It's probably better to start with the RHS and get the LHS.Consider two finite dimensional vector spaces and over complex numbers , their bases and , their dual spaces and , and their dual bases and .
Two linear maps and are formally dual if:
Prove that the linear maps and are formally dual.
so that for each is an element of defined by .
defined by (I know that )
ie.
It's also clear that or
Take a basis vector of , call it . Do the same for , call the basis vector .
I have to get
We get:
since and i'm now working on a vector in . The linear map also has to act on a vector, so it's acting on some fixed .
Here but linear maps are defined on how they work on a vector so I need to put a in there.
Apply to both sides to get:
From here i'm absolutely lost! Does anyone have any ideas??
(P.S This is very hard!)
Let U be finite dimention Hilbert space, for any , associates defined by . (Define Similarly in V)
Denote the mapping :
,Since Hilbert space is self-dual,the mapping is injective,
then RHS
= LHS