EQUICONTINUITY AND SINGULARITIES OF FAMILIES OF MONOMIAL MAPPINGS
WOLFGANG W. BRECKNER and TIBERIU TRIF
Dedicated to Professor S¸tefan Cobza¸s at his60th anniversary
Abstract. The starting-point for the present paper is the principle of condensation of the singularities of families consisting of continuous linear mappings that act between normed linear spaces. It is proved that this basic functional analytical principle can be generalized for families of con- tinuous monomial mappings of degreenbetween topological linear spaces.
The obtained principle yields a generalization of the principle of uniform boundedness published by I. W. Sandberg [IEEE Trans. Circuits and Sys- tems CAS-32 (1985), 332–336] and recently rediscovered by R. Miculescu [Math. Reports (Bucharest) 5 (55) (2003), 57–59]. Furthermore, by ap- plying the new nonlinear principle there are revealed Baire category prop- erties of certain subsets of the normed linear spaceC[a, b] involved with Riemann-Stieltjes integrability.
1. Introduction
One of the most important and most useful results in the theory of real or complex normed linear spaces is the following theorem, known as the principle of condensation of the singularities.
Theorem 1.1. Let X and Y be normed linear spaces, and let(Fj)j∈J be a family of continuous linear mappings fromX intoY such that
sup{ kFjk |j ∈J}=∞.
Received by the editors: 06.04.2006.
2000Mathematics Subject Classification.Primary 47H60, secondary 54E52, 26A42.
Key words and phrases. Topological linear spaces, monomial mappings, equicontinuity, residual sets, Riemann-Stieltjes integrability.
Then the set of allx∈X satisfying
sup{ kFj(x)k | j∈J}=∞ is residual, i.e. its complement is a set of the first category.
This theorem immediately provides the next theorem called the principle of uniform boundedness and considered to represent also a major functional analytical result.
Theorem 1.2. LetX andY be normed linear spaces of whichX is complete, and let(Fj)j∈J be a family of continuous linear mappings fromX intoY. Then
sup{ kFj(x)k | j∈J}<∞ for allx∈X if and only if
sup{ kFjk |j ∈J}<∞.
Both these theorems have been extensively investigated and have been gener- alized in several directions. In some papers more general spaces have been considered instead of the normed linear spacesXandY. For instance, S¸. Cobza¸s and I. Muntean [5] dealt with the topological structure of the set of singularities associated with a nonequicontinuous family of continuous linear mappings from a topological linear space into another topological linear space and pointed out cases when this set of singularities is an uncountable infinite Gδ-set. In other papers the linear mappings Fj (j ∈J) have been replaced by nonlinear mappings of a certain type. Moreover, W. W. Breckner [1] has proved a very general principle of condensation of the singu- larities which does not require any algebraic structure of the considered spaces and neither assumptions as to the shape of the mappings that are concerned.
For a detailed information on diverse generalizations of the Theorems 1.1 and 1.2 the reader is referred to the surveys by W. W. Breckner [2, 3] as well as to T. Trif [14, 15].
In the present paper we deal with the equicontinuity of families of monomial mappings. Moreover, by following the general line of proving principles of condensa- tion of the singularities we show that the Theorems 1.1 and 1.2 can be generalized for families of continuous monomial mappings of degreenacting between topological linear spaces. Consequently, these generalizations integrate well into the framework described in [1]. Besides, it should be mentioned that the new principle of uniform boundedness turns out to be a generalization of the principle of uniform boundedness proved by I. W. Sandberg [11] and recently rediscovered by R. Miculescu [9]. The paper ends with an application of the obtained nonlinear principle of condensation of singularities that directly reveals Baire category properties of certain subsets of the normed linear spaceC[a, b] involved with Riemann-Stieltjes integrability.
2. Monomial mappings
All linear spaces as well as all topological linear spaces that will occur in this paper are over K, where K is either the field R of real numbers or the field C of complex numbers. IfX is a linear space, then its zero-element is denoted byoX. The set of all positive integers isN.
Throughout this section letX andY be linear spaces. Furthermore, letnbe a positive integer. A mappingF:Xn →Y is said to be:
(i)symmetric if
F(x1, . . . , xn) =F(xσ(1), . . . , xσ(n))
for each (x1, . . . , xn)∈Xn and each bijectionσ: {1, . . . , n} → {1, . . . , n};
(ii)n-additive if for eachi∈ {1, . . . , n} the mapping
∀x∈X 7−→ F(x1, . . . , xi−1, x, xi+1, . . . , xn)∈Y is additive wheneverx1, . . . , xi−1, xi+1, . . . , xn∈X are fixed.
