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Vol. 11, No. 3, 2018, 740-750

ISSN 1307-5543 – www.ejpam.com Published by New York Business Global

The Product-Normed Linear Space

Benedict Barnes1∗, I. A. Adjei2, C. Sebil3, E. Harris4

1,2,3,4Department of Mathematics, Kwame Nkrumah University of Science and Technology,

Kumasi, Ghana

Abstract. In this paper, both the product-normed linear spaceP−N LS(product-Banach space) and product-semi-normed linear space (product-semi-Banch space) are introduced. These normed linear spaces are endowed with the first and second product inequalities, which have a lot of applications in linear algebra and differential equations. In addition, we showed that P−N LS

admits functional properties such as completeness, continuity and the fixed point.

2010 Mathematics Subject Classifications: 4B47, 44B49

Key Words and Phrases: product-normed linear space, product-semi-normed linear space, completeness, fixed point theorem

1. Introduction

In recent times, the solution space of a mathematical problem has become necessary condition for its existence. Without the solution space, the qualitative, as well as, quan-titative property of the mathematical structure cannot be established. In [1], the author introduced the n−metrics. About 30 decades afterwards, the author in [2] narrowed the

n−metric space to two-dimensional space and also, obtained some topological properties associated with this space.

Definition 1 (Norm). Let V be a linear space over the real number field R. A norm on V is a real-valued function

k · k:V →[0,∞)

such that for any u, v∈V and α∈R the following conditions are met:

1.kuk ≥ 0, andkuk= 0, if f u= 0 2.kαuk = |α|kuk, ∀u∈V and α∈R 3.ku+vk ≤ kuk+kvk, ∀u, v∈V See [3].

Corresponding author.

DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3284

Email addresses: ewiekwamina@gmail.com bbarnes.cos@knust.edu.gh(B. Barnes)

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In [4], the author generalized the normed linear space by replacing the third axiom of the norm, triangle inequality, with tetrahedral inequality.

The authors in [5] and [6] independently extended the linear normed space to quasi-normed linear space. They replaced the third axiom of the quasi-normed linear space with quasi-norm

ku+vk ≤K

kuk+kvk, ∀ u, v∈V,

where K ≥1 is the quasi-number. In [7], the authors extended the quasi-normed linear space to the complex version and also, included a parameter p ∈ (0,∞) in this space, which is used to establish p− uniformly P L− convex. The author in [8], observed that the quasi-normed linear space becomes the p−normed linear space if the quasi number

K = 21p−1. In [9], the author established that the quasi-normed linear space is endowed with a unique fixed point. In [10], he narrowed the quasi-normed linear space to quasi-2-normed linear space Q-2-NLS. Addition properties of the Q-2-NLS such as the convergence of the two Cauchy sequences and a pseudo-quasi-2-norm of the Q-2-NLS were observed by the authors in [11].

Undoubtedly, the normed linear space is called a p−normed linear if ku+vkp≤ kukp+kvkp, ∀ u, v∈V, and0< p≤1,

is included to the three axioms of the linear normed space. For example, see the author in [12]. There are, however, situations where it is more convenient to work with a subspace than the entire functional space. For example, the imposing of absolute value of a variable

x as an inhomogeneous boundary condition of the Laplace equation does not yield a solution in the Hilbert space, but in Sobolev spaces. Thus, the continuity of the partial differential operator is not guaranteed in the Hilbert space. In order for a functional space to admit such a solution, we impose some continuity conditions on the partial differential operator which lead to analysis in Sobolev spaces. Again, much attention has been paid to the extension of the triangle inequality, the third axiom of the normed linear space, to obtain the different normed linear spaces with their functional properties.

In this paper, we include another axiom to the normed linear space as a fourth axiom, first product inequality, from which we coined the name product-normed linear space

P −N LS, k.kpn. This functional space narrows the normed linear space which allows the vectors to be defined on [0,2]. TheP −N LS admits the first product inequality, see [13], and useful properties which we shall establish them later in this paper. Also, the product-semi-normed linear spacek.kpn is introduced in this paper.

