## 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,:

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

## 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:

 $\ds \innerprod {\paren {\lambda A} h} k_\KK$ $=$ $\ds \lambda \innerprod {A h} k_\KK$ Property $(2)$ of Inner Product $\ds$ $=$ $\ds \lambda \innerprod h {A^*k}_\HH$ Definition of Adjoint Linear Transformation $\ds$ $=$ $\ds \innerprod h {\paren {\overline \lambda A^*} k}_\HH$ Properties $(1)$ and $(2)$ of Inner Product

Thus, by Existence and Uniqueness of Adjoint:

$\paren {\lambda A}^* = \overline \lambda A^*$

$\Box$

### Proof of $(2)$

Let $h \in \HH, k \in \KK$.

Then:

 $\ds \innerprod {\paren {A + B} h} k_\KK$ $=$ $\ds \innerprod {A h} k_\KK + \innerprod {B h} k_\KK$ Property $(3)$ of Inner Product $\ds$ $=$ $\ds \innerprod h {A^*k}_\HH + \innerprod h {B^*k}_\HH$ Definition of Adjoint Linear Transformation $\ds$ $=$ $\ds \innerprod h {\paren {A^* + B^*} k}_\HH$ Properties $(1), (3)$ of Inner Product

Thus, by Existence and Uniqueness of Adjoint:

$\paren {A + B}^* = A^* + B^*$

$\blacksquare$