IfF:Xn→Y is ann-additive mapping, then it can be shown that F(r1x1, . . . , rnxn) =r1· · ·rnF(x1, . . . , xn)
for allx1, . . . , xn ∈X and all rational numbersr1, . . . , rn. If in additionX andY are topological linear spaces andF is continuous, then we even have
F(a1x1, . . . , anxn) =a1· · ·anF(x1, . . . , xn) for allx1, . . . , xn ∈X and alla1, . . . , an∈R.
Given a mappingF:Xn→Y, the mapping F∗:X →Y, defined by F∗(x) :=F(x, . . . , x
| {z } ntimes
) for allx∈X,
is said to be the diagonalization of F. Any symmetricn-additive mapping can be expressed by means of its diagonalization as the following proposition points out (see A. M. McKiernan [8, Lemma 1] or D. ˇZ. Djokovi´c [6, Lemma 2]).
Proposition 2.1. If F:Xn→Y is a symmetricn-additive mapping, then F(u1, . . . , un) = 1
n!(∆u1· · ·∆unF∗) (x)
for allu1, . . . , un, x∈X, where∆u:YX →YX is defined for eachu∈X by (∆uf) (x) :=f(x+u)−f(x) for allf ∈YX and all x∈X.
A mappingQ:X →Y is said to be amonomial mapping of degree nif there exists a symmetric n-additive mapping F:Xn → Y such that Q = F∗. In virtue of Proposition 2.1 there exists for each monomial mappingQ:X →Y of degreena single symmetricn-additive mappingF:Xn→Y such thatQ=F∗.
A monomial mappingQ: X →Y of degreenhas the homogeneity property Q(rx) =rnQ(x) for everyx∈X and every rational number r. If in additionX and Y are topological linear spaces andQis continuous, this property implies
Q(ax) =anQ(x) for every x∈X and everya∈R.
Finally, we mention a useful characterization of the monomial mappings of degreen(see A. M. McKiernan [8, Corollary 3] or D. ˇZ. Djokovi´c [6, Corollary 3]).
Proposition 2.2. A mappingQ:X →Y is a monomial mapping of degree nif and only if
1
n!(∆nuQ) (x) =Q(u) for all u, x∈X.
The monomial mappings of degree 1 coincide with the additive mappings, while the monomial mappings of degree 2 are calledquadratic.
3. Equicontinuity of families of monomial mappings
LetX andY be topological linear spaces, and let F:= (Fj)j∈J be a family of mappings fromX intoY. Ifxis a point inX, thenF is said to beequicontinuous at xif for every neighbourhoodV ofoY there exists a neighbourhoodU ofoX such that
{Fj(x+u)−Fj(x)|j∈J} ⊆V for allu∈U.
IfF is equicontinuous at each point ofX, thenF is said to beequicontinuous on X.
For families of symmetricn-additive mappings the following characterization of the equicontinuity is valid.
Theorem 3.1. Let nbe a positive integer, letX andY be topological linear spaces, letF := (Fj)j∈J be a family of symmetric n-additive mappings fromXn into Y, and letF∗:= (Fj∗)j∈J. Then the following assertions are equivalent:
1◦ F∗ is equicontinuous onX. 2◦ F∗ is equicontinuous at oX. 3◦ F is equicontinuous at oXn. 4◦ F is equicontinuous onXn.
Proof. Since the implications 1◦⇒2◦ and 4◦ ⇒3◦ are obvious, it remains to prove that 2◦⇒3◦, 3◦⇒1◦ and 1◦⇒4◦.
We start by proving the implication 2◦⇒3◦. Let V be any neighbourhood ofoY. Choose a balanced neighbourhoodV0ofoY such that
V0+· · ·+V0
| {z } 2nterms
⊆V. (1)
The equicontinuity ofF∗ at oX ensures the existence of a neighbourhood U0 of oX such that
{Fj∗(u)|j∈J} ⊆V0 for allu∈U0. (2) Now select a neighbourhoodU ofoX such that
U+· · ·+U
| {z } nterms
⊆U0. (3)
We claim that
{Fj(u1, . . . , un)|j∈J} ⊆V for all (u1, . . . , un)∈Un. (4) Indeed, letj be any index in J and let (u1, . . . , un) be any point in Un. According to Proposition 2.1 we have
Fj(u1, . . . , un) = 1
n! ∆u1· · ·∆unFj∗ (oX)
= 1 n!
