Double dual/Double Transpose Question

nyisles131

The question that I am stuck on is:
Show that if X" (double dual of X) is identified with X and U" (double dual of U) with U via the duality relation, then T" (double transpose) = T.
(Duality relation is f(L) = L (x) where f is in X", L is in X', and x is in X)

So far, here is my work:

We know:

T: X --> U is a linear homogeneous map
Therefore,
T': U' --> X' where U' is the dual of U and X' is the dual of X
Then,
T": X" --> U" where X" is the double dual of X and U" is the double dual of U.

Also, X" is isomorphic to X, and U" is isomorphic to U.

I am missing something here, however. This is where I am stuck. How can one deduce that, in fact, T" = T? How do we show that two linear homogenoue maps are the equivalent?
The idea of a double dual has left me slightly confused and any help would REALLY be appreciated.

Thanks.

Hurkyl

You can show two linear maps are equal in exactly the same way as any other function: by proving that T(x) = T"(x) for all x.

nyisles131

Let me check with you if this is correct...

Let T(x) = u
Then, we have to show that T"(f) = u where f is in X"?

We know the following:
For f in X", L in X', T"(f)(L) = fT'(L)
This implies: T"(f)(L) = T' (Lx) (by duality relation)
Then, T"(f)(L) = LT(x)
Then, T"(f)(L) = Lu
But, this does not equal u. What am I doing wrong here?

Hurkyl

Strictly speaking, just as f is assumed to be the double dual of x, you need to show that T"(f) is the double dual of u.


That's wrong.

T"(f) is an element of U".
L is an element of X'.
Therefore, one cannot evaluate T"(f) at L.

nyisles131

I'm very confused right now.
By definition, isn't (T"(f))(L) = f(T'(l)?


...Can you give me any suggestions as to how to approach this problem?

Hurkyl

Short answer: you're using something from the wrong space. You need to consider an element of U', since T"(f) is an element of U".



Honestly, I too find functions of functions confusing. (And worse, you're dealing with functions of functions of functions!)

My solution is simply to try and be extra precise and write everything down that I know -- in particular, I try to (at least mentally) write down what set everything lives in, and if it's a function, I write down what it's domain and its range is.

One cute little trick that works in this particular example is to have elements of X" act on the right. In particular, you would write:

(L)f

and not

f(L)

This has the benefit of the suggestively similar notation:

Lf = Lx

I'm not going to do that in what follows, though.


(Changing letters to reduce possible confusion)

If you have a map:

S : Y --> Z

then you have a map

S' : Z' --> Y'

Suppose g is an element of Z'. Then S'(g) is an element of Y'. In particular:

S'(g) : Y ---> F (where F is your base field)

So, S'(g) has to take something in Y as its argument. (Not something in Z -- that's the mistake you were making)

And, IIRC, the definition is:

S'(g)(y) = g(S(y))

(look at the type of everything, and make sure that the whole expression makes sense. e.g. note that S(y) is an element of Z)


(P.S. you have to assume that X and U are finite dimensional spaces. I just want to make sure you know that)

nyisles131

Ok, so let me start over again....
(FORGET ALL THE VARIABLES I USED BEFORE)

I have:
T: X --> U
T': U' --> X'
T": X" --> U"

Then, take f in X"
Therefore, T"(f) is in U".
Then, T"(f)(L) = f(T'(L)) for L in U'
Then T"(f)(L) = f(LT) because T'(L) = LT
Thus, T"(f)(L) = LT(x) by duality relation
Finally, T"(f)(L) = Lu, which is in U" by the duality relation
So, T"(f) does indeed map to Lu, which is in U".

Now, is it enough to say that since X" is isomorphic to X (dim X" = dim X) and U" is isomorphic to U (dim U" = dim U), then T" = T. Or is there a step I am missing here? (I feel like I am missing something, but I'm not sure)

nyisles131

Is there anything else I need to do for this problem other than what I stated in the previous post?

nyisles131

Does anyone know if what I have stated 2 posts ago is the correct answer?

Hurkyl

This last line is correct. T"(f) isn't mapped to Lu. (T"(f) maps L to Lu) The important thing is that, since L is arbitrary, T"(f) is the dual of u.

So, you've proven that if T(x) = u, then T"(x") = u". When they say:

X" is identified with X

they mean that (wave hands a bit) we are considering x" = x to be an actual equality. (similarly, that u" = u) And that is what's required to show T" = T.


If you feel uncomfortable with that level of imprecision, it should be okay to simply remember that:
T"(x") = u" iff T(x) = u

nyisles131

I finally get it...
Thanks for your help!!!!