# Properties of adjoints of linear maps

I am studying linear algebra by myself, and I came across the need to prove the following, for adjoints of linear maps:

• ($f^\dagger)^\dagger=f$
• $(f\circ g)^\dagger=g^\dagger\circ f^\dagger$
• $\langle v|fw\rangle=\langle f^\dagger v|w\rangle$

Thank you very much!

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Do you know the definition of the adjoint? I strongly recommend that you do these yourself, because they're very easy. If you still can't do them, tell us what you have tried. –  wj32 Oct 26 '12 at 12:59
Thank you. Off course, the first 2 are obvious if I accept the third. But if I don't, I don't know where to start. –  dagger Oct 26 '12 at 15:42
The third is usually the definition of the adjoint. What is your definition? –  wj32 Oct 26 '12 at 20:53