X
(a1,...,an)∈A
(−1)n−(a1+···+an)Fj∗(a1u1+· · ·+anun), (5)
whereA:={0,1}n. Since
a1u1+· · ·+anun∈U+· · ·+U
| {z } nterms
⊆U0 for all (a1, . . . , an)∈A,
we conclude in virtue of (2) that
Fj∗(a1u1+· · ·+anun)∈V0 for all (a1, . . . , an)∈A.
Taking into account that cardA= 2n and thatV0 is balanced, we get by (5) Fj(u1, . . . , un)∈V0+· · ·+V0
| {z } 2nterms
⊆V.
Consequently, (4) is true. From (4) it follows thatF is equicontinuous atoXn. Next we prove that 3◦ ⇒ 1◦. Let x be any point inX, and let V be any neighbourhood ofoY. Choose a balanced neighbourhoodV0 ofoY such that
V0+· · ·+V0
| {z } 2n−1 terms
⊆V.
The equicontinuity of F at oXn ensures the existence of a balanced neighbourhood U0 ofoX such that
{Fj(u1, . . . , un)|j∈J} ⊆V0 for all (u1, . . . , un)∈U0n. (6) Select a rational numberr∈]0,1] such thatrx∈U0. We assert that
{Fj∗(x+rn−1u)−Fj∗(x)| j∈J} ⊆V for allu∈U0. (7) Indeed, letj∈J andu∈U0 be arbitrarily chosen. Then we have
Fj∗(x+rn−1u)−Fj∗(x)
=Fj(x+rn−1u, . . . , x+rn−1u
| {z }
ntimes
)−Fj(x, . . . , x
| {z } ntimes
)
=
n
X
k=1
n k
Fj( x, . . . , x
| {z } n−ktimes
, rn−1u, . . . , rn−1u
| {z } k times
)
=
n
X
k=1
n k
Fj(rx, . . . , rx
| {z } n−ktimes
, rk−1u, rn−1u, . . . , rn−1u
| {z } k−1 times
). (8)
SinceU0 is balanced andr∈]0,1], we see that (6) implies Fj(rx, . . . , rx
| {z } n−ktimes
, rk−1u, rn−1u, . . . , rn−1u
| {z } k−1 times
)∈V0
for eachk∈ {1, . . . , n}. Therefore it follows from (8) that Fj∗(x+rn−1u)−Fj∗(x)∈V0+· · ·+V0
| {z } 2n−1 terms
⊆V.
Hence (7) is true. If we setU :=rn−1U0, thenU is a neighbourhood ofoX satisfying {Fj∗(x+u)−Fj∗(x)|j∈J} ⊆V for allu∈U.
Consequently,F∗is equicontinuous atx.
Finally, we prove that 1◦ ⇒ 4◦. Let (x1, . . . , xn) be any point in Xn, and letV be any neighbourhood ofoY. Choose a balanced neighbourhoodV0 ofoY such
that (1) holds. Let A:={0,1}n. SinceF∗ is equicontinuous onX, there exists for each (a1, . . . , an)∈Aa neighbourhoodUa ofoX such that
{Fj∗(a1x1+· · ·+anxn+u)−Fj∗(a1x1+· · ·+anxn)|j∈J} ⊆V0 (9) for allu∈Ua. Next we choose a neighbourhoodU ofoX such that
U+· · ·+U
| {z } nterms
⊆ \
a∈A
Ua.
Then it results from (9) that
{Fj∗(a1(x1+u1) +· · ·+an(xn+un))−Fj∗(a1x1+· · ·+anxn)|j ∈J} ⊆V0 (10) for all (a1, . . . , an)∈A and all (u1, . . . , un)∈Un. But, according to Proposition 2.1 we have
Fj(x1+u1, . . . , xn+un)−Fj(x1, . . . , xn)
= 1 n!
∆x1+u1· · ·∆xn+unFj∗
(oX)− ∆x1· · ·∆xnFj∗ (oX)
= 1 n!
X
(a1,...,an)∈A
(−1)n−(a1+···+an)
Fj∗(a1(x1+u1) +· · ·+an(xn+un))
−Fj∗(a1x1+· · ·+anxn) for allj ∈J and all (u1, . . . , un)∈Un. By (10) it follows that
{Fj(x1+u1, . . . , xn+un)−Fj(x1, . . . , xn)|j ∈J} ⊆V0+· · ·+V0
| {z } 2nterms
⊆V
for all (u1, . . . , un)∈Un. Consequently, F is equicontinuous at (x1, . . . , xn).
