tensor product (idea)

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 {u₁,u₂,...,u_m} and V has basis {v₁,v₂,...,v_n} then simply define U×V to be a space with basis {u₁×v₁,u₁×v₂,...,u₁×v_n, u₂×v₁,u₂×v₂,...,u₂×v_n,... u_m×v₁,u_m×v₂,...,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.