Prove that the Reisz map defined by for all y in V, is an isomorphism.

Let

I am not sure how to show it is onto and one to one with the Reisz map.

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- Nov 5th 2011, 10:12 AMdwsmithRiesz Map
Prove that the Reisz map defined by for all y in V, is an isomorphism.

Let

I am not sure how to show it is onto and one to one with the Reisz map. - Nov 5th 2011, 07:20 PMDrexel28Re: Riesz Map
Is this really what the Riesz map is? Usually the Riesz map is defined by where . Is this what you mean? Regardless, most likely to show bijectivity you just need to show that the map is injective, and since the dual space is of equal dimension to the space, this implies bijectivity.

- Nov 5th 2011, 07:45 PMdwsmithRe: Riesz Map
Graduate Texts in Mathematics-Advanced Linear Algebra by Roman 3rd edition.

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? - Nov 5th 2011, 08:21 PMDevenoRe: Riesz Map
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). - Nov 5th 2011, 08:28 PMdwsmithRe: Riesz Map
- Nov 5th 2011, 09:22 PMDevenoRe: Riesz Map
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.

(amplification:

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). - Nov 5th 2011, 11:25 PMDrexel28Re: Riesz Map
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 .

- Nov 6th 2011, 02:00 AMDevenoRe: Riesz Map
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).