Corollary 3.2. Let nbe a positive integer, letX andY be topological linear spaces, and letQ:= (Qj)j∈J be a family of monomial mappings of degreenfrom X intoY. Then Q is equicontinuous on X if and only if it is equicontinuous at some point ofX.
Proof. Necessity. Obvious.
Sufficiency. Suppose that x ∈ X is a point at which Q is equicontinuous.
ThenQis equicontinuous atoX. Indeed, whenx=oX, then this assertion is trivial.
Whenx6=oX, then it can be proved as follows. LetV be any neighbourhood ofoY. Choose a balanced neighbourhoodV0 ofoY such that
V0+· · ·+V0
| {z } nterms
⊆V.
The equicontinuity ofQatxensures the existence of a neighbourhoodU0ofoX such that
{Qj(x+u)−Qj(x)|j∈J} ⊆V0 for allu∈U0. (11) Select a neighbourhoodU ofoX that satisfies (3). Taking into account that
n
X
k=0
(−1)n−k n
k
= 0, the Proposition 2.2 implies
Qj(u) = 1
n!(∆nuQj) (x)
= 1 n!
n
X
k=0
(−1)n−k n
k
Qj(x+ku)
= 1 n!
n
X
k=1
(−1)n−k n
k
[Qj(x+ku)−Qj(x)] (12) for allj ∈J and allu∈U. Since
ku∈U +· · ·+U
| {z } kterms
⊆U0
for allk∈ {1, . . . , n}and allu∈U, it follows from (11) that {Qj(x+ku)−Qj(x)|j∈J} ⊆V0
for allk∈ {1, . . . , n}and allu∈U. SinceV0 is balanced, (12) implies that {Qj(u)|j∈J} ⊆V0+· · ·+V0
| {z } nterms
⊆V for allu∈U.
Consequently,Q is equicontinuous atoX.
By applying now the implication 2◦ ⇒1◦ stated in Theorem 3.1, it follows
thatQis equicontinuous onX.
Corollary 3.3. Let nbe a positive integer, letX andY be topological linear spaces, and letQ:X →Y be a monomial mapping of degreen. ThenQis continuous onX if and only if it is continuous at some point ofX.
In the special case whenn= 1 this corollary is well-known. Whenn= 2 it generalizes a result stated by S. Kurepa [7, Theorem 2] under the assumption thatXis a normed linear space andY =R. In addition we note that a similar continuity result involving quadratic set-valued mappings was obtained by W. Smajdor [12, Theorem 4.2].
Proposition 3.4. Letn be a positive integer, letX andY be normed linear spaces, and let (Fj)j∈J be a family of symmetric n-additive mappings fromXn into Y. Then the following assertions are equivalent:
1◦ (Fj∗)j∈J is equicontinuous at oX.
2◦ sup{ kFj(x1, . . . , xn)k |j∈J, kx1k ≤1, . . . , kxnk ≤1}<∞.
3◦ sup{ kFj∗(x)k |j∈J, kxk ≤1}<∞.
Proof. 1◦ ⇒2◦ According to the implication 2◦ ⇒3◦ in Theorem 3.1, the family (Fj)j∈J is equicontinuous at oXn. Therefore there exists a neighbourhoodU ofoX such that
kFj(u1, . . . , un)k ≤1 for allj∈J and all (u1, . . . , un)∈Un.
Letrbe a positive rational number such that{x∈X | kxk ≤r} ⊆U. Then we have kFj(x1, . . . , xn)k= 1
rnkFj(rx1, . . . , rxn)k ≤ 1 rn
for all j ∈J and all (x1, . . . , xn)∈Xn satisfyingkx1k ≤ 1, . . . , kxnk ≤1. Conse- quently, assertion 2◦ is true.
2◦⇒3◦ Obvious.
3◦ ⇒1◦ LetV be a neighbourhood ofoY. Choose a positive real numbera such that{y ∈Y | kyk ≤a} ⊆V. In addition, choose a positive rational numberr such thatbrn ≤a, where
b:= sup{ kFj∗(x)k |j∈J, kxk ≤1}.
Then we have
kFj∗(u)k=rn
Fj∗ 1
ru
≤brn ≤a
for allj ∈ J and all u∈ X with kuk ≤ r. Consequently, the neighbourhood U :=
{x∈X | kxk ≤r} ofoX satisfies
{Fj∗(u)|j∈J} ⊆V for allu∈U.
Hence (Fj∗)j∈J is equicontinuous atoX.
