February 4th, 2010, 12:21 AM  #1 
Newbie Joined: Feb 2010 Posts: 1 Thanks: 0  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 finitedimensional 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 08:18 AM. 
February 6th, 2010, 06:11 AM  #2 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  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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