k0 : ‖ỹk − ỹk0
‖ < ε .
Since
y(s + h) − y(s)
h
(t) =
1
h
, for s � t � s + h ,
0 , otherwise ,
we obtain
‖ỹk − ỹk0
‖ =
∥∥∥
y(s0 + hk) − y(s0)
hk
−
y(s0 + hk0
) − y(s0)
hk0
∥∥∥
=
∫ 1
0
∣∣∣
(y(s0 + hk) − y(s0)
hk
−
y(s0 + hk0
) − y(s0)
hk0
)
(t)
∣∣∣dt
=
∫ s0+hk
s0
∣∣∣
1
hk
−
1
hk0
∣∣∣dt +
∫ s0+hk0
s0
dt
hk0
= hk ·
( 1
hk
−
1
hk0
)
+
hk0
− hk
hk0
= 2 − 2
hk
hk0
→ 2 as k → ∞ ,
which contradicts (2.9) for ε < 2. So, y(·) is nowhere K-subdifferentiable. At the same
time, V (y) < +∞ because y(·) is Lipschitz.
Example 2.4. Let E(I) be the space of the all measurable bounded real-valued functions
x = ξ(t) on I =
[
1
3 ; 2
3
]
with the norm ‖ξ‖ = supt∈I |ξ(t)|. Define a mapping y(·) : I →
E(I) by
y(s)(t) =
s
t
for s � t �
2
3
, y(s)(t) =
s − 1
t − 1
for
1
3
� t � s .
Repeating the arguments from the preceding example, choose a sequence {ỹk}∞k=1 in
E(I) satisfying (2.9). Since, in this case,
y(s + h) − y(s)
h
(t) =
1
t−1 + 1
h
· s−t
t(t−1) , for s � t � s + h ,
1
t−1 , for t � s ,
we obtain
‖ỹk − ỹk0
‖ � sup
t∈I
∣∣∣
s − t
t(t − 1)
∣∣∣ ·
∣∣∣
1
hk
−
1
hk0
∣∣∣ → +∞ as k → ∞ ,
which contradicts (2.9). So, like in the preceding example, y(·) is nowhere K-subdiffe-
rentiable.
84 I. V. ORLOV AND F. S. STONYAKIN
Note, in the conclusion of this section, that a compact variation mapping also can
be nowhere K-subdifferentiable. The corresponding example will be considered in the
following section.
3. Description of the indefinite Bochner integral in terms of compact
variation and compact subdifferentiability
The main results obtained in this section are representation of each (strongly) ab-
solutely continuous and K-subdifferentiable a.e. mapping F : I = [a; b] → E, where E is
an arbitrary real LCS, as an indefinite Bochner integral,
F (x) = F (a) + (B)
∫ x
a
f(t) dt (a � x � b)
(Theorem 3.1) and that the class of the all indefinite Bochner integrals in Frechet spaces
coincides with the class of all mappings absolutely continuous and differentiable a.e. on
I (Theorem 3.2). First, we need several auxiliary statements.
Lemma 3.1. Let E be a real LCS, F : I → E be K-subdifferentiable at a point x ∈ I.
If a sequence [αn; βn] contracts to x, then ∂KF (x) contains all possible partial limits of
the sequence
(3.1)
F (βn) − F (αn)
βn − αn
(n ∈ N) .
Moreover, in the case of where E is a Frechet space, such partial limits do exist.
Proof. The identity
F (βn) − F (αn)
βn − αn
=
F (βn) − F (x)
βn − x
·
βn − x
βn − αn
+
F (αn) − F (x)
αn − x
·
αn − x
αn − βn
implies
F (βn) − F (αn)
βn − αn
∈ co
{
F (βn) − F (x)
βn − x
,
F (αn) − F (x)
αn − x
}
⊂ ∂KF (x, δ)
for |βn − x| � δ, |αn − x| � δ. Hence, each possible partial limit of the sequence (3.1) is
contained in each partial K-subdifferential ∂KF (x, δ > 0) and therefore is contained in
∂KF (x).
Next, let E be a Frechet space. Since, for some sequence of εn-neighborhoods Uεn
(0)
of zero in E, εn → 0,
F (βn) − F (αn)
βn − αn
⊂ ∂KF (x, βn − αn) ⊂ ∂KF (x) + Uεn
(0) ,
we have (F (βn) − F (αn)
βn − αn
+ Uεn
(0)
) ⋂
∂KF (x) �= ∅ .