4. Singularities of families of monomial mappings
LetX andY be topological linear spaces, and let F:= (Fj)j∈J be a family of mappings fromX intoY. Ifxis a point inX, thenFis said to bebounded at xif the set{Fj(x)|j ∈J}is bounded, i.e. for each neighbourhoodV ofoY there exists a positive real numberasuch that{Fj(x)|j∈J} ⊆aV. IfM is a subset ofX and F is bounded at each point ofM, thenF is said to be pointwise bounded on M.
Any point inX at which F is not bounded is said to be a singularity ofF.
The set of all singularities of F is denoted bySF. Clearly,F is pointwise bounded onX if and only ifSF =∅.
Theorem 4.1. Let nbe a positive integer, letX andY be topological linear spaces, and letQ:= (Qj)j∈J be a family of monomial mappings of degreenfrom X intoY which is equicontinuous at oX. Then Qis pointwise bounded on X.
Proof. Letxbe any point inX. We prove thatQis bounded atx. LetV be a neighbourhood ofoY. SinceQis equicontinuous atoX, there exists a neighbourhood U ofoX such that
{Qj(u)| j∈J} ⊆V for allu∈U.
Choose a rational numberr 6= 0 such that rx ∈ U. Then {Qj(rx) | j ∈J} ⊆V, whence
{Qj(x)| j∈J} ⊆ 1 rnV.
Consequently,Q is bounded atx.
The converse of Theorem 4.1 is not true. The pointwise boundedness ofQ on X does not imply the equicontinuity of Q at oX, not even when n = 1. But, taking into consideration the next theorem, which is a principle of condensation of the singularities of families of continuous monomial mappings between topological linear spaces, we will be able to point out cases when the pointwise boundedness of Qimplies the equicontinuity ofQatoX (and therefore on the whole spaceX).
Theorem 4.2. Let nbe a positive integer, letX andY be topological linear spaces, and letQ:= (Qj)j∈J be a family of continuous monomial mappings of degree n from X into Y which is not equicontinuous at oX. Then the following assertions are true:
1◦ SQ is a residual set.
2◦ If, in addition, X is of the second category, then SQ is of the second category, dense inX and with cardSQ≥ ℵ.
Proof. 1◦SinceQis not equicontinuous atoX, there exists a neighbourhood V ofoY such that for every neighbourhoodU ofoX there is au∈U satisfying
{Qj(u)|j∈J} 6⊆V.
Choose a closed balanced neighbourhoodV0 ofoY such that V0+· · ·+V0
| {z } n+ 1 terms
⊆V.
For each positive integermput Sm:= \
j∈J
{x∈X |Qj(x)∈mV0}.
Since V0 is closed and all the mappings Qj (j ∈ J) are continuous, it follows that all the sets Sm are closed. We claim that all these sets are nowhere dense. Indeed, otherwise there exists a positive integerm such that intSm 6=∅. Choose any point x0∈intSm. Next select a neighbourhoodU0ofoXsuch thatx0+U0⊆Smand after that select a neighbourhoodU ofoX satisfying (3). Fix anyj∈J and anyu∈U. In
virtue of Proposition 2.2 we have Qj(u) = 1
n!
n
X
k=0
(−1)n−k n
k
Qj(x0+ku). (13)
Since
x0+ku∈x0+U+· · ·+U
| {z } kterms
⊆x0+U0⊆Sm
for allk∈ {0,1, . . . , n}, it follows that
Qj(x0+ku)∈mV0 for allk∈ {0,1, . . . , n}.
Taking into account thatV0 is balanced, we obtain from (13) that Qj(u)∈m(V0+· · ·+V0
| {z } n+ 1 terms
)⊆mV, whence
Qj
1 mu
= 1
mn Qj(u)∈ 1
mn−1V ⊆V.
Sincej∈J andu∈U were arbitrarily chosen, we have
Qj 1
mu
j∈J
⊆V for allu∈U,
which contradicts the choice ofV. Consequently, all the setsSm are nowhere dense, as claimed.
It is immediately seen that X\SQ⊆
∞
[
m=1
Sm.
ThereforeX\SQ is a set of the first category, i.e. SQ is a residual set.
2◦SinceXis of the second category, it follows in virtue of a well-known result in the theory of topological linear spaces thatX is a Baire space. Consequently, the residual setSQ is of the second category and dense. ThereforeSQ is not empty. Let xbe any point inSQ. SinceQis bounded atoX, we havex6=oX. Besides we have
{Qj(ax)|j∈J}=an{Qj(x)|j∈J} for alla∈R.
