The above is certainly formidable, but would not scare most

applied mathematicians. They are most happy in defining the product space in terms of bases:

If U has basis {u_{1},u_{2},...,u_{m}} and V has basis {v_{1},v_{2},...,v_{n}} then simply define U×V to be a space with basis {u_{1}×v_{1},u_{1}×v_{2},...,u_{1}×v_{n}, u_{2}×v_{1},u_{2}×v_{2},...,u_{2}×v_{n},... u_{m}×v_{1},u_{m}×v_{2},...,u_{m}×v_{n}}

while the basis independent definition is more general and makes no assumption of such a space existing, this definition is sometimes easier to stomach.

An example of this use of this in Quantum Mechanics is used for dealing with states of particles with spin: if |m> is a spin state and |**x**> is the usual position ket, then define product kets as |**x**,m> = |**x**> × |m>. This can then be used to define the complete wave function of the particle, as detailed in *Angular Momentum in Quantum Mechanics*.

Also, if some operator acts on U, its action naturally be specified on U×V by U(u × v) = (U(u)) × v ie. it acts as the identity on V.