ON THE TRIANGLE INEQUALITY IN
QUASI-BANACH SPACES
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
Cong Wu and Yongjin Li
vol. 9, iss. 2, art. 41, 2008
Title Page
stslyj@mail.sysu.edu.cn
Contents
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 quasiBanach spaces.
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Contents
1
Introduction
3
2
Main Results
4
Triangle Inequality in
Quasi-Banach Spaces
Cong Wu and Yongjin Li
vol. 9, iss. 2, art. 41, 2008
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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 elements x, y in a normed linear space X with kxk ≥
kyk,
x
y kyk
kx + yk+ 2 − +
kxk kyk ≤ kxk + kyk
x
y
≤ kx + yk + 2 − +
kxk kyk kxk.
We recall that a quasi-norm k · k defined on a vector space X (over a real or
complex field K) is a map X → R+ such that:
(i) kxk > 0 for x 6= 0;
(ii) kαxk = |α|kxk for α ∈ K, x ∈ X;
(iii) kx + yk ≤ C(kxk + kyk) for all x, y ∈ X, where C is a constant independent
of x, y.
If k · k is a quasi-norm on X defining a complete metrizable topology, then X is
called a quasi-Banach space.
In the present paper we will present the triangle inequality in quasi-normed spaces.
Triangle Inequality in
Quasi-Banach Spaces
Cong Wu and Yongjin Li
vol. 9, iss. 2, art. 41, 2008
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2.
Main Results
Theorem 2.1. For all nonzero elements x, y in a quasi-Banach space X with kxk ≥
kyk
x
y
kx + yk+C 2 − +
kxk kyk kyk
(2.1)
≤ C(kxk + kyk)
x
y 2
kxk,
(2.2)
≤ kx + yk + 2C − +
kxk kyk where C ≥ 1.
Proof. Let kxk ≥ kyk. We first show the inequality (2.1).
x
y
x
x kx + yk = kyk
+
+ kxk
− kyk
kxk kyk
kxk
kxk x
y
x
x
≤C
kyk kxk + kyk + C kxk kxk − kyk kxk x
y
= Ckyk kxk + kyk + C(kxk − kyk)
x
y
+
= Ckyk kxk kyk + C(kxk + kyk − 2kyk)
x
y
+
− 2 + C(kxk + kyk).
= Ckyk kxk kyk Triangle Inequality in
Quasi-Banach Spaces
Cong Wu and Yongjin Li
vol. 9, iss. 2, art. 41, 2008
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Since
x
y
y
y kx + yk = kxk
+
− kxk
− kyk
kxk kyk
kyk
kyk x
1 y x
x ≥ kxk
+
− kxk
− kyk
C
kxk kyk kxk
kxk x
1
y − (kxk − kyk)
= kxk +
C
kxk kyk x
1
y + (kxk + kyk − 2kxk)
= kxk +
C
kxk kyk x
y
1 − 2 + (kxk + kyk).
= kxk
+
C kxk kyk Triangle Inequality in
Quasi-Banach Spaces
Cong Wu and Yongjin Li
vol. 9, iss. 2, art. 41, 2008
Title Page
Contents
we have
x
y kxk
C(kxk + kyk) ≤ Ckx + yk + 2C − +
kxk kyk x
y
= kx + yk + (C − 1)kx + yk + 2C − kxk + kyk kxk
x
y
≤ kx + yk + (C − 1)C(kxk + kyk) + 2C − kxk + kyk kxk
x
y
≤ kx + yk + (C − 1)C(2kxk) + 2C − +
kxk kyk kxk
x
y
+
= kx + yk + 2C 2 − kxk 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 exists 0 < p ≤ 1 and an equivalent quasi-norm k| · k| on X that satisfies for
every x, 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 any x, y ∈ X, kx + yk ≤ K(kxk + kyk)} and p is such that 21/p = 2k. It is
shown [3] that the function k| · k| defined on X by:


! p1
n
n
 X

X
p
:x=
xi
k|xk| = inf
kxi k


i=1
Triangle Inequality in
Quasi-Banach Spaces
Cong Wu and Yongjin Li
vol. 9, iss. 2, art. 41, 2008
Title Page
i=1
Contents
is an equivalent quasi-norm on X that satisfies the required inequality.
Next, we will prove the p-triangle inequality in quasi-Banach spaces.
Theorem 2.3. For all nonzero elements x, y in a quasi-Banach space X with kxk ≥
kyk,
p y p
p
p
p
p x
+
kx + yk + kxk + kyk − (kxk − kyk) − kyk kxk kyk ≤ kxkp + kykp
p y p
p
p
p
p x
,
≤ kx + yk + kxk + kyk + (kxk − kyk) − kxk +
kxk kyk where 0 < p ≤ 1.
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Proof. We have
p
y
x
x
x
kx + yk = +
−
kyk
kyk
+
kxk
kxk kyk
kxk
kxk p
p x
y
x
x
+ kxk
≤
kyk
+
−
kyk
kxk kyk kxk
kxk p
y p x
+ (kxk − kyk)p
= kyk +
kxk kyk p
y p x
+ kxkp
= kyk +
kxk kyk + kykp − (kxkp + kykp ) + (kxk − kyk)p .
p
Triangle Inequality in
Quasi-Banach Spaces
Cong Wu and Yongjin Li
vol. 9, iss. 2, art. 41, 2008
Title Page
Thus
Contents
p y p
p
p
p
p x
≤ kxkp + kykp
kx + yk + kxk + kyk − (kxk − kyk) − kyk +
kxk kyk JJ
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and
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p
x
y
y
y
p
kx + yk = kxk kxk + kyk − kxk kyk − kyk kyk p
p x
y
x
x
− kxk
≥
kxk
+
−
kyk
kxk kyk kxk
kxk p
x
y
p
+
= kxkp kxk kyk − (kxk − kyk)
p
y p x
= kxk +
kxk kyk + kxkp + kykp − (kxkp + kykp ) − (kxk − kyk)p .
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Hence
p x
y
+
kxkp +kykp ≤ kx+ykp + kxkp + kykp + (kxk − kyk)p − kxkp kxk kyk .
This completes the proof.
Triangle Inequality in
Quasi-Banach Spaces
Cong Wu and Yongjin Li
vol. 9, iss. 2, art. 41, 2008
Title Page
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References
[1] T. AOKI, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo, 18
(1942), 588–594.
[2] H. HUDZIK AND T.R. LANDES, Characteristic of convexity of Köthe function
spaces, Math. Ann., 294 (1992), 117–124.
[3] N.J. KALTON, N.T. PECK AND J.W. ROBERTS, An F-Space Sampler, London
Math. Soc. Lecture Notes 89, Cambridge University Press, Cambridge, 1984.
Triangle Inequality in
Quasi-Banach Spaces
Cong Wu and Yongjin Li
vol. 9, iss. 2, art. 41, 2008
[4] K.-I. MITANI, K.-S. SAITO, M.I. KATO AND T. TAMURA, On sharp triangle
inequalities in Banach spaces, J. Math. Anal. Appl., 336 (2007), 1178–1186.
Title Page
[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.
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