Norm of Cartesian Product

If we have two normed linear spaces $Z$ and $U$ , their Cartesian product $Z\oplus U$ can also be normed, such as by setting $|(z,u)|=|z|+|u|$ , $|(z,u)|'=\max\{|z|,|u|\}$ , or $|(z,u)|''=(|z|^2+|u|^2)^{1/2}$ . Note that we are following Lax’s Functional Analysis, where a norm is denoted as $|\cdot |$ , rather than $\|\cdot\|$ which is clearer but more cumbersome to write.

It is routine to check that all the above 3 are norms, satisfying the positivity, subadditivity, and homogeneity axioms. Minkowski’s inequality is useful to prove the subadditivity of the last norm.

We may check that all of the above 3 norms are equivalent. This follows from the inequalities $\frac{1}{2}(|z|+|u|)\leq\max\{|z|,|u|\}\leq |z|+|u|$ , and $M=(M^2)^{1/2}\leq (|z|^2+|u|^2)^{1/2}\leq (2M^2)^{1/2}=\sqrt 2 M$ , where $M:=\max\{ |z|,|u|\}$ . In general, we have that all norms are equivalent in finite dimensional spaces.