Since the set{Qj(x)|j∈J}is not bounded, it follows that {ax |a∈R\ {0} } ⊆SQ,
whence cardSQ≥ ℵ.
Together the Theorems 4.1 and 4.2 yield the following theorem revealing cases when the equicontinuity atoX of a family Qof continuous monomial mappings of degree n from a topological linear space X into a topological linear space Y is equivalent to the pointwise boundedness ofQonX.
Theorem 4.3. Let nbe a positive integer, letX andY be topological linear spaces, and letQ be a family of continuous monomial mappings of degree nfrom X intoY. Then the following assertions are equivalent:
1◦ X is of the second category andQ is pointwise bounded on X.
2◦ There exists a subset M ⊆X of the second category on whichQ is point- wise bounded.
3◦ X is of the second category andQ is equicontinuous at oX. Proof. 1◦⇒2◦ Obvious.
2◦ ⇒ 3◦ SinceM ⊆X, it follows that X is of the second category. Analo- gously, it follows fromM ⊆X\SQ that X\SQ is of the second category. In other words,SQis not a residual set. According to assertion 1◦in Theorem 4.2 the family Qmust be equicontinuous atoX.
3◦⇒1◦ Results by Theorem 4.1.
Corollary 4.4. Let n be a positive integer, let X and Y be normed linear spaces, and let(Fj)j∈J be a family of continuous symmetricn-additive mappings from Xn intoY. Then the following assertions are equivalent:
1◦ X is of the second category and
sup{ kFj∗(x)k |j∈J}<∞ for allx∈X.
2◦ There exists a subset M ⊆X of the second category such that
sup{ kFj∗(x)k |j∈J}<∞ for all x∈M. (14) 3◦ X is of the second category and
sup{ kFj(x1, . . . , xn)k |j ∈J, kx1k ≤1, . . . , kxnk ≤1}<∞. (15)
Proof. 1◦⇒2◦ Obvious.
2◦ ⇒ 3◦ The inequality (14) expresses that the family (Fj∗)j∈J is pointwise bounded on M. By applying the implication 2◦ ⇒ 3◦ from Theorem 4.3 it follows thatX is of the second category and that (Fj∗)j∈J is equicontinuous atoX. Therefore, by the implication 1◦⇒2◦ in Proposition 3.4, the inequality (15) is true.
3◦⇒1◦ Letx∈X be arbitrarily chosen. Whenx=oX, then sup{ kFj∗(x)k |j∈J}= 0.
Whenx6=oX, then the number a:= 1/kxksatisfies kFj∗(x)k= 1
ankFj(ax, . . . , ax
| {z } ntimes
)k ≤ b an
for allj ∈J, where
b:= sup{ kFj(x1, . . . , xn)k |j∈J, kx1k ≤1, . . . , kxnk ≤1}.
Consequently, assertion 1◦ is true.
It should be remarked that Theorem 4.2 is a generalization of Theorem 1.1, while Theorem 4.3 and Corollary 4.4 are generalizations of Theorem 1.2. Besides, Corollary 4.4 is also a generalization of the principle of uniform boundedness proved by I. W. Sandberg [11, Theorem 2], which recently was rediscovered by R. Miculescu [9, Theorem 2].
5. An application to the theory of the Riemann-Stieltjes integral
Throughout this section a and b are real numbers satisfying the inequality a < b. Any finite sequence (x0, x1, . . . , xn) of points of the interval [a, b] such that a=x0< x1<· · ·< xn=bis called asubdivision of [a, b].
If ∆ := (x0, x1, . . . , xn) is a subdivision of [a, b], then the number µ(∆) := max{x1−x0, . . . , xn−xn−1}
is called themesh of ∆ and any finite sequence (c1, . . . , cn) such that cj ∈[xj−1, xj] for all j ∈ {1, . . . , n} is called a selection assigned to ∆. The set of all selections assigned to ∆ will be denoted byS∆.
