Existence and Uniqueness of Adjoint/Lemma 3

Lemma
Let $\mathbb F \in \set {\R, \C}$.

Let $\HH$ a be Hilbert space over $\mathbb F$ with inner product ${\innerprod \cdot \cdot}_\HH$.

Let $\KK$ a be Hilbert space over $\mathbb F$ with inner product ${\innerprod \cdot \cdot}_\KK$.

Let $A : \HH \to \KK$ be a bounded linear transformation.

Let $B : \KK \to \HH$ be the unique mapping satisfying:


 * $\innerprod {A x} y_\KK = \innerprod x {B y}_\HH$

for each $x \in \HH$ and $y \in \KK$.

Then $B$ is a bounded linear transformation.

Proof
Let $\norm \cdot_\HH$ be the inner product norm of $\HH$.

Let $\norm \cdot_\KK$ be the inner product norm of $\KK$.

From Existence and Uniqueness of Adjoint: Lemma 2, we have that:


 * $B$ is a linear transformation.

It remains to show that $B$ is bounded.

Let $\norm A$ denote the norm on $A$.

We have that $A$ is a bounded linear transformation.

From Norm on Bounded Linear Transformation is Finite:


 * $\norm A$ is finite.

Then:

Note that for all $y \in \KK$ such that:


 * $B y \ne 0$

we have:


 * $\norm {B y}_\HH \le \norm A \norm y_\KK$

From the definition, a norm is positive definite.

Hence this inequality holds