Table Of ContentOn functions of bounded semivariation
G. A. Monteiro∗
5
1
Abstract
0
2
The concept of bounded variation has been generalized in many ways. In the
n
frame of functions taking values in Banach space, the concept of bounded semi-
a
J variation is a very important generalization. The aim of this paper is to provide
1 an accessible summary on this notion, to illustrate it with an appropriate body of
2
examples, andtooutline its connection with theintegration theory duetoKurzweil.
]
A
2010 Mathematics Subject Classification: 26A45
C
Key words. Semivariation, Kurzweil integral, variation, regulated functions.
.
h
t
a
m 1 Introduction
[
1 Different notions of variation appear when dealing with problems in infinite dimension.
v
Among them, the semivariation is very frequent, being commonly found in studies involv-
6
9 ing convolution, Stieltjes type integration, and also in topics related to vector measures.
1
Initially called w-property, the concept of bounded semivariation for operator-valued
5
0 functions was introduced in 1936 by M. Gowurin in his paper on the Stieltjes integral
1. in Banach space [17]. Some decades later, the Gowurin w-property revealed to be very
0 useful in the investigation of integral representations of continuous linear transformations
5
(see [41] and [13]).
1
: Nowadays, a handful of papers make use of the concept of bounded semivariation.
v
i However, in its majority, the results on such type of variation are only stated with no
X
proofs or no proper references. Besides that, we can observe in the literature a lack of
r
a material collecting basic results on such a concept.
In view of this, the purpose of this survey is to summarize the present knowledge on
semivariation. The presentation does not reflect the chronological order of the discoveries,
but rather attempts to organizeresults ina logical framework. Moreover, inorder to make
∗
MathematicalInstitute, AcademyofSciencesofthe CzechRepublic, Prague,CzechRepublic (email:
[email protected]).Supportedby RVO: 67985840and by the Academic Human Resource Programof the
Academy of Sciences of the Czech Republic.
1
these notes self-contained, most results are presented with a detailed proof and some
illustrative examples are given. We trust that our citations and bibliography sufficiently
identify the appropriate antecedent.
This survey includes, besides basic results and properties, also a section dedicated to
the investigation of the relation between semivariation and non-absolute integrals.
First, let us fix some notation.
Throughout this survey X and Y denote Banach spaces and L(X,Y) stands for the
Banach space of bounded linear operators from X to Y. By k·k and k·k we denote
X L(X,Y)
the norm in X and the usual operator norm in L(X,Y), respectively. In particular, we
write L(X) = L(X,X) and X∗ = L(X,R).
For an arbitrary function f : [a,b] → X we set kfk = sup kf(t)k .
∞ t∈[a,b] X
Consider a nondegenerate closed interval [a,b] and denote by D[a,b] the set of all finite
divisions of [a,b] of the form
D = {α ,α ,...,α }, a=α <α < ... <α =b,
0 1 ν(D) 0 1 ν(D)
where ν(D) ∈ N corresponds to the number of subintervals in which [a,b] is divided.
With these concepts in hand we are ready to define the semivariation of an operator-
valued function.
Definition 1.1. Given a function F : [a,b] → L(X,Y) and a division D ∈ D[a,b], let
ν(D)
V(F,D,[a,b]) = sup F(α )−F(α ) x : x ∈ X, kx k ≤ 1 .
j j−1 j j j X
(cid:13) (cid:13)
(cid:13)Xj=1 (cid:2) (cid:3) (cid:13)Y
(cid:13) (cid:13)
(cid:13) (cid:13)
The semivariation of F on[a(cid:13),b] is then defined by (cid:13)
SVb(F) = sup{V(F,D,[a,b]) : D ∈ D[a,b]}.
a
If SVb(F) < ∞, we say that the function F is of bounded semivariation on [a,b]. The
a
set of all functions F : [a,b] → L(X,Y) of bounded semivariation on [a,b] we denote by
SV([a,b],L(X,Y)).
If no misunderstanding can arise, we write simply V(F,D) instead of V(F,D,[a,b]).