Let us choose a sequence of the points xn from the intersections above. Since ∂KF (x)
is sequentially compact set, {xn} contains a convergent subsequence xni
→ x0, x0 ∈
∂KF (x). Then
F (βni
) − F (αni
)
βni
− αni
→ x0 , too. �
Lemma 3.2. Let E be a real LCS, F : I → E be absolutely continuous and K-
subdifferentiable a.e. on I. Then, for all ℓ ∈ E∗ and a selector ∂̂KF of ∂KF ,
(3.2) ℓ(F (x))′ = ℓ(∂̂KF (x)) for a.e. x ∈ I .
COMPACT VARIATION, COMPACT SUBDIFFERENTIABILITY AND BOCHNER INTEGRAL 85
Proof. By the properties of the K-subdifferential ([12]), ∂Kℓ(F (x)) = ℓ(∂KF (x)). But
in view of absolute continuity, the real-valued function ℓ(F (x)) is differentiable a.e. on
I. Hence, ℓ(∂KF (x)) is one-point a.e. on I and
ℓ(F (x))′ = ℓ(∂KF (x)) a.e. on I ,
whence (3.2) follows. �
Lemma 3.3. Let E be a real Banach space, F : I = [a; b] → E be continuous on I and
K-subdifferentiable a.e. on I. Then each selector ∂̂KF is almost separable-valued.
Proof. Similarly to [2], note that F is continuous and hence F (I) is contained in a closed
separable subspace E0 of E. Since limits of the elements from E0 are contained in E0,
and so almost all values of ∂̂KF will lie in E0. �
Lemma 3.4. ([20], Ch. III, § 6). Let (S, µ) be a finite measure space, {fn}∞n=1 be a
sequence of the real-valued µ-integrable functions on S such that fn � 0 (mod µ) and∫
S
fndµ � M < ∞, n ∈ N. Then the function h(t) = limn→∞ fn(t) is also µ-integrable
on S.
Let us now pass to the main theorems. Recall that Bochner integrability in LCS
implies the same for all the factor spaces defined by kernels of the defining seminorms
and then completed in factor norms.
Theorem 3.1. Let E be a real LCS, F : I = [a; b] → E be a mapping (strongly) absolutely
continuous and K-subdifferentiable a.e. on I. Then an arbitrary selector ∂̂KF of ∂KF
is Bochner integrable over I and
(3.3) F (x) = F (a) + (B)
∫ x
a
∂̂KF (t) dt (a � x � b) .
Proof. (i). Choose an arbitrary selector ∂̂KF : I → E (a.e.) and ℓ ∈ E∗. Taking into
account absolute continuity of ℓ(F ) and equality (3.2), we obtain
ℓ(F (x) − F (a)) = ℓ(F (x)) − ℓ(F (a)) =
∫ x
a
[ℓ(F (t))]′dt =
∫ x
a
ℓ(∂̂KF (t)) dt (a � x � b)
whence ∂̂KF is Pettis integrable over I and
(3.4) F (x) = F (a) + (P )
∫ x
a
∂̂KF (t) dt .
(ii). Let us now show that an arbitrary selector ∂̂KF in (3.4) is Bochner integrable.
Let {‖ · ‖j}j∈J be a defining system of seminorms in E, Êj be completions of the factor
spaces Ej = E/ ker ‖ · ‖j in the factor norms ‖ · ‖�j , ϕj : E → Ej be the canonical
embeddings, Fj = ϕj ◦ F : I → Êj .
First, applying ϕj to the both sides of (3.4), we obtain
Fj(x) − Fj(a) = (P )
∫ x
a
∂̂KFj(t) dt (j ∈ J) .
It follows from here and Lemma 3.3 that each selector ∂̂KFj = ϕj(∂̂KF ) is almost
separable-valued and, hence ([2], Th. 3.5.2), strongly measurable.
Next, introduce the following function sequence on I:
(3.5)
fn(t) =
2n
b − a
[
F
(
a + k
b − a
2n
)
− F
(
a + (k − 1)
b − a
2n
)]
for t ∈
[
a + (k − 1)
b − a
2n
; a + k
b − a
2n
]
, k = 1, 2n, n ∈ N .
86 I. V. ORLOV AND F. S. STONYAKIN
In view of absolute continuity of F it is easy to see that for all n ∈ N, j ∈ J ,
(3.6)
∫ b
a
‖fn(t)‖j dt � Vj(F ) < +∞ .
Further, for an arbitrary point t ∈ I of K-subdifferentiability of F (and, hence of
Fj ; ∂KFj(t) = ϕj(∂KF (t))) there exists a sequence of the segments of type (3.5) that
contracts to t. Then, by Lemma 3.1, the set of partial limits of the sequence {fn(t)}∞n=1
is nonempty and is contained in ∂KFj(t). Denote by
ℓj(t) = lim
n→∞
‖fn(t)‖j (j ∈ J)
and choose f j
ni
(t) for each i ∈ N such that
‖f j
ni
(t)‖j < ℓj(t) +
1
i
.