A. Pelczynski and S. Rolewicz [10, Corollary] proved that a function f : [a, b] → R is Riemann-Stieltjes integrable with respect to itself over [a, b] if and only if for each ε ∈]0,∞[ there exists a δ ∈]0,∞[ such that for any subdivision
∆ := (x0, x1, . . . , xn) of [a, b] withµ(∆)< δ the inequality
n
X
j=1
[f(xj)−f(xj−1)]2< ε
holds. According to this result each function f : [a, b] → R, which is Riemann- Stieltjes integrable with respect to itself over [a, b], has to be continuous. On the other hand, the main result established by A. Pelczynski and S. Rolewicz [10, Theorem 3], concerning the Riemann-Stieltjes integrals of the form
Z b a
Φ(f(x))df(x),
reveals that not every continuous function f : [a, b] → R is Riemann-Stieltjes inte- grable with respect to itself over [a, b]. Actually, the set consisting of all continuous functionsf : [a, b]→Rhaving the property thatf is not Riemann-Stieltjes integrable with respect to itself over [a, b] is very large. More precisely, the following theorem holds.
Theorem 5.1. Let C[a, b] be the linear space of all real-valued continuous functions defined on [a, b]endowed with the usual uniform norm
kfk= max{ |f(x)| |x∈[a, b]} (f ∈C[a, b]),
and let RS[a, b]f be the set of all functions f: [a, b] → R having the property that f is Riemann-Stieltjes integrable with respect to itself over [a, b]. Then the following assertions are true:
1◦ The set C[a, b]\RS[a, b]f is residual, whence of the second category, dense inC[a, b]and with card (C[a, b]\RS[a, b])f ≥ ℵ.
2◦ The set RS[a, b]f is of the first category and dense in C[a, b].
Proof. Letϕ: [0,1]→[a, b] be defined byϕ(t) :=a+t(b−a). Taking into account that the mapping
∀f ∈C[a, b] 7−→ f◦ϕ∈C[0,1]
is an isometric isomorphism as well as that a function f: [a, b] → R is Riemann- Stieltjes integrable with respect to itself over [a, b] if and only if f ◦ϕ is Riemann- Stieltjes integrable with respect to itself over [0,1], it suffices to prove the theorem in the special case whena= 0 andb= 1.
LetD be the set consisting of all subdivisions of [0,1]. Given a subdivision
∆ := (x0, x1, . . . , xn)∈ D and a selectionξ:= (c1, . . . , cn)∈ S∆, letQ∆,ξ:C[0,1]→ Rbe the mapping defined by
Q∆,ξ(f) :=
n
X
j=1
f(cj)[f(xj)−f(xj−1)] for allf ∈C[0,1].
It is easily seen that Q∆,ξ is continuous. Besides, we notice that Q∆,ξ is a qua- dratic mapping, because it is the diagonalization of the symmetric bilinear mapping F∆,ξ:C[0,1]×C[0,1]→R, defined by
F∆,ξ(f, g) := 1 2
n
X
j=1
f(cj)[g(xj)−g(xj−1)] +1 2
n
X
j=1
g(cj)[f(xj)−f(xj−1)]
for allf, g∈C[0,1].
1◦ Passing to the proof of the first assertion of the theorem, we consider for every positive integernthe family
Qn:={Q∆,ξ |∆∈ D, ξ∈ S∆, µ(∆)≤1/n}.
We claim that for every positive integernthe familyQn is not equicontinuous at the zero-element ofC[0,1].
Indeed, letnbe any positive integer. Define the functionf: [0,1]→Rby
f(x) :=
0 if x= 0
√x cosπx
if 0< x≤ n1
√1
n if n1 < x≤1.
For each positive integerpset
∆p:=
0, 1
n+p, 2
2(n+p)−1, 1
n+p−1, 2
2(n+p)−3, 1
n+p−2, . . . , 1
n+ 1, 2 2n+ 1, 1
n, 2 n, 3
n, . . . , n−1 n ,1
and
ξp:=
0, 2
2(n+p)−1, 1
n+p−1, 2
2(n+p)−3, 1
n+p−2, . . . , 2
2n+ 1, 1 n, 2
n, . . . , n−1 n ,1
.
Obviously, ∆pis a subdivision of [0,1] withµ(∆p)≤1/nandξpis a selection assigned to ∆p. Since
Q∆p,ξp(f) =
n+p−1
X
j=n
f 1
j f
1 j
−f 2
2j+ 1
= 1
n+ 1
n+ 1 +· · ·+ 1 n+p−1 for everyp∈N, it follows that
sup{Q∆p,ξp(f)|p∈N}=∞.
Consequently,f is a singularity ofQn. By applying Theorem 4.1 we conclude that Qn is not equicontinuous at the zero-element ofC[0,1], as claimed.
By virtue of Theorem 4.2 we deduce that all the setsSQn(n∈N) are residual, hence the set
S:=
∞
\
n=1
SQn is residual, too.
