I am having trouble understanding this remark on page 192 of Linear Algbera Done Right about the Riesz representation theorem.
The author claims that part d of this problem provides an alternate proof of the Riesz representation theorem in the case that F = R. However I am struggling to see why this is true. The Riesz representation theorem proves the existence and uniqueness of a vector u in V such that for all v and for any linear functional φ, φ(v) = <v,u>, and I don’t see how part d allows us to draw the same conclusion about any linear functional on V.