Choosing, if necessary, without loss of generality, a subsequence from {f j
ni
(t)}, we
obtain f j
ni
(t) → ∂̂KF (t) ∈ ∂KF (t). It follows that
‖∂̂KF (t)‖j � lim
n→∞
‖fn(t)‖j (a.e. on I)
for all j ∈ J , whence ∀j ∈ J we obtain
(3.7) ‖∂̂KFj(t)‖
�
j � lim
n→∞
‖f j
n(t)‖�j (a.e. on I) .
Since ∂̂KFj is strongly measurable, from (3.6), (3.7), Lemma 3.4, and Fatou’s theorem
it follows that
∫ b
a
‖∂̂KFj(t)‖
�
jdt � lim
n→∞
∫ b
a
‖f j
n(t)‖�jdt � Vj(F ) < +∞ ,
whence Bochner integrability of ∂̂KFj over I immediately follows ([2], Th. 3.7.4). So,
the equality (3.4) implies
(3.8) Fj(x) = Fj(a) + (B)
∫ x
a
∂̂KFj(t) dt .
Since Êj is a Banach space, in view of (3.8) and property of differentiability a.e. of
the Bochner integral, we have
∂KFj(t) =
d
dt
Fj(t) a.e. on I ,
i.e., ∂KFj(t) = ϕj(∂KF (t)) is one-point a.e. on I. Therefore all selectors ∂̂KFj(t) are
coincide a.e. on I and an arbitrary selector ∂̂KFj(t) = ϕj(∂KF (t)) is Bochner integrable
on I. This implies Bochner integrability on I of an arbitrary selector ∂̂KF (t) of K-
subdifferential ∂KF (t). So, the equality (3.4) takes form of (3.3). �
Lemma 3.5. Let E be a real Frechet space, f : I = [a; b] → E be Bochner integrable.
Then the mapping F (x) = (B)
∫ x
a
f(t)dt is a.e. differentiable on I; moreover, F ′(x) =
f(x) a.e. on I.
Proof. Let {‖ · ‖j}j∈N be a countable system of defining seminorms in E. Since, in the
notation from the preceding proof, Êj are Banach spaces, for all j ∈ N we have
lim
h→0
∥∥∥
1
h
(B)
∫ x+h
x
f(t) dt − f(x)
∥∥∥
j
= 0 ∀x ∈ [a; b]\ej ,
where ej has measure zero. Then for all x ∈ [a; b]\
⋃
j∈N
ej, that is a.e. on [a; b], the last
equality holds for all j ∈ N simultaneously. This precisely means that F ′(x) = f(x) a.e.
on [a; b]. �
COMPACT VARIATION, COMPACT SUBDIFFERENTIABILITY AND BOCHNER INTEGRAL 87
The next result of this section easily follows from Theorem 3.1, Remark 2.2, (strong)
absolute continuity and differentiability a.e. of the indefinite Bochner integral for the
case of real Frechet spaces (Lemma 3.5).
Theorem 3.2. Let E be a real Frechet space, F : I = [a; b] → E. Then the following
assertions are equivalent:
(i) F is an indefinite Bochner integral, i.e.,
F (x) = F (a) + (B)
∫ x
a
f(t) dt (a � x � b ) ;
(ii) F is (strongly) absolutely continuous and differentiable a.e. on I ;
(iii) F is absolutely continuous and K-subdifferentiable a.e. on I .
Remark 3.1. Note that, as it immediately follows from Example 2.2, the result of Theo-
rem 3.2 is not valid in general if E is not a Frechet space and therefore condition of K-
subdifferentiability a.e. in Theorem 3.1 cannot be replaced with that of differentiability
a.e. on I. This means that the K-subdifferential describes the differential properties of
the Bochner integral of LCS-valued mappings better then the usual derivative.
To prove another main result, we need preliminary some equivalent form of Defini-
tion 1.3.
Definition 3.1. Let E be a real LCS, F : I → E. For every P ∈ P(I) let us introduce
a partial C0-variation,
C0V (F, P ) =
n∑
k=1
△F (Ik) .
The set
C0V (F ) =
⋃
P∈P(I)
C0V (F, P )
is called the (total) C0-variation of F on I. If C0V (F ) is a compact set, we say that F
is compact C0-variation mapping (F ∈ C0V (I)).
It is obvious that C0V (F ) ⊂ CV (F ), whence (F ∈ CV (I)) ⇒ (F ∈ C0V (I)). Let us
establish the converse implication.
