Problem:
I'm wondering what could be a realization for algebraic tensor product of infinite dimensional vector spaces?
Any ideas are welcome, of course!
Attempts:
My first guess would be the space of bilinear forms:
$$\phi:U\times V\to\mathcal{B}(U^*\times V^*; \mathbb{R}):\phi(u,v)(\mu,\nu):=\mu(u)\nu(v)$$
Within this realization the tensor product becomes the linear span of the image:
$$U\otimes V\cong\langle\mathrm{im}\phi\rangle$$
This description is rather abstract. So how to prescribe it concretely?
Disclaimer:
I'm neither concerned with the topological tensor product nor with monoids, groups, rings or modules but only with the algebraic tensor product of vector spaces. My interest in here is not about a categorical description (though it should of course satisfy the universal property).
I'm asking since I'd like to have a realization applicable in general since many authors give a variety of realizations - most of them not working for infinite dimensional vector spaces.