2. Some Preliminary Results

In this section, we give the theorems and definitions regarding to the product-normed linear space and product-semi-normed linear space.

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u, v of vectors inV a scalar, denoted(u, v), is called an inner product on V if and only if (i) (u, u)>0 whenever u6= 0, and (u, u) = 0 if and only if u= 0

(ii) (u, v) = (v, u), ∀u, v∈V

(iii) (αu1+βu2, v) =α(u1, v) +β(u2, v), ∀α, β∈R, andu1, u2, v∈V. See[14]

Definition 3(Subspace). Let V be a vector space and Ωa nonempty subset of V. IfΩis a vector space with respect to the operations in V, then Ω is called a subspace of V. See

[15].

Definition 4 (Continuity at a Point). Let U and V be two normed linear spaces. An operatorA:U →V is continuous at uo∈U, if for every >0, there exists a δ =δ(, uo) such that

kAu−Auok< , whenever ku−uok< δ.

See [16].

Definition 5 (Convergent Sequence). A sequence {Un}∞n=1 in a normed linear space

Rn,k · k converges to a point u ∈ Rn, if and only if, for every real number > 0, there exists an integer N such that

kUn−uk< , n > N.

See [17].

Definition 6 (Closed set). A set u= [0,2]in a normed linear space Rn,k · k is closed if and only if every sequence of points {Un}∞n=1 ⊂U convergent in Rn has a limit in U. For example, see [18]

Definition 7 (Contractive Mapping). Let T : U → U be a mapping from a complete normed linear spaceU into itself. The Lipschitz continuity on T is said to be contraction if

kT(u)−T(v)k ≤λku−vk, ∀u, v∈U, and λ <1. See [19].

3. Main Result

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Theorem 1 (Product-normed Linear Space). Let U be a linear space over [0,2]⊆R. A vector space with a product-norm u→ kUk satisfying real-valued function k · k,

k · kpn:U →[0,∞),

such that for arbitrary u, v∈U, α∈[0,2], the following conditions are satisfied: P1.kuk ≥ 0, andkuk= 0, if and only if u= 0

P2.kαuk = |α|kuk, α∈[0,2], and u∈U P3.ku+vkpn ≤ kukpn+kvkpn, ∀u, v∈U

We call k · kpn a product norm if in addition toP1−P3, the u and v satisfy

P4(a).kukpnkykpn ≤ kukpn+kvkpn ∀u, v∈[0,2] (f irst product inequality) P4(b).kuk+kvk ≤ kukkvk, ∀u, v∈[2,∞) (second product inequality). P roof: We start atP1. Suppose thatu= 0, then

kuk2pn = h0,0i ⇒ kuk2pn = 0 ⇒ kukpn = 0.

On the other hand, we suppose that kukpn= 0. But,

kuk2

pn = hu, ui

⇒0 = huui.

The above equality holds if u = 0. Hence, kuk= 0 if and only if u= 0. Without loss of generality, let the

kukpn=|u|pn,

then|u|pn= 0, which implies thatkukpn = 0. But kuk2pn = hu, ui.

Since the hu, ui cannot be negative. This implies thatkukpn>0. Thus,P1 is satisfied. In order to prove axiomP2, we take the expression on the left hand side yields

kαuk2 = hαu, αui

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⇒ kαuk2 =

|α|kuk2 ⇒ kαuk = |α|kuk.

Also, the axiom P3 is proved as follows.

ku+vk2 = hu+v, u+vi

⇒ ku+vk2 = hu, ui+hu, vi+hv, ui+hv, vi ⇒ ku+vk2 = hu, ui+hu, vi+hu, vi+hv, vi ⇒ ku+vk2 = hu, ui+ 2|hu, vi|+hv, vi ⇒ ku+vk2 = kuk2+ 2|hu, vi|+kvk2.

Applying the Cauchy-Schwarz inequality to the middle term on the right hand side, we obtain

ku+vk2 ≤ kuk2+ 2kukkvk+kvk2 ⇒ ku+vk2 = kuk+kvk2

⇒ ku+vk ≤ kuk+kvk.

