Exa 7

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ A.7}$

Show that the binary operator $f\otimes g: (a,b) \mapsto f(a)g(b)$ for $f\in {\bf R}^A$ and $g\in {\bf R}^B$ satisfies the relations defining a tensor product structure on ${\bf R}^{A\times B}$ given in Section 3.1.2.