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.

Author: mathtuition88

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