Remark 1.2. The concept presented in Definition 1.1, called w-property in [17], is also
known as (B)-variation, with respect to the bilinear triple B = (L(X,Y),X,Y). For
details, see [33] and [14]. The terminology used in this paper is consistent with that
found in the book by H¨onig [22] and seem to be the most frequent in literature. However,
we call the readers attention to the fact that the term ‘semivariation’ might also appear
with slight different formulation - for example, when applied to measure theory or to
functions with values in a general Banach space. See, for instance, [7], [9] or, in the
frames of functions with values in locally convex spaces, [11].
2
It is not hard to see that the semivariation is more general than the notion of variation
in the sense of Jordan. Indeed, for F : [a,b] → L(X,Y), we have
SVb(F) ≤ varb(F)
a a
where varb(F) stands for the variation of F on [a,b] and is given by
a
ν(D)
varb(F) = sup kF(α )−F(α )k : D ∈ D[a,b] .
a j j−1 L(X,Y)
Xj=1
Denoting by BV([a,b],L(X,Y)) the set of all functions F : [a,b]→ L(X,Y) of
bounded variation on [a,b] (i.e varb(F) < ∞), clearly,
a
BV([a,b],L(X,Y)) ⊆ SV([a,b],L(X,Y)).
The relation between these two sets will be analysed in more details in Section 4.
The following example of a function of bounded semivariation was inspired by some
ideas found in [40].
Example 1.3. Let ℓ be the Banach space of sequences x = {x } in R such that the
2 n n
series ∞ |x |2 converges, equipped with the norm
n=1 n
P ∞ 1/2
kxk = |x |2
2 n
(cid:16)Xn=1 (cid:17)
Denote by e , k ∈ N, the canonical Schauder basis of ℓ , where e is the sequence whose
k 2 k
k-th term is 1 and all other terms are zero.
For each k ∈ N, consider y ∈ ℓ given by y = 1e , that is,
k 2 k k k
1
y = {y(k)} with y(k) = and y(k) = 0 for n 6= k.
k n n k k n
Note that the series ∞ y converges in ℓ and denote by S its sum.
k=1 k 2
Let F : [0,1] → L(R,ℓ ) be given by
P 2
n
x y if t ∈ ( 1 , 1], n ∈ N,
F(t) x = k n+1 n
k=1
XxS if t = 0
(cid:0) (cid:1)
for t ∈ [0,1] and x ∈ R.
In order to prove that F ∈ SV([0,1],L(R,ℓ )), let us consider D ∈ D[0,1] with
2
D = {α ,α ,...,α }. Put
0 1 ν(D)
n = max{k ∈ N : kα ≤ 1} for j = 1,...,ν(D),
j j
3
and Λ = {j : n < n } ⊂ {2,...,ν(D)}. For x ∈ R, j = 1,...,ν(D) with |x | ≤ 1 we
j j−1 j j
have F(α )x = nj y and consequently
j j k=1 k
ν(D) P n1 ν(D) nj nj−1
[F(α )−F(α )]x = x y −S + x y − y
j j−1 j 1 k j k m
Xj=1 (cid:16)Xk=1 (cid:17) Xj=2 (cid:16)Xk=1 mX=1 (cid:17)
∞ nj−1
= −x y − x y
1 k j k
(cid:16)k=Xn1+1 (cid:17) Xj∈Λ (cid:16)k=Xnj+1 (cid:17)
For k ∈ N, define
−x if n < k ≤ n , j ∈ Λ
j j j−1
λ = −x if k > n . (1.1)
k 1 1
0 otherwise
Using this sequence and thedefinition of y , we can write
k
ν(D) ∞ ∞ ∞
2 2 λ 2 1
[F(α )−F(α )]x = λ y = k ≤ .