Proposition 3.1. Let E be a real separable LCS, F : I → E. Then
(3.9) CV (F ) = co (C0V (F )) .
If, in addition, E is quasicomplete, then the conditions F ∈ CV (I) and F ∈ C0V (I) are
equivalent.
Proof. For all P ∈ P(I) and ℓ ∈ E∗, the identities
sup ℓ(CV (F, P )) = sup ℓ
( n∑
k=1
wF (Ik)
)
= sup
n∑
k=1
ℓ(wF (Ik)) =
n∑
k=1
sup ℓ(wF (Ik))
= sup
( n∑
k=1
ℓ(△F (Ik))
)
= sup ℓ
( n∑
k=1
△F (Ik)
)
= sup ℓ(C0V (F, P ))
hold true. Hence, passing to the closures of unions by P , we obtain
sup ℓ(CV (F )) = sup ℓ(C0V (F )) (∀ℓ ∈ E∗) ,
and so
sup ℓ(CV (F )) = sup ℓ(co C0V (F )) (∀ℓ ∈ E∗) .
Using functional separability of E ([21], 2.1.4), this implies (3.9).
88 I. V. ORLOV AND F. S. STONYAKIN
Finally, in the case of quasicomplete E, compactness of C0V (F ) implies compactness
for its closed convex hull ([18], II.4.3) and (F ∈ C0V (I)) ⇒ (F ∈ CV (I)) by virtue of
(3.9). �
Theorem 3.3. Let E be a real complete LCS, f : I = [a; b] → E be Bochner integrable
over I. Then, for an arbitrary C ∈ E, the mapping
(3.10) F (x) = C + (B)
∫ x
a
f(t) dt
has compact variation on I.
Proof. First, suppose that E is a Banach space. Since, for each partition P ∈ P(I),
C0V (F, P ) =
n∑
k=1
{
(B)
∫ xk
2
xk
1
f(t) dt
∣∣∣ xk
1 , xk
2 ∈ Ik
}
=
{
(B)
∫
�
n
k=1
[xk
1
,xk
2
]
f(t) dt
∣∣∣ [xk
1 , xk
2 ] ⊂ Ik
}
,
it follows from Definition 3.1 that
(3.11) C0V (F ) ⊂
{
± (B)
∫
A
f(t) dt
∣∣∣ A ∈ B(I)
}
,
where B(I) is the Borel σ-algebra on I.
Let us now check that C0V (F ) is totally bounded in E.
Given ε > 0, choose, by virtue of the definition of Bochner integrability ([21], X.3), a
simple mapping
fε(t) =
n∑
k=1
ck · χTk
(t)
(
ck ∈ E , I =
n⋃
k=1
Tk , Tk are disjoint
)
such that ∫
I
‖f(t) − fε(t)‖dt <
ε
2(b − a)
.
Hence, for each A ∈ B(I),
(3.12)
∥∥∥(B)
∫
A
f(t) dt − (B)
∫
A
fε(t) dt
∥∥∥ �
∫
A
‖f(t) − fε(t) |dt <
ε
2(b − a)
.
In addition,
(B)
∫
A
fε(t) dt =
n∑
k=1
ck · mes(Tk
⋂
A) = mes(A) ·
n∑
k=1
ck ·
mes(Tk
⋂
A)
mes(A)
∈ mes(I) · abs.co{ck}
n
k=1 =: K .
The set K is compact in E, not depending on the choice of A. Let us choose a finite
ε/2-net N = {x1, . . . , xm} for K. Then, in view of (3.12), N is ε-net for the set in the
right-hand side of (3.11) and, therefore, N is an ε-net for C0V (F ). This implies, since
C0V (F ) is closed, compactness of this set and thus, by Proposition 3.1, compactness of
CV (F ).
Secondly, let E be a real complete LCS with a defining system of seminorms {‖·‖j}j∈J .
In the notation from the preceding, we see that separation of E implies
(3.13)
⋂
j∈J
ϕ−1
j (ϕj(y)) = {y} for each y ∈ E .
COMPACT VARIATION, COMPACT SUBDIFFERENTIABILITY AND BOCHNER INTEGRAL 89
Applying ϕj(j ∈ J) to (3.10) we obtain
F̂j(x) = F̂j(a) + ϕj
(
(B)
∫ x
a
f(t) dt
)
= F̂j(a) + (B)
∫ x
a
f̂j(t) dt ,
where f̂j(t) = ϕj(f(t)). Hence, according to the first part of the proof, the sets CV (F̂j)
are compact in Êj . But E is a projective limit of the Banach spaces Êj [18] and, therefore,
can be closely and continuously embedded into the product Πj∈J Êj ([18], II.5.3). Hence,
CV (F ) can be injectively (in view of (3.13)) continuously and closely embedded into the
product
∏
j∈J
CV (F̂j) .
