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February 3rd, 2010, 11:21 PM   #1
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Adjoint problem

I recently teach myself linear algebra with Friedberg's textbook.
And I have a question about adjoint operator, which is on p.367.

Definition Let T : V ? W be a linear transformation where V and W are finite-dimensional inner product spaces with inner products <?,?> and <?,?>' respectively. A function T* : W ? V is called an adjoint of T if <T(x),y>' = <x,T*(x)> for all x in V and y in W.

Then, my question is how to prove that there is a unique adjoint T* of T ?

Is it right to mimic the textbook which discuss the problem at the condition of T 's being an operator ?

Last edited by skipjack; November 4th, 2016 at 07:18 AM.
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February 6th, 2010, 05:11 AM   #2
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Re: Adjoint problem

I'll pretend to be a pure functional analyst throughout, so that the inner product is linear in the first slot and antilinear in the second.

Suppose that both satisfy for all Then,



so



for all

EDIT:
In particular, we can take to give for all Therefore
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