Some hermitian operators relations

[merkamerka]

How can I formally demonstrate this relations with hermitian operators?(A^{\dagger})^{\dagger}=A
(AB)^{\dagger}=B^{\dagger}A^{\dagger}
\langle x|A^{\dagger}y \rangle=\langle y|Ax \rangle ^*
If \ A \ is \ hermitian \ and \ invertible, \ then \ A^{-1} \ is \ hermitian

I've tried to prove them taking the definition of hermitian operator or/and considering matrices while operating, but I want something more formal.

Thanks

 

[lanedance]

how about using
<br /> AA^{-1}= \mathbb{I}<br />