The axiom P4 has been discussed by the authors in [13]. We present the proof as follows.

(u+v)n≥0

⇒ hu, uin2 +nC1hu, vihu, ui

n−2

2 +nC2hu, ui

n−2

2 hv, vi+nC3hu, ui

n−4

2 hv, vihu, vi

+ nC4hu, ui n−4

2 hv, vi2+nC5hu, ui

n−6

2 hv, vi2hu, vi+nC6hu, ui

n−6 2 hv, vi3

+ nC7hu, ui n−8

2 hv, vi3hu, vi+. . .+nCn

2hu, ui

n

4hv, vi

n

4 +. . .+hu, ui

n

2 ≥0

⇒ −nCn

2hu, ui

n

4hv, vi

n

4 ≤ {hu, ui

n

2 +nC1hu, vihu, ui

n−2

2 +nC2hu, ui

n−2 2 hv, vi

+ nC3hu, ui n−4

2 hv, vihu, vi+nC4hu, ui

n−4

2 hv, vi2+nC5hu, ui

n−6

2 hv, vi2hu, vi

+ nC6hu, ui n−6

2 hv, vi3+nC7hu, ui

n−8

2 hv, vi3hu, vi+. . .+hu, ui

n

2}

⇒ −nCn

2(u, u)

n

4(v, v)

n

4 ≤

n

(u, u)n2 +nC1(u, v)(u, u)

n−2

2 +nC2(u, u)

n−2 2 (v, v)

+ nC3(u, u) n−4

2 (v, v)(u, v) +nC4(u, u)

n−4

2 (v, v)2+nC5(u, u)

n−6

2 (v, v)2(u, v)

+ nC6(u, u) n−6

2 (v, v)3+nC7(u, u)

n−8

2 (v, v)3(u, v) +. . .+nCn

2(u, u)

n

4(v, v)

n

4 +. . .+ (u, u)

n

2 o

⇒ −nCn

2(u, u)

n

4(v, v)

n

4 1

nCn

2

(u, u)n4(v, v)

n

4 ≤(u+v)n

⇒ k −(u, u)n2(v, v)

n

2k=k(u+v)nk

⇒ {kukkvk}n=k(u+v)kn

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The proof ofP4(b), see [13].

We show that the product-normed space is valid for 1−norm, 2−norm,p−norms and ∞−norm. We give the proof forp−norm which can be used to generate the other norms. Theorem 2. Suppose thatf(x)and g(x)are measurable functions over the domain [0,2], then

kfkpkgkp≤ kfkp+kgkp, where 1p +1q = 1.

P roof: We see that

(kfkkgk)p = (kfkkgk)p−1(kfkkgk)

Applying the first product inequality to the right hand side of the above equation yields (kfkkgk)p ≤ (kfk+kgk)p−1(kfk+kgk)

(kfkkgk)p = (kfk+kgk)p−1kfk+ (kfk+kgk)p−1kgk

Taking the Lebesgue integral of both sides of the above equation with respectµ, we obtain

Z

(kfkkgk)pdµ =

Z n

(kfk+kgk)p−1kfk+ (kfk+kgk)p−1kgkodµ

Z

(kfkkgk)pdµ =

Z

(kfk+kgk)p−1kfkdµ+

Z

(kfk+kgk)p−1kgkdµ

Z

(kfkkgk)p

1

p

dµ =

hZ

(kfk+kgk)p−1kfkdµ+

Z

(kfk+kgk)p−1kgkdµ i1

p

⇒ kfkpkgkp =

hZ

((kfk+kgk)p−1kfk)pdµ p1

+

Z

((kfk+kgk)p−1kgk)pdµ 1pi1p

⇒ kfkpkgkp ≤ hn

Z

kfkp

1

p +

Z

kgkp

1

ponZ

(kfk+kgk)pdµ 1

pop−1i

1

p

⇒ kfkpkgkp ≤ hkfkp+kgkpkfkp+kgkpp−1i

1

p

⇒ kfkpkgkp = kfkp+kgkp

1

p

kfkp+kgkp

p−1

p

⇒ kfkpkgkp ≤ kfkp+kgkp. (1) Choosingp= 1 and p= 2 in inequality (1), we obtain the following

kfk1kgk1 ≤ kfk1+kgk1

and

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respectively. But if,p→ ∞ then the inequality (1) becomes kfk∞kgk∞≤ kfk∞+kgk∞.

