[consilient]

> Tensor product V⊗V is space of all bilinear forms on V*

This is only true for finite-dimensional V. In general bilinear forms on V* are linear maps `V*⊗V* -> F` (where F is the base field) which is isomorphic to `(V*⊗V*)*`. For infinite-dimensional V this is much larger than `V⊗V`.

The general characterization is that `V⊗W` is the vector space (and associated linear map `⊗: V x W -> V⊗W`) such that for every bilinear map `h: V x W -> T` there's a unique linear g such that `h(v,w) = g(v⊗w)`. Intuitively, it adds "just enough" elements to `V x W` to encode arbitrary bilinear maps as linear ones.