j j−1 j k k k k2
2 2
(cid:13)Xj=1 (cid:13) (cid:13)Xk=1 (cid:13) Xk=2(cid:12) (cid:12) Xk=2
(cid:13) (cid:13) (cid:13) (cid:13) (cid:12) (cid:12)
(cid:13) ∞ (cid:13) (cid:13) (cid:13) (cid:12) (cid:12)
1 1/2 π2
This shows that SV1(F) ≤ = −1
0 k2 6
r
(cid:16)Xk=2 (cid:17)
We claim that SV1(F) = π2 −1. Indeed, fixed an arbitrary N ∈ N, consider
0 6
D ∈ D[0,1] given by q
N
1 1 1
D = 0, , ,..., ,1 ,
N
N N −1 2
Thus, for x ∈ R, j = 1,...,N withn|x | ≤ 1 we have o
j j
N−1
[F(1)−F( 1 )]x +[F(1)−F(0)]x
ℓ ℓ+1 ℓ N N
2
(cid:13)Xℓ=1 (cid:13)
(cid:13) (cid:13)
N−1 ∞ ∞
(cid:13) (cid:13) x˜ 2 1/2
= x y + x y = k
ℓ ℓ+1 N k k
2
(cid:13)Xℓ=1 k=XN+1 (cid:13) (cid:16)Xk=2(cid:12) (cid:12) (cid:17)
(cid:13) (cid:13) (cid:12) (cid:12)
where x˜ = x if k = 2,..(cid:13).,N, and x˜ = x for k ∈(cid:13) N, k > N(cid:12). T(cid:12)aking the supremum
k k−1 k N
over all possible choices of x ∈ R, j = 1,...,N with |x | ≤ 1 we obtain
j j
∞
1 1/2
V(F,D ,[0,1]) = ,
N k2
(cid:16)Xk=2 (cid:17)
which proves the claim.
4
Remark 1.4. In the particular case X = R the space SV([a,b],L(R,Y)) can be regarded
as the space of the functions of weak bounded variation, usually denoted by BW([a,b],Y)
(c.f. [20]). This is clear once we recall that the weak variation of a function f : [a,b] → Y
is given by Wb(f) = sup{W(f,D) : D ∈ D[a,b]} where
a
ν(D)
W(f,D) = sup f(α )−f(α ) λ : λ ∈ R, |λ | ≤ 1 for D ∈ D[a,b].
j j−1 j j j
Y
(cid:13)Xj=1 (cid:2) (cid:3) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
Therefore, wecansaythatthesemivariationofthefunctionF : [0,1] → L(R,ℓ )inthe
2
Example 1.3 coincides with the weak variation of f : [0,1] → ℓ given by f(t) = (F(t))1
2
for t ∈ [0,1].
2 Semivariation: basic results
This section summarizes basic properties of the semivariation that are often mentioned
without proof in papers which are directly or indirectly connected to such a notion. In
order to make this work as complete as possible, all the proofs are included. Most of the
results can be found, for instance, in [20], [22] and [37].
We start by noting that SV([a,b],L(X,Y)) is a vector space.
Proposition 2.1. Let F, G ∈ SV([a,b],L(X,Y)) and λ ∈ R be given. Then both func-
tions (F +G) and (λF) are of bounded semivariation on [a,b], and
SVb(F +G) ≤ SVb(F)+SVb(G) and SVb(λF) = |λ| SVb(F). (2.1)
a a a a a
Proof. The assertions follow from the fact that the relations
V(F +G,D) ≤ V(F,D)+V(G,D) and V(λF,D) = |λ|V(F,D)
hold for every division D ∈ D[a,b].
According to (2.1), SVb(·) defines a seminorm on the space of functions of bounded
a
semivariation. On the other hand, if we put
kFk = kF(a)k +SVb(F) for F ∈ SV([a,b],L(X,Y), (2.2)
SV L(X,Y) a
then SV([a,b],L(X,Y)) becomes a normed space. This fact is a consequence of (2.1)
together with the following assertion.
Proposition 2.2. Let F ∈ SV([a,b],L(X,Y)). Then SVb(F) = 0 if and only if F ≡ C
a
for some fixed operator C ∈ L(X,Y).
5
Proof. Clearly, the semivariation of a constant function is zero. Conversely, assume that
SVb(F) = 0. Given t ∈ (a,b], if we consider the division D = {a,t,b} of [a,b], for any
a
x ∈ X with kxk ≤ 1 we have
X
kF(t)x−F(a)xk = [F(t)−F(a)]x+[F(b)−F(t)]0 ≤ V(F,D,[a,b]).
Y Y
(cid:13) (cid:13)
Therefore [F(t)−F(a)] = 0(cid:13)∈ L(X,Y), that is, F is a consta(cid:13)nt function.
Remark 2.3. It is worth mentioning that in the definition of the norm k · k we can
SV
use the fixed value of the function in any point of the interval, that is, taking c ∈ [a,b],
we can consider
kFk = kF(c)k +SVb(F), F ∈ SV([a,b],L(X,Y)).