Theorem 3 (Completeness of the Product-normed linear space). The product-normed linear space (U,k · kpn) is complete.

P roof: Let {Un}∞n=1 be a sequence in (U,k · kpn) and u ∈ U. For every >0, there

exists an integerno >N such thatkUn−uk< , for alln > no, then {Un}∞n=1 converges u. Thus, limn→∞kUn−ukpn = 0. Also, we see that for every >0, there exists integers m, n > N such that limm,n→∞kUm−Unk = 0. In this case, we see that every Cauchy

sequence converges to a point (U,k · kpn), then (U,k · kpn) is a complete product-normed

linear space. Alternatively, we the following result shows the product-normed linear space is complete.

Theorem 4 (Completeness of the Product-normed linear space Using Closed Set). Let U be a linear space over [0,2]⊆R. A subspace U,k · kpn

of a complete normed linear

space Rn,k · k pn

is complete if and only if U is a closed set.

P roof: Let{Un}∞n=1 be a sequence of points of U. We see that {Un}∞n=1 has limit u

inU, sinceU is closed. Thus, lim

n→∞kUn−ukpn= 0. Since u∈U then

U,k · kpn

is complete. Conversely, assume that

U,k · kpn

is complete. Let u be an accumulation point of

U. Then each open ball centred atu, B1/n(u) contains a pointun∈U. That is,

kUn−ukpn <

1

n.

This implies that{Un}∞n=1converges tou. However,

U,k·kpn

is complete on the grounds thatu is in U. Hence,U is closed.

Theorem 5 (Continuity). Let (U,k · kpn1) and (U,k · kpn2) be two product normed linear spaces, and T : U1 → U2 be a linear operator. Then T is bounded if and only if T is continuous.

P roof: LetAbe a bounded linear operator. Then there exists a constantM such that kA(u)kU2 ≤MkxkU1, ∀ u∈U1.

Let{Un}∞n=1 be a sequence in U1 such that

lim

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Now,

kA(Un)−A(u)kU2 = A

Un−u

U2

kA(Un)−A(u)kU2 ≤ M Un−u

U

1

lim

n→∞kA(Un)−A(u)kU2 ≤ 0

lim

n→∞A(Un)−nlim→∞A(u) = 0 lim

n→∞A(Un)−A(u) = 0

A( lim

n→∞Un) → A(u).

This implies thatA is continuous onU1.

Conversely, suppose that A is a continuous linear operator. Then A is continuous at 0. Thus, for everyδ >0, kukU1, there exists an >0 such that

kAuk ≤. (2)

Setting uo 6= 0, be an arbitrary element in U1 and u =αuo, where α = kuoδkU

1

. We can see that

kAukU2 = kA(αuo)kU2

kAukU2 = |α|kAuokU2, ∀uo ∈U1. (3)

Comparing (2) and (3), we obtain

|α|kAuokU2 ≤

⇒ kAuokU2 ≤

|α|

⇒ kAuokU2 ≤

|δ|kuokU1.

Hence Ais bounded. This completes the proof. Theorem 6 (Unique Fixed Point). Let U,k · kpn

be a product-normed linear space and setting T :U →U be a contractive mapping. Then T has a unique fixed point u¯∈U.

P roof: Lettinguo ∈U, then

U n = T(Un)

⇒U n = Tn(uo), ∀n∈N.

Setting {U}∞

n a Cauchy sequence in

U,k · kpn. Since U,k · kpnis a complete, which implies thatu∈U such that

lim

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Again, we see thatT is a contractive mapping, then it is continuous. Thus, lim

n→∞T(Un) =T u. We can see that

T u = lim

n→∞T(Un)

⇒T u = lim

n→∞Un+1

⇒T u = u.

