If we have two normed linear spaces and , their Cartesian product can also be normed, such as by setting , , or . Note that we are following Lax’s Functional Analysis, where a norm is denoted as , rather than 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 , and , where . In general, we have that all norms are equivalent in finite dimensional spaces.