Adjoining is Linear

Theorem
Let $\HH$ and $\KK$ be Hilbert spaces over $\Bbb F \in \set {\R, \C}$.

Let $\map \BB {\HH, \KK}$ be the set of bounded linear transformations from $\HH$ to $\KK$.

Let $A, B \in \map \BB {\HH, \KK}$ be bounded linear transformations.

Then the operation of adjoining $^*$ satisfies, for all $\lambda \in \Bbb F$:


 * $(1): \quad \paren {\lambda A}^* = \overline \lambda A^*$
 * $(2): \quad \paren {A + B}^* = A^* + B^*$

where $\overline \lambda$ denotes the complex conjugate of $\lambda$.

That is,:
 * $^*: \map \BB {\HH, \KK} \to \map \BB {\KK, \HH}$

is a linear transformation.

Proof
Let $\innerprod \cdot \cdot_\HH$ and $\innerprod \cdot \cdot_\KK$ be inner products on $\HH$ and $\KK$ respectively.

Proof of $(1)$
Let $\lambda \in \Bbb F$, $h \in \HH, k \in \KK$.

Then:

Thus, by Existence and Uniqueness of Adjoint:
 * $\paren {\lambda A}^* = \overline \lambda A^*$

Proof of $(2)$
Let $h \in \HH, k \in \KK$.

Then:

{{eqn | l = \innerprod {\paren {A + B} h k_\KK | r = \innerprod {A h} k_\KK + \innerprod {B h} k_\KK | c = Property $(3)$ of Inner Product }}

Thus, by Existence and Uniqueness of Adjoint:
 * $\paren {A + B}^* = A^* + B^*$