Journal of Mathematics ISSN 2094-7380
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Julius V. Benitez . . . 20 An Integer Programming Approach for Optimizing Street Food Snack Nutrient Intake Among College Students Rhoda P. Agdeppa-Namoco . . . 28 On the AB-Generalized Fibonacci, Pell and Jacobsthal Sequences by Hessenberg Matrices
Mhelmar A. Labendia, Michael B. Frondoza . . . 41 Another Look at Geodetic and Hull Numbers of a Graph Dennis Tarepe, Sergio R. Canoy, Jr. . . . 52 Intersection Number of Some Graphs
Jesrael B. Palco, Rolando N. Paluga . . . 63 Bridging the Gap: Remediating Mathematics Compe- tencies of College Entrants
Leorence C. Tandog . . . 76 A Proposed Two-Stage Sequential Point Estimation of the Hazard Rate Function
Aljo Clair P. Pingal, Daisy Lou L. Polestico . . . 85 McShane Integral in Locally Convex Topological Vector Spaces
Juniven Acapulco, Julius V. Benitez . . . 91
McShane Integral in Locally
Convex Topological Vector Spaces
Juniven P. Acapulcoa, Julius V. Benitezb
aMSU-General Santos, bMSU-Iligan Institute of Technology
Abstract: In this note, we show that the definition of the McShane integral in [2] coincides with the definition of the McShane integral introduced in [3] whenever the functions are taking values in a locally convex topological vector space.
Keywords/phrases: topological vector space, locally con- vex, Mcshane integral
1 Introduction
In the late 1960’s, Edward James McShane introduced an integral which was a modified version of the Henstock inte- gral and proved that its equivalent to the Lebesgue integral.
He broadened the class of tagged partitions by not insisting that the tag of an interval belong to the interval. In 1990, Gordon [1] extend the definition of the McShane integral for functions taking values in Banach spaces and since then attention has been paid to this field. Paluga [2] also intro- duced the definition of the McShane integral for functions taking values in topological vector spaces (TVS). He proved that the TVS version of the McShane integral coincides with the Banach version whenever the functions are taking values in Banach spaces. Just recently, Tato and Temaj [3] defined the McShane integral for functions taking values in locally convex topological vector spaces (LCTVS).
Here we prove that the McShane integral introduced in
[2] and [3] coincide whenever the functions take values in LCTVS. Minkowski functionals play a major role in estab- lishing the objective of this paper.
2 Preliminaries
Definition 2.1 LetA be a non-empty subset of a real vec- tor space X. Then
(i) A is convexif tA+ (1 t)A ✓A for any t 2[0,1].
(ii) A is balanced if tA✓A for any t 2[ 1,1].
(iii) A is absorbing if for each x 2 X there exists s > 0 such that for any t2R with |t|> s, we have x2tA.
Definition 2.2 Let A be a non-empty absorbing subset of a real vector space X. The Minkowski’s functional of A is given by the function
µA : X ! [0,+1) x 7! inf Ax where Ax ={k >0 :x2kA}.
Definition 2.3 Let X be a real vector space. A function
⇢:X !R is said to be a seminorm onX if (i) ⇢(x+y)⇢(x) +⇢(y) for all x, y 2X, (ii) ⇢(↵x) = |↵|⇢(x) for all x2X and all↵2R.
If⇢has the additional property that⇢(x) = 0 impliesx=✓, then ⇢ is a norm onX.
Definition 2.4 Let X be a real vector space. A family P of seminorms on X is said to be separating if for every x2X with x6=✓, there exists ⇢ 2P such that ⇢(x)6= 0.
Definition 2.5 Suppose T is a topology on a real vector space X such that
(i) {x} is closed for everyx2X and
(ii) the vector space operations are continuous with re- spect to T.
Under these conditions, T is said to be a vector topology onX and X is a topological vector space.
Althroughout the paper, ✓-nbdW means thatW is open in a TVS X, and✓ 2W, where✓ is the zero vector in X.
Definition 2.6 Let X be a topological vector space. We say that the functionf :X !R iscontinuous atc2X if for every ✏ >0 we can find a ✓-nbd V such that for any x inX with x c2V, we have |f(x) f(c)|<✏.
Remark 2.7 If X is a TVS, then continuity of the vector space operations mean that the functions
+ : X⇥X ! X
(x, y) 7! x+y and · : R⇥X ! X (↵, x) 7! ↵x are continuous where R carries the usual topology and the spaces R⇥ X and X ⇥ X carry their respective product topologies.
Definition 2.8 LetX be a TVS. ThenX is said to belo- cally convexif there exists a local baseBat the zero vector
✓ whose members are convex. In this case, we say that X is locally convex topological vector space(LCTVS).
The following results are useful in this paper. See [4] for details.
Theorem 2.9 [4] LetX be a TVS, a2X and t6= 0. Then for any open set G in X, a+G and tG are also open in X.
Theorem 2.10 [4] Let X be a TVS. Then every ✓-nbd is absorbing.
Theorem 2.11 [4]If X is LCTVS, then there exists a local base B at ✓ whose members are both convex and balanced.
Theorem 2.12 [4] Suppose A is a convex balanced absorb- ing set in a real vector space X. Then µA is a seminorm on X.
Theorem 2.13 [4]SupposeP is a separating family of semi- norms on a real vector space X. For each ⇢ 2 P and each
✏>0, let
V(⇢,✏) ={x2X :⇢(x)<✏}.
Let B(P) be the collection of all finite intersections of the sets V(⇢,✏). Define
T(P) ={G✓X :8x2G,9B 2B(P) such that x+B ✓G}. Then T(P)is a vector topology on X and B(P) is a convex balanced local base for T(P), which turns X into LCTVS with the property that every member ofP isT(P)-continuous.
