NORMS ON CARTESIAN PRODUCT OF 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.