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How can I formally demonstrate this relations with hermitian operators?

[tex](A^{\dagger})^{\dagger}=A [/tex]

[tex](AB)^{\dagger}=B^{\dagger}A^{\dagger} [/tex]

[tex]\langle x|A^{\dagger}y \rangle=\langle y|Ax \rangle ^*[/tex]

[tex]If \ A \ is \ hermitian \ and \ invertible, \ then \ A^{-1} \ is \ hermitian[/tex]

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

[tex](A^{\dagger})^{\dagger}=A [/tex]

[tex](AB)^{\dagger}=B^{\dagger}A^{\dagger} [/tex]

[tex]\langle x|A^{\dagger}y \rangle=\langle y|Ax \rangle ^*[/tex]

[tex]If \ A \ is \ hermitian \ and \ invertible, \ then \ A^{-1} \ is \ hermitian[/tex]

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

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