Adjoining is Linear

Theorem
Let $H, K$ be Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$.

Let $A, B \in B \left({H, K}\right)$ be bounded linear transformations.

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


 * $(1): \qquad \left({\lambda A}\right)^* = \overline \lambda A^*$
 * $(2): \qquad \left({A + B}\right)^* = A^* + B^*$

That is, $^*: B \left({H, K}\right) \to B \left({K, H}\right)$ is a linear transformation.

Proof of $(1)$
Let $\lambda \in \Bbb F$, $h \in H, k \in K$. Then:

Thus, by Existence and Uniqueness of Adjoint, $\left({\lambda A}\right)^* = \overline \lambda A^*$.

Proof of $(2)$
Let $h \in H, k \in K$. Then:

Thus, by Existence and Uniqueness of Adjoint, $\left({A + B}\right)^* = A^* + B^*$.