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
$$AA^{-1}= \mathbb{I}$$