SV L(X,Y) a
The choice of the left-ending point of the interval seems to be the most common in the
literature, though. Therefore, in this work, we assume the norm in SV([a,b],L(X,Y)) as
introduced in (2.2).
Note that, for F : [a,b] → L(X,Y) and t ∈ [a,b] we have
kF(t)k ≤ kF(a)k +SVb(F).
L(X,Y) L(X,Y) a
Hence, every function F ∈ SV([a,b],L(X,Y)) is bounded and
kFk ≤ kFk .
∞ SV
In view of this, we can say that the topology induced in the space SV([a,b],L(X,Y)) by
the supremum norm is weaker than the one induced by k·k .
SV
Inthesequel weprove thatthespace offunctions ofboundedsemivariation iscomplete
when equipped with the norm k·k (c.f. [37, Proposition 4] or [20, I.3.3]). To this aim,
SV
we will need the following convergence result.
Lemma 2.4. Let F : [a,b] → L(X,Y), a sequence {F } ⊂ SV([a,b],L(X,Y)) and a
n n
constant M > 0 be such that
SVb(F ) ≤ M for every n ∈ N,
a n
and
lim kF (t)x−F(t)xk = 0 for every t ∈ [a,b] and x ∈ X.
n Y
n→∞
Then SVb(F) ≤ M.
a
6
Proof. Let D = {α ,α ,...,α } be a division of [a,b] and let x ∈ X, j = 1,...,ν(D)
0 1 ν(D) j
with kx k ≤ 1. Note that, for each n ∈ N, we have
j X
ν(D)
[F(α )−F(α )]x
j j−1 j
Y
(cid:13)Xj=1 (cid:13)
(cid:13) (cid:13)
(cid:13) ν(D) (cid:13) ν(D)
≤ [F (α )−F (α )]x + [F(α )−F (α )−F(α )+F (α )]x
n j n j−1 j j n j j−1 n j−1 j
Y Y
(cid:13)Xj=1 (cid:13) (cid:13)Xj=1 (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) ν(D) (cid:13) (cid:13)ν(D) (cid:13)
≤ M + k[F(α )−F (α )]x k + k[F(α )−F (α )]x k (2.3)
j n j j Y j−1 n j−1 j Y
j=1 j=1
X X
Given ε > 0, there is N ∈ N such that
D
ε
k[F(α )−F (α )]x k < for j = 0,1,...,ν(D),
j ND j i Y 2ν(D)
where i = j,j+1 (whenever it has a sense). Therefore, taking n = N in (2.3) we obtain
D
ν(D)
[F(α )−F(α )]x < M +ε
j j−1 j
Y
(cid:13)Xj=1 (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
which implies that
V(F,D) ≤ M +ε.
Since ε > 0 is arbitrary, it follows that V(F,D) ≤ M for every D ∈ D[a,b], and conse-
quently SVb(F) ≤ M.
a
The previous convergence result usually appears applied to some integration theory
This type of result, often mentioned as Helly-Bray theorem (cf. [22, Theorem I.5.8] or
[10]), will be study in Section 5 in the frames of Kurzweil-Stieltjes integral.
Now, we are ready to prove the completeness of the space SV([a,b],L(X,Y)).
Theorem 2.5. SV([a,b],L(X,Y)) is a Banach space with respect to the norm k·k .
SV
Proof. Let {F } be a Cauchy sequence in SV([a,b],L(X,Y)) with respect to the norm
n n
k·k . This means that given ε > 0 there exists n ∈ N such that
SV 0
kF (t)−F (t)k ≤ kF −F k < ε, n, m ≥ n and t ∈ [a,b]. (2.4)
n m L(X,Y) n m SV 0
Hence, for each t ∈ [a,b], {F (t)} is a Cauchy sequence in L(X,Y) which implies that
n n
there exists F(t) ∈ L(X,Y) such that
lim kF (t)−F(t)k = 0.
n L(X,Y)
n→∞
7
Moreover, due to (2.4), this convergence is uniform on [a,b]. By the fact that {F } is a
n n
Cauchy sequence there exists M > 0 such that SVb(F ) ≤ M for every n ∈ N. Therefore,
a n
by Lemma 2.4 F ∈ SV([a,b],L(X,Y)).