Definition 2.14 A divisionof the interval [a, b] is a finite collection {Ii : 1 i n} of non-overlapping closed inter- vals Ii such that [a, b] =
[n i=1
Ii. Let be a positive function defined on [a, b]. We say that the collection D = {(Ii,⇠i) : 1 i n} of interval-point pairs is a -fine free tagged division of [a, b] if
(i) {Ii : 1in} is a division of [a, b],
(ii) Ii ✓ (⇠i (⇠i),⇠i + (⇠i)) and ⇠i 2 [a, b] for i = 1,· · ·, n.
We shall denote the length of Ii by|Ii|.
Definition 2.15 [2] Let X be a topological vector space.
A function f : [a, b]!X isMcShane integrable on [a, b]
if there is an ↵ 2 X such that for any ✓-nbd W, there is a positive function on [a, b] such that for any -fine free tagged division D={(Ii,⇠i) : 1 in} of [a, b], we have
f(D) ↵ 2W where f(D) =
Xn i=1
f(⇠i)|Ii|. In this case, we call ↵ the Mc- Shane integral of f on [a, b] and we write
(M) Z b
a
f =↵.
3 Main Result
First, we consider the following lemmas.
Lemma 3.1 Let X be a real vector space and let A ✓ X.
If A is convex and absorbing then for each r >0, we have {x2X :µA(x)< r}✓rA.
Proof: Since A is absorbing, using Definition 2.1(iii), we get ✓ 2 A. Let x 2 X with µA(x) < r. We will show that x 2 rA. By Definition 2.2, µA(x) = infAx, where Ax = {k > 0 : x 2 kA}. Thus, there must exists t 2 Ax such that t < r. So, t >0 and x2 tA. Now, 0 < tr 1 <1 and sinceA is convex with✓ 2A, it follows from Definition 2.1(i) thattr 1A✓A. Thus, x2tA✓rA. Hence,x2rA.
The result follows. ⇤
Lemma 3.2 Suppose A is an open convex balanced ✓-nbd in a TVS X. Then µA is continuous.
Proof: SinceAis a✓-nbd, by Theorem 2.10,Ais absorbing.
So, µA is well defined. Using Theorem 2.12, the Minkowski functional µA is a seminorm on X. Let ✏ > 0. Take V =
✏2 1A. Then by Theorem 2.9, V is a ✓-nbd. Thus for any x in X with x c2V, we get
|µA(x) µA(c)|µA(x c) ✏ 2 <✏.
Hence, by Definition 2.6, the function µA is continuous. ⇤ In what follows, (X,T) is a locally convex topological vector space and P(X) is a family of T-continuous semi- norms on X so that the topology is generated by P(X).
Definition 3.3 [3] A function f : [a, b] ! X is said to be McShane integrable on [a, b] if there exists ↵ 2 X such that for each ⇢ 2 P(X) and for any ✏ > 0 there exists
⇢ : [a, b]! (0,+1) such that if D= {(Ii,⇠i) : 1i n} is a ⇢-fine free tagged division of [a, b], we have
⇢(f(D) ↵)<✏.
We now state and prove our main result.
Theorem 3.4 LetX be LCTVS. Then Definitions2.15and 3.3 are equivalent.
Proof: First, we will show that Definition 2.15 implies Def- inition 3.3. Assume that f is McShane integrable on [a, b]
in the sense of Definition 2.15. Let ✏>0 and let ⇢2P(X).
Take
W ={x2X :⇢(x)<1}.
Since ⇢ is a seminorm on X, ⇢(✓) = 0 < 1 and so, ✓ 2 W. Now,
W =⇢ 1(( 1,1))
and because⇢isT-continuous,W 2T. ThusW is a✓-nbd.
Thus, there exists a gauge ⇢: [a, b]!(0,+1) such that if D={(Ii,⇠i) : 1in} is a ⇢-fine free tagged division of [a, b], we have
f(D) (M) Z b
a
f 2✏W.
This implies that
⇢
✓
f(D) (M) Z b
a
f
◆
<✏.
Next, assume that f is McShane integrable on [a, b] in the sense of Definition 3.3. Let W be ✓-nbd. Then by Theorem 2.11, there exists a convex balanced✓-nbdV such that V ✓ W. Since V is a ✓-nbd, by Theorem 2.10, V is absorbing.
So,µV is well defined. Thus, by using Theorem 2.12,µV is a seminorm on X. Also, by Lemma 3.2, µV is T-continuous.
Hence, µV 2 P(X). Thus, corresponding to ✏ = 1, there exists a gauge : [a, b]!(0,+1) such that ifD={(Ii,⇠i) : 1in} is a -fine free tagged division of [a, b], we have
µV(f(D) ↵)<1.
Note that by Lemma 3.1, we get
{x2X :µV(x)<1}✓V.
Thus,f(D) ↵2V ✓W. ⇤
References
[1] R.A. Gordon, The McShane integral of Banach-valued functions, Illinois J. Math, 34(1990), 557-567.
[2] S.R. Canoy and R.N. Paluga, On the McShane inte- gral in topological vector spaces, Chamchuri Journal of Mathematics, Vol.2(2010)No.2, pp.45-56.
[3] A.Tato and I. Timaj, The Convergence Theorems for the McShane Integral of Functions Taking Values in a Locally Convex Space, Int. Journal of Math. Analysis, Vol.6(2012)No.40, pp.1965-1976.
[4] W. Rudin, Functional Analysis, McGraw-Hill Inc, 1973.
