ON THE TRIANGLE INEQUALITY IN QUASI-BANACH SPACES

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Triangle Inequality in Quasi-Banach Spaces Cong Wu and Yongjin Li vol. 9, iss. 2, art. 41, 2008

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ON THE TRIANGLE INEQUALITY IN QUASI-BANACH SPACES

CONG WU AND YONGJIN LI

Department of Mathematics Sun Yat-Sen University Guangzhou, 510275, P. R. China

EMail:congwu@hotmail.com stslyj@mail.sysu.edu.cn

Received: 11 May, 2007

Accepted: 10 June, 2008

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 26D15.

Key words: Triangle inequality, Quasi-Banach spaces.

Abstract: In this paper, we show the triangle inequality and its reverse inequality in quasi- Banach spaces.

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1. Introduction

The triangle inequality is one of the most fundamental inequalities in analysis. The following sharp triangle inequality was given earlier in H. Hudzik and T. R. Landes [2] and also found in a recent paper of L. Maligranda [5].

Theorem 1.1. For all nonzero elementsx, yin a normed linear spaceXwithkxk ≥ kyk,

kx+yk+

2−

x

kxk+ y kyk

kyk

≤ kxk+kyk

≤ kx+yk+

2−

x

kxk + y kyk

kxk.

We recall that a quasi-norm k · k defined on a vector space X (over a real or complex fieldK) is a mapX →R⁺such that:

(i) kxk>0forx6= 0;

(ii) kαxk=|α|kxkforα∈K, x∈X;

(iii) kx+yk ≤ C(kxk+kyk)for allx, y ∈X, whereCis a constant independent ofx, y.

Ifk · kis a quasi-norm onX defining a complete metrizable topology, thenX is called a quasi-Banach space.

In the present paper we will present the triangle inequality in quasi-normed spaces.

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2. Main Results

Theorem 2.1. For all nonzero elementsx, yin a quasi-Banach spaceXwithkxk ≥ kyk

kx+yk+C

2−

x

kxk + y kyk

kyk

≤C(kxk+kyk) (2.1)

≤ kx+yk+

2C²−

x

kxk + y kyk

kxk, (2.2)

whereC ≥1.

Proof. Letkxk ≥ kyk. We first show the inequality (2.1).

kx+yk=

kyk x

kxk+ y kyk

+kxk x

kxk − kyk x kxk

≤C

kyk x

kxk+ y kyk

+C

kxk x

kxk − kyk x kxk

=Ckyk

x

kxk + y kyk

+C(kxk − kyk)

=Ckyk

x

kxk + y kyk

+C(kxk+kyk −2kyk)

=Ckyk

x

kxk + y kyk

−2

+C(kxk+kyk).

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Since

kx+yk=

kxk x

kxk + y kyk

−

kxk y

kyk− kyk y kyk

≥ 1 C

kxk x

kxk + y kyk

−

kxk x

kxk − kyk x kxk

= 1 Ckxk

x

kxk+ y kyk

−(kxk − kyk)

= 1 Ckxk

x

kxk+ y kyk

+ (kxk+kyk −2kxk)

=kxk 1

C

x

kxk + y kyk

−2

+ (kxk+kyk).

we have

C(kxk+kyk)≤Ckx+yk+

2C−

x

kxk + y kyk

kxk

=kx+yk+ (C−1)kx+yk+

2C−

x

kxk + y kyk

kxk

≤ kx+yk+ (C−1)C(kxk+kyk) +

2C−

x

kxk + y kyk

kxk

≤ kx+yk+ (C−1)C(2kxk) +

2C−

x

kxk + y kyk

kxk

=kx+yk+

2C²−

x

kxk + y kyk

kxk.

Thus the inequality (2.2) holds.

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T. Aoki [1] and S. Rolewicz [6] characterized quasi-Banach spaces as follows:

Theorem 2.2 (Aoki-Rolewicz Theorem). Let X be a quasi-Banach space. Then there exists0 < p ≤ 1and an equivalent quasi-normk| · k|onX that satisfies for everyx, y ∈X

k|x+yk|^p ≤ k|xk|^p+k|yk|^p.

Idea of the proof. Letk · kbe the original quasi-norm onX, denote byk = inf{K ≥ 1 : for anyx, y ∈X,kx+yk ≤K(kxk+kyk)}andpis such that2^1/p = 2k. It is shown [3] that the functionk| · k|defined onX by:

k|xk|= inf







n

X

i=1

kx_ik^p

!_p¹ :x=

n

X

i=1

x_i





 is an equivalent quasi-norm onXthat satisfies the required inequality.

Next, we will prove thep-triangle inequality in quasi-Banach spaces.

Theorem 2.3. For all nonzero elementsx, yin a quasi-Banach spaceXwithkxk ≥ kyk,

kx+yk^p+

kxk^p +kyk^p−(kxk − kyk)^p− kyk^p

x

kxk + y kyk

p

≤ kxk^p+kyk^p

≤ kx+yk^p +

kxk^p+kyk^p+ (kxk − kyk)^p− kxk^p

x

kxk + y kyk

p , where0< p ≤1.

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Proof. We have

kx+yk^p =

kyk x

kxk + y kyk

+kxk x

kxk − kyk x kxk

p

≤

kyk x

kxk + y kyk

p

+

kxk x

kxk − kyk x kxk

p

=kyk^p

x

kxk + y kyk

p

+ (kxk − kyk)^p

=kyk^p

x

kxk + y kyk

p

+kxk^p

+kyk^p−(kxk^p +kyk^p) + (kxk − kyk)^p. Thus

kx+yk^p+

kxk^p+kyk^p−(kxk − kyk)^p− kyk^p

x

kxk + y kyk

p

≤ kxk^p+kyk^p and

kx+yk^p =

kxk x

kxk + y kyk

−

kxk y

kyk − kyk y kyk

p

≥

kxk x

kxk + y kyk

p

−

kxk x

kxk − kyk x kxk

p

=kxk^p

x

kxk + y kyk

p

−(kxk − kyk)^p

=kxk^p

x

kxk + y kyk

p

+kxk^p+kyk^p−(kxk^p +kyk^p)−(kxk − kyk)^p.

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Hence

kxk^p+kyk^p ≤ kx+yk^p+

kxk^p+kyk^p+ (kxk − kyk)^p− kxk^p

x

kxk + y kyk

p . This completes the proof.

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References

[1] T. AOKI, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo, 18 (1942), 588–594.

[2] H. HUDZIKANDT.R. LANDES, Characteristic of convexity of Köthe function spaces, Math. Ann., 294 (1992), 117–124.

[3] N.J. KALTON, N.T. PECKANDJ.W. ROBERTS, An F-Space Sampler, London Math. Soc. Lecture Notes 89, Cambridge University Press, Cambridge, 1984.

[4] K.-I. MITANI, K.-S. SAITO, M.I. KATOANDT. TAMURA, On sharp triangle inequalities in Banach spaces, J. Math. Anal. Appl., 336 (2007), 1178–1186.

[5] L. MALIGRANDA, Simple norm inequalities, Amer. Math. Monthly, 113 (2006), 256–260.

[6] S. ROLEWICZ, On a certain class of linear metric spaces, Bull. Acad. Polon.

Sci. Sér. Sci. Math. Astrono. Phys., 5 (1957), 471–473.