It remains to show that the convergence is true also in the topology induced by the
norm k · k . To this aim, consider a division D = {α ,α ,...,α } of [a,b] and
SV 0 1 ν(D)
arbitrary x ∈ X, j = 1,...,ν(D) with kx k ≤ 1. By (2.4), for n, m ≥ n , we have
j j X 0
ν(D)
[F (α )−F (α )−F (α )+F (α )]x < ε.
n j m j n j−1 m j−1 j
Y
(cid:13)Xj=1 (cid:13)
(cid:13) (cid:13)
Thus taking the(cid:13)limit m → ∞ we obtain (cid:13)
ν(D)
[F (α )−F(α )−F (α )+F(α )]x ≤ ε,
n j j n j−1 j−1 j
Y
(cid:13)Xj=1 (cid:13)
(cid:13) (cid:13)
that is, V((F −(cid:13)F),D) ≤ ε, for n ≥ n . Since the division D (cid:13)∈ D[a,b] is arbitrary, it
n 0
follows that lim SV(F −F) = 0, concluding the proof.
n→∞ n
The following theorem proves that the functions of bounded variation are multipliers
for the space SV([a,b],L(X,Y)) (see [22, Lemma I.1.11]).
Theorem 2.6. Let F ∈ SV([a,b],L(X,Y)) and G ∈ BV([a,b],L(X)). Consider the
function FG : [a,b] → L(X,Y) given by (FG)(t) = F(t)G(t) for t ∈ [a,b]. Then
F G ∈ SV([a,b],L(X,Y)) and
SVb(F G) ≤ kFk varb(G)+kGk SVb(F).
a ∞ a ∞ a
Proof. Consider adivisionD = {α ,α ,...,α }of[a,b]andletx ∈ X,j = 1...,ν(D)
0 1 ν(D) j
with kx k ≤ 1. Therefore
j X
ν(D)
[(F G)(α )−(F G)(α )]x
j j−1 j
Y
(cid:13)Xj=1 (cid:13)
(cid:13) (cid:13)
(cid:13) ν(D) (cid:13)
= [F(α )G(α )−F(α )G(α )+F(α )G(α )−F(α )G(α )]x
j j j j−1 j j−1 j−1 j−1 j
Y
(cid:13)Xj=1 (cid:13)
(cid:13) (cid:13)
(cid:13)ν(D) ν(D) (cid:13)
≤ F(α )[G(α )−G(α )]x + [F(α )−F(α )]G(α )x
j j j−1 j j j−1 j−1 j
Y Y
(cid:13)Xj=1 (cid:13) (cid:13)Xj=1 (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) ν(D) (cid:13) (cid:13) ν(D) (cid:13)
G(α )x
≤ kFk kG(α )−G(α )k +kGk [F(α )−F(α )] j−1 j
∞ j j−1 L(X) ∞(cid:13) j j−1 kGk (cid:13)
j=1 (cid:13)j=1 ∞ (cid:13)
X (cid:13)X (cid:13)Y
≤ kFk∞ varba(G)+kGk∞ SVba(F). (cid:13)(cid:13) (cid:13)(cid:13)
(cid:13) (cid:13)
8
This implies that
V((FG),D) ≤ kFk varb(G)+kGk SVb(F)
∞ a ∞ a
for every D ∈ D[a,b], wherefrom the result follows.
The next theorem presents some algebraic properties of the semivariation.
Theorem 2.7. If F : [a,b] → L(X,Y) and [c,d] ⊂ [a,b], then
SVd(F) ≤ SVb(F).
c a
Moreover,
SVb(F) ≤ SVc(F)+SVb(F) for c ∈ [a,b]. (2.5)
a a c
Proof. It is easy to see that, for every division D of [c,d], taking D˜ = D∪{a,b}, we have
D˜ ∈ D[a,b] and
V(F,D,[c,d]) ≤ V(F,D˜,[a,b]) ≤ SVb(F),
a
Therefore SVd(F) ≤ SVb(F).
c a
To prove the superadditivity, given c ∈ [a,b] and an arbitray division D ∈ D[a,b],
consider D = (D ∩ [a,c]) ∪ {c} and D = (D ∩ [c,b]) ∪ {c}. Clearly, D and D are
1 2 1 2
divisions of [a,c] and [c,b], respectively. In addition,
V(F,D,[a,b]) ≤ Vb(F,D∪{c},[a,b]) ≤ V(F,D ,[a,c])+V(F,D ,[c,b]).
a 1 2
Hence
V(F,D,[a,b]) ≤ SVc(F)+SVb(F) for every D ∈ D[a,b],
a c
which leads to the inequality (2.5).
