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

(1)

Volume 9 (2008), Issue 2, Article 41, 4 pp.

ON THE TRIANGLE INEQUALITY IN QUASI-BANACH SPACES

CONG WU AND YONGJIN LI DEPARTMENT OFMATHEMATICS

SUNYAT-SENUNIVERSITY

GUANGZHOU, 510275 P. R. CHINA

[email protected] [email protected]

Received 11 May, 2007; accepted 10 June, 2008 Communicated by S.S. Dragomir

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

Key words and phrases: Triangle inequality, Quasi-Banach spaces.

2000 Mathematics Subject Classification. 26D15.

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 spaceX withkxk ≥ kyk, kx+yk+

2−

x

kxk + y kyk

kyk

≤ kxk+kyk

≤ kx+yk+

2−

x

kxk + y kyk

kxk.

We recall that a quasi-normk · kdefined on a vector spaceX(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.

If k · k is a quasi-norm on X 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.

157-07

(2)

2 CONGWU ANDYONGJINLI

2. MAINRESULTS

Theorem 2.1. For all nonzero elementsx, yin a quasi-Banach spaceX withkxk ≥ kyk kx+yk+C

2−

x

kxk + y kyk

kyk

≤C(kxk+kyk) (2.1)

≤ kx+yk+

2C²−

x

kxk + y kyk

(2.2) kxk,

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).

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

J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 41, 4 pp. http://jipam.vu.edu.au/

(3)

TRIANGLEINEQUALITY INQUASI-BANACHSPACES 3

≤ 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.

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 exists 0< 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. Let k · k be the original quasi-norm on X, denote by k = 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 onXby:

k|xk|= inf







n

X

i=1

kx_ik^p

!¹_p :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 spaceX withkxk ≥ 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.

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

(4)

4 CONGWU ANDYONGJINLI

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. 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.

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 Ba- nach 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.

http://jipam.vu.edu.au/