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 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 finite-dimensional inner product spaces with inner products and ' respectively. A function T* : W ? V is called an adjoint of T if ' = 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 $U_1,U_2\,:\,W\to V$ both satisfy $\langle Tx,y\rangle_W=\langle x,U_iy\rangle_V$ for all $x\in V,\ y\in W,\ i=1,2.$ Then, $\langle x,U_1y\rangle_V=\langle Tx,y\rangle_W=\langle x,U_2 y\rangle_V,$ so $\langle x,(U_1-U_2)y\rangle_V=0$ for all $x\in V,\ y\in W.$ EDIT: In particular, we can take $x=(U_1-U_2)y,$ to give $\left\|(U_1-U_2)y\right\|^2=0$ for all $y\in W.$ Therefore $U_1=U_2.$

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