I think this is far too simplistic, for one because the values of this putative function depend on the chosen coordinate system.
So I completely agree with the comment you are replying to: when a physicist says "tensor" they really mean a "tensor field" and the definition of the latter is quite a bit more involved than just specifying a multilinear map at each point of a manifold.
I think this is far too simplistic, for one because the values of this putative function depend on the chosen coordinate system.
So I completely agree with the comment you are replying to: when a physicist says "tensor" they really mean a "tensor field" and the definition of the latter is quite a bit more involved than just specifying a multilinear map at each point of a manifold.