Let \(\mathbf{A}\in M_{n,n}(\mathbb{C}).\) Show that there are two unique Hermitian matrices \(\mathbf{B}\) and \(\mathbf{C}\) such that \(\mathbf{A}=\mathbf{B}+\mathrm i \mathbf{C}.\)

Show that the set of Hermitian \(n\times n\)-matrices admits the structure of an \(\mathbb{R}\)-vector space \(V.\) Find a basis for \(V.\)

Provide the details of the proof of Lemma 11.4 (iii) and, as a consequence, show that the determinant of a Hermitian matrix is a real number.

Let \[\ell^2(\mathbb{C})=\left\{z=(z_n)_{n\in\mathbb{N}}\in \mathbb{C}^\infty \left| \|z\|^2= \sum_{i=1}^\infty |z_i|^2<\infty\right\}\right.\] be the complex vector space of square summable complex sequences equipped with the sesquilinear form \[\langle z, w\rangle = \sum_{i=1}^\infty \overline{z_i}w_i .\]

1. Show that \((\ell^2(\mathbb{C}),\langle \cdot{,}\cdot\rangle)\) is a unitary space.

2. Let \(f:\ell^2(\mathbb{C})\to\ell^2(\mathbb{C}), z=(z_1,z_2,\ldots)\mapsto (0,z_1,z_2,\ldots).\) Determine the adjoint map \(f^*:\ell^2(\mathbb{C})\to\ell^2(\mathbb{C})\) which – as in the real case – is uniquely determined by \(\langle f(z),w\rangle = \langle z,f^*(w)\rangle.\)