SinceS⊆C[0,1]\RS[0,f 1], it follows that the setC[0,1]\RS[0,f 1] is residual, whence of the second category, dense inC[0,1] and with
card (C[0,1]\RS[0,f 1])≥ ℵ.
2◦The fact thatRS[0,f 1] is of the first category follows from assertion 1◦. On the other hand, sinceRS[0,f 1] contains the restrictions to [0,1] of all polynomials, it
follows thatRS[0,f 1] is dense inC[0,1].
Remark. There is also another way to prove Theorem 5.1. The characteriza- tion of the functions that are Riemann-Stieltjes integrable with respect to themselves over [a, b], recalled at the beginning of this section, yields thatRS[a, b]f ⊆CBV2[a, b], where CBV2[a, b] denotes the set of all real-valued functions defined on [a, b] that are continuous and of bounded variation of order 2. Taking into consideration that CBV2[a, b] is a set of the first category inC[a, b] (see, for instance, [4, Corollary 2.8]), it follows thatRS[a, b] is also of the first category inf C[a, b]. Apparently this proof avoids the condensation of singularities, but in reality the property ofCBV2[a, b] to be of the first category in C[a, b] is a consequence of a principle of condensation of the singularities of a family of nonnegative functions as shown in [4].
References
[1] Breckner, W. W.,A principle of condensation of singularities for set-valued functions.
Mathematica – Rev. Anal. Num´er. Th´eor. Approx., S´er. L’Anal. Num´er. Th´eor. Approx.
12(1983), 101 – 111.
[2] Breckner, W. W., On the condensation of the singularities of families of nonlinear functions. In: Approximation and Optimization (eds.: D. D. Stancu et al.). Vol. I.
Cluj-Napoca: Transilvania Press 1997, pp. 35 – 44.
[3] Breckner, W. W.,A short introduction to the condensation of the singularities of families of nonlinear functions.In: Proceedings of the 2nd International Summer School “Op- timal Control and Optimization Theory”, Halle, 2002 (eds.: C. Tammer et al.). Halle:
Martin-Luther-Universit¨at, Reports of the Institute of Optimization and Stochastics, Report No. 25 (2002), pp. 59 – 83.
[4] Breckner, W. W., Trif, T. and Varga, C.,Some applications of the condensation of the singularities of families of nonnegative functions.Anal. Math.25(1999), 15–32.
[5] Cobza¸s, S¸. and Muntean, I., Condensation of singularities and divergence results in approximation theory.J. Approx. Theory31(1981), 138–153.
[6] Djokovi´c, D. ˇZ., A representation theorem for (X1 −1)(X2−1)· · ·(Xn−1) and its applications.Ann. Polon. Math.22(1969/1970), 189 – 198.
[7] Kurepa, S.,On the quadratic functional.Publ. Inst. Math. Acad. Serbe Sci.13(1959), 57 – 72.
[8] McKiernan, A. M.,On vanishingn-th ordered differences and Hamel bases.Ann. Polon.
Math.19(1967), 331 – 336.
[9] Miculescu, R.,A uniform boundedness principle type result.Math. Reports (Bucharest) 5 (55)(2003), 57 – 59.
[10] Pelczynski, A. and Rolewicz, S., Remarks on the existence of the Riemann-Stieltjes integral.Colloq. Math.5(1957), 74–77.
[11] Sandberg, I. W.,Multilinear maps and uniform boundedness.IEEE Trans. Circuits and Systems CAS–32 (1985), 332 – 336.
[12] Smajdor, W., Subadditive and Subquadratic Set-Valued Functions. Katowice: Prace Naukowe Uniwersytetu ´Sl¸askiego w Katowicah, 1987.
[13] Tabor, J.,Monomial selections of set-valued functions.Publ. Math. Debrecen56(2000), 33 – 42.
[14] Trif, T., Singularities and equicontinuity of certain families of set-valued mappings.
Comment. Math. Univ. Carolin.39(1998), 353 – 365.
[15] Trif, T.-V.,Contributions concerning the study of the singularities of families of func- tions(in Romanian). PhD Thesis. Cluj-Napoca: Universitatea “Babe¸s-Bolyai”, 2000.
Babes¸-Bolyai University,
Faculty of Mathematics and Computer Science,
Str. Kog˘alniceanu Nr. 1, RO-400084 Cluj-Napoca, Romania E-mail address: [email protected]
Babes¸-Bolyai University,
Faculty of Mathematics and Computer Science,
Str. Kog˘alniceanu Nr. 1, RO-400084 Cluj-Napoca, Romania E-mail address: [email protected]