Hence,u is a fixed point of the mappingT.

In addition, we show that the fixed point is unique in U,k · kpn. Settingu1, u2∈U

be the fixed points of T, then

kT(u1)−T(u2)kpn=ku1−u2kpn. (2)

Also, we see that:

kT(u1)−T(u2)kpn ≤λku1−u2kpn, ∀λ <1 (3)

Substituting equation (2) into inequality (3) yields

1−λku1−u2kpn ≤ 0

⇒ ku1−u2kpn = 0.

The above equation holds if u1 =u2. Hence, the fixed point x is unique in the

product-normed linear space.

Another normed linear space is introduced as:

Theorem 7 (Product Semi-Normed Linear Space). Let U be a linear space over [2,∞]⊆ R. A product norm on U is a real-valued function k · k,

k · k:U →[0,∞),

such that for arbitrary u, v∈U, α∈[0,2], the following conditions are satisfied:

1.kuk > 0, andkuk= 0, if and only if u= 0 2.kαuk = |α|kuk, α∈[0,2], and u∈U

3.ku+vk ≤ kuk+kvk, ∀u, v∈U

4.kuk+kvk ≤ kukkvk ∀u, v∈V (product inequality).

4. Conclusion

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References

[1] I. K. Menger, Untersuchungen veber allgeine matrik, Mathematische Annalen, vol. 100, (1928).

[2] V. S. G¨ahler, Lineare 2−normierte r¨aumeDiese Nachr, 28, 1-2, (1963).pp.1 - 43 [3] J. T. Scheick, Linear Algebra with Applications, Prentice-Hall, Inc, New York; 1997. [4] A. Branciari, A fixed point theorem of Banach-Caccippoli type on a class of

general-ized metric spaces,Publ. Math. Debrecen 57(2000). pp 31-37.

[5] M. Pavel, On quasi normed spaces Bull. Acad. Polon Sci. , CI III, 5, no. 5, (1957).pp. 479-487

[6] S. Rolewiez, On a certain class of linear metric spaceBull. Acad. Polon Sci. ,CI III, 5, no. 5, (1957).pp. 471-473

[7] W. J. Davis, D. J. H. Garling and N. Tomczak-Jaegermann, The complex of quasi-normed linear spacesJournal of Functional Analysis, 55 , (1984). pp. 110-150. [8] S. Rolewicz, Metric linear spaces, PMN, Warsaw(1972).

[9] G. Rano and T. Bag, Bounded linear operators in quasi-normed linear space,Journal of the Egyptian Mathematical Society,23 , (2015).pp. 303-308.

[10] C. Park, Generalized quasi-Banach spaces and quasi-(2, p) normed space,Journal of the Chungcheong matematical society,19, No. 2 (2006).

[11] M. Kir and M. Acikgoz, A study involving the completion of a quasi-2-normed space,

International Journal of Analysis ,(2013).http://dx.doi.org/10.1155/2013/512372. [12] M. Mastylo, On interpolation of some quasi-Banach spaces,Journal of Mathematical

Analysis and Applications,147 , (1990).pp. 403-419.

[13] B. Barnes, E. D. J. Owusu-Ansah, S. K. Amponsah and C. Sebil, The proofs of product inequalities in a generalized vector spaces, European journal of pure and applied mathematics,11, No. 2(2018). pp. 375-389.

[14] B. Kolman and D. R. Hill, Elementary Linear Algebra, Prentice-Hall, Inc, New Jersey; 2000.

[15] H. Royden and P. Fitxpatrick, Real Analysis. Pearson Education, Inc, 4th ed., Upper saddle River; 2010.

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[17] B. D. MacCluer, Graduate texts in mathematics: elementary functional analysis, Springer science + business media, New York; 2009.

[18] G. Bachman and L. Narici, Functional Analysis, Academic Press, New York and London; 1966.

References

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