Algebraic Properties of the Adjoint in Finite Dimensions

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space that has an orthonormal basis $e\in V^{n}$ for some natural number $n$, where $V^{n}$ is the set of $n$-tuples in $V$. By Uniqueness of the Adjoint, and Existence in Finite Dimensions every linear operator on $V$ has exactly one adjoint, written with a star.

Let $S$ and $T$ be linear operators on $V$ and let $\lambda$ be a complex number with conjugate $\overline{\lambda}$. The sum, scalar multiple, product and identity operator are as in that definition. Then the following hold.

1. (Sum) $(S+T)^{*}=S^{*}+T^{*}$.

2. (Scalar multiple) $(\lambda S)^{*}=\overline{\lambda}\,S^{*}$.

3. (Product) $(ST)^{*}=T^{*}S^{*}$.

4. (Involution) $(S^{*})^{*}=S$.

5. (Identity) $(\mathrm{id}_{V})^{*}=\mathrm{id}_{V}$.