Prove that the Reisz map defined by for all y in V, is an isomorphism.
I am not sure how to show it is onto and one to one with the Reisz map.
Look at (2) on page 222 under the Riesz Representation Theorem. The map you wrote is (1) but only (2) is being called the Reisz map.
How do I show this is one to one?
Would I start if or would it start differently?
well, you would suppose first that R(f) = R(g) (these are the riesz vectors for f and g).
therefore <R(f),y> = <R(g),y> for all vectors y. but this means that f(y) = g(y) for all y in V, so f = g.
EDIT: drexel28's map is the inverse riesz map. since we have an isomorphism, we're really talking about the same thing: a 1-1 correspondence (bijection), it's a "dual" thing (and that is a million dollar pun).
you don't like alex's suggestion (using the fact that dim V = dim V*)?
very well, for any v in V, we need to find some f in V* such that R(f) = v.
how about f = <v,_>? then R(f) is defined by <R(f),y> = f(y) = <v,y>, and since <R(f),y> = <v,y> for every y,
we conclude R(f) = v.
if <R(f),y> = <v,y>, then <R(f) - v,y> = 0 for all y in V. but R(f) - v is some vector in V,
so <R(f) - v,R(f) - v> = 0, so by positive definiteness of the inner product, R(f) - v = 0, so R(f) = v).
I make only one remark. The reason why, to me at least, it makes more sense to define the Riesz map as I did is that it produces a monomorphism regardless of what kind of vector space is. Conversely, the map can't possibly hope to be a monomorphism if is infinite dimensional since in that case .
well, you and Mr. Roman better have a l'il talk then. i mean, sheesh calling the wrong map the Riesz map!
i agree with you, it makes more "sense" to think of V as the natural starting place (domain), and V* as the "derived" thing (image),
although his way of defining things does make the name "Riesz vector" seem more meaningful.
(myself, i find myself wondering how we really know which vectors are really vectors, and which are co-vectors. what if the math elves switched them while we were sleeping? i guess that shows why you should keep an infinite-dimensional vector space on hand, so you can embed your finite-dimensional space in it, and see if its dual is bigger).