NORMS ON CARTESIAN PRODUCT OF LINEAR SPACES

Authors

  • Chi-Kwong Li Department of Mathematics, The College of William and Mary, Williamsburg, VA 23185.
  • Nam-Kiu Tsing Systems Research Center and Electrical Engineering Department, University of Maryland, College Park, MD 20742.

DOI:

https://doi.org/10.5556/j.tkjm.21.1990.4692

Keywords:

norm, cartesian product, linear spaces

Abstract

Let $X_i, (i=1, \cdots, n)$ be real or complex linear spaces, each equipped with a norm $||\cdot||_i$. Standard ways of constructing norms $||\cdot||$ on the Cartestian product $X =X_1 \times \cdots \times X_n$ are to define

\[ ||(x_1, \cdots, x_n)||=\phi(||x_1||_1, \cdots, ||x_n||_n)\]

via some functions $\phi$ on $\mathbb{R}^n$. Common examples of $\phi$ in standard texbooks are norms on $\mathbb{R}^n$. This may mislead peoples to think that any norm $\phi$ on $\mathbb{R}^n$ can induce a norm on the product space $X$ in the above way. In this note we show that this is actually false and characterize the functions $\phi$ that can give rise to norms on $X$ in the above manner. It turns out that a necessary and sufficient condition on $\phi$ is :

for any $a_1, \cdots, a_n, b_1, \cdots, b_n\ge 0$,

(I) $\phi(a_1, \cdots, a_n)>0$ if $(a_1, \cdots, a_n)\neq (0, \cdots,0)$;

(II) $\phi(\alpha(a_1, \cdots, a_n))= \alpha \phi(a_1, \cdots, a_n)$ if $\alpha\ge 0$;

(III) $\phi(c_1, \cdots, c_n)\le \phi(a_1, \cdots, a_n)+ \phi(b_1, \cdots, b_n)$ if $(c_1, \cdots, c_n)= (a_1, \cdots, a_n)+ (b_1, \cdots, b_n)$;
(IV) $\phi(a_1, \cdots, a_n) \le \phi(b_1, \cdots, b_n)$ if $a_i \le b_i$ for all $i$.

Several interesting consequences of the result are discussed.

References

R. A. Hom and C. R. Johnson, Matrix Analysis, Combridge Univ. Press, New York, 1985.

A. A. Kirillov and A. D. Gvishiani, Theorems and Problems in Functional Analysis, Springer-Varlag, New York, 1982.

J. R. Leigh, Functional Analysis and Linear Control Theory, Academic Press, London, 1980.

B. V. Limaye, Functional Analysis, Wiley Eastern Ltd., New Delhi, 1981.

R. D. Milne, Applied Functional Analysis - An introductory Treatment, Pitman Publishing Inc., Massachusetts, 1980.

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Published

1990-03-01

How to Cite

Li, C.-K., & Tsing, N.-K. (1990). NORMS ON CARTESIAN PRODUCT OF LINEAR SPACES. Tamkang Journal of Mathematics, 21(1), 35–39. https://doi.org/10.5556/j.tkjm.21.1990.4692

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Section

Papers