According to the previous theorem: if F ∈ SV([a,b],L(X,Y)), then F is of bounded
semivariation on each closed subinterval of [a,b]. As a consequence we have the following.
Corollary 2.8. Let F ∈ SV([a,b],L(X,Y)) be given. Then
1. the mapping t ∈ [a,b] 7−→ SVt(F) is nondecreasing;
a
2. the mapping t ∈ [a,b] 7−→ SVb(F) is nonincreasing.
t
Theorem2.7indicatesthat, unlikethevariation,thesemivariationneednotbeadditive
with respect to intervals. Next example shows that the inequality in (2.5) may be strict.
Example 2.9. Let F : [0,1] → L(R,ℓ ) be the function given on Example 1.3, that is,
2
for t ∈ [0,1] and x ∈ R,
n
x y if t ∈ ( 1 , 1], n ∈ N,
F(t) x = k n+1 n
k=1
XxS if t = 0
(cid:0) (cid:1)
where yk = k1ek ∈ ℓ2 1for k ∈ N,and S = ∞k=1yk.
1For k ∈N, ek denotes an element of the canoPnical Schauder basis of ℓ2.
9
We will prove that
1
SV1(F) < SV2(F)+SV1(F). (2.6)
0 0 1
2
1
First, let us calculate SV2(F).
0
Given a division D = {α ,α ,...,α } of [0, 1], as in Example 1.3, put
0 1 ν(D) 2
n = max{k ∈ N : kα ≤ 1} for j = 1,...,ν(D),
j j
and Λ = {j : n < n } ⊂ {2,...,ν(D)}. For x ∈ R, j = 1,...,ν(D) with |x | ≤ 1 we
j j−1 j j
have
ν(D) ∞ ∞ ∞
λ 2 1/2 1 1/2
[F(α )−F(α )]x = λ y = k ≤ .
j j−1 j k k k k2
2 2
(cid:13)Xj=1 (cid:13) (cid:13)Xk=1 (cid:13) (cid:16)Xk=3(cid:12) (cid:12) (cid:17) (cid:16)Xk=3 (cid:17)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:12) (cid:12)
where(cid:13)λ for k ∈ N, k ≥ 3, is g(cid:13)iven (cid:13)as in (1.1)(cid:13)(note that(cid:12)the(cid:12) corresponding n satisfies
k j
n ≥ 2, j = 1,...,ν(D)). In view of this, it is clear that
j
∞
1 1 1/2 π2 5
SV2(F) ≤ = − .
0 k2 6 4
r
(cid:16)Xk=3 (cid:17)
1
The equality SV2(F) = π2 − 5 is a consequence of the fact that
0 6 4
q
∞
1 1 1/2
V(F,D ,[0, ]) =
N 2 k2
(cid:16)Xk=3 (cid:17)
for any division D = 0, 1, 1 ,..., 1 with N ∈ N.
N N N−1 2
On the other hand,nit is not hard toosee that SV1(F) = 1. Indeed, for any division
1 2
2
D = {α ,α ,...,α } of [1,1] and for any choice of x ∈ R, j = 1,...,ν(D) with
0 1 ν(D) 2 j
|x | ≤ 1 we have
j
ν(D)
1 x
[F(α )−F(α )]x = [F(α )−F( )]x = −x y = − 1 e
j j−1 j 1 1 1 2 2
2 2
j=1
X
Hence, V(F,D,[1,1]) = 1.
2 2
Recalling that SV1(F) = π2 −1, we conclude that (2.6) holds.
0 6
q
In the sequel we provide some further characterizations of the semivariation of a func-
tion. The first one, Theorem 2.10, can be found for instance in [34, Proposition 1.1] or
[22, Theorem I.4.4]. Basically, it connects the notions of semivariation and B∗-variation,
with respect to the bilinear triple B∗ = (L(X,Y),L(X),L(X,Y)) (for definition see [22]).
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