But the last set is compact by Tychonoff theorem ([22], I.9.5), whence compactness of
CV (F ) follows. �
From the definition of Radon-Nikodym property (see Introduction) and Theorem 3.3,
we immediately get the following.
Corollary 3.1. Let E be a real Radon-Nikodym space [7]. Then each absolutely contin-
uous mapping F : I → E has a compact variation on I. This is true, in particular, for
all reflexive Banach spaces E.
In the conclusion, let us give an example of a nowhere K-subdifferentiable mapping
having compact variation. First, recall the definition of a strongly Pettis integrable
mapping [23].
Definition 3.2. Let E be a real LCS, f : I = [a; b] → E, B(I) be the Borel σ-algebra
over I. We say that f is strongly Pettis integrable over I if f is Pettis integrable over I
and the set
mf (B(I)) =
{
(P )
∫
B
f(x) dx | B ∈ B(I)
}
is relatively compact in E.
The following proposition can be proved quite similarly to Theorem 3.3.
Theorem 3.4. Let E be a real complete LCS, f : I = [a; b] → E be strongly Pettis
integrable over I. Then for an arbitrary C ∈ E, the mapping
(3.14) F (x) = C + (P )
∫ x
a
f(t) dt
has compact variation on I.
Example 3.1. At first note that, according to ([23], Theorem 7), every mapping into
a Banach space, which is strongly measurable and Pettis integrable over I, is strongly
Pettis integrable and, therefore, has compact variation.
Secondly (see [24], remark to Theorem 1), for an arbitrary infinite-dimensional Banach
space E, there exists a strongly measurable and Pettis integrable mapping f : I → E
such that
lim
h→0
∥∥∥
1
h
(P )
∫ t+h
t
f(x) dx
∥∥∥ = ∞ (∀t ∈ I).
This immediately implies that the corresponding mapping (3.14) is nowhere K-subdif-
ferentiable on I.
90 I. V. ORLOV AND F. S. STONYAKIN
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Taurida National V. Vernadsky University, 4 Vernadsky ave., Simpheropol, 95007, Ukraine
E-mail address: old@crimea.edu
E-mail address: oiv@crimea.edu
Taurida National V. Vernadsky University, 4 Vernadsky ave., Simpheropol, 95007, Ukraine
E-mail address: fedyor@mail.ru
Received 22/05/2008; Revised 05/11/2008
|
| id | nasplib_isofts_kiev_ua-123456789-5697 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1029-3531 |
| language | English |
| last_indexed | 2025-11-30T13:23:29Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Orlov, I.V. Stonyakin, F.S. 2010-02-02T12:56:37Z 2010-02-02T12:56:37Z 2009 Compact variation, compact subdifferentiability and indefinite Bochner integral / I.V. Orlov, F.S. Stonyakin // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 1. — С. 74-90. — Бібліогр.: 24 назв. — англ. 1029-3531 https://nasplib.isofts.kiev.ua/handle/123456789/5697 The notions of compact convex variation and compact convex subdifferential for the mappings from a segment into a locally convex space (LCS) are studied. In the case of an arbitrary complete LCS, each indefinite Bochner integral has compact variation and each strongly absolutely continuous and compact subdifferentiable a.e. mapping is an indefinite Bochner integral. en Інститут математики НАН України Compact Variation, Compact Subdifferetiability and Indefinite Bochner Integral Article published earlier |
| spellingShingle | Compact Variation, Compact Subdifferetiability and Indefinite Bochner Integral Orlov, I.V. Stonyakin, F.S. |
| title | Compact Variation, Compact Subdifferetiability and Indefinite Bochner Integral |
| title_full | Compact Variation, Compact Subdifferetiability and Indefinite Bochner Integral |
| title_fullStr | Compact Variation, Compact Subdifferetiability and Indefinite Bochner Integral |
| title_full_unstemmed | Compact Variation, Compact Subdifferetiability and Indefinite Bochner Integral |
| title_short | Compact Variation, Compact Subdifferetiability and Indefinite Bochner Integral |
| title_sort | compact variation, compact subdifferetiability and indefinite bochner integral |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/5697 |
| work_keys_str_mv | AT orloviv compactvariationcompactsubdifferetiabilityandindefinitebochnerintegral AT stonyakinfs compactvariationcompactsubdifferetiabilityandindefinitebochnerintegral |