For any two vector spaces [math]V[/math] and [math]W[/math], the tensor product [math]V\bigotimes W[/math] is the space of bilinear functions on [math]V\times W[/math] .

If [math]V[/math] and [math]W[/math] are inner product spaces then for any [math]v\in V[/math] and [math]w\in W[/math] we can define the pure tensor [math]v\otimes w[/math] in [math]V\bigotimes W[/math] by [math] v\otimes w (v*,w*)=(v\cdot v* )(w\cdot w*)[/math] also often written by physicists as [math]\langle v|v*\rangle \langle w|w*\rangle[/math]. But there are also elements of [math]V\bigotimes W[/math] that are not of the pure tensor form.

For example if [math]v_1\otimes w_1+v_2\otimes w_2[/math] could be written in the form [math](a_1v_1+a_2v_2)\otimes(b_1w_1+b_2w_2)[/math], then for all [math]v\in V[/math] and [math]w\in W[/math], we’d need [math]\begin{align}&(a_1v_1+a_2v_2)\otimes(b_1w_1+b_2w_2)(v,w)\\ &=(a_1v_1\cdot v)(b_1w_1\cdot w)+(a_1v_1\cdot v)(b_2w_2\cdot w)\\&+(a_2v_2\cdot v)(b_1w_1\cdot w)+(a_2v_2\cdot v)(b_2w_2\cdot w)\\&=(v_1\cdot v)(w_1\cdot w)+(v_2\cdot v)(w_2\cdot w)\end{align}[/math] .

But this is only true if [math]a_1b_1=a_2b_2=1[/math] and [math]a_1b_2=a_2b_1=0[/math], but if one of [math]a_1[/math] or [math]b_2[/math] is zero then one of [math]a_1b_1[/math] or [math]a_2b_2[/math] must be also.

Source: *(405) What is a tensor product in simple words? – Quora*