Physical space is the vector space $\mathbb{R}^3$ equipped with an inner product (dot product $\cdot$), a bilinear product (cross product $\times$) and is complete (all limits of vector sequences approach some other vector).
Is this a "complete algebra over the Reals with an inner product"? How about "a Hilbert space with a bilinear product?"
Is there a standard name for this structure? What would you call this structure?
P.S did I miss any other operators/structure on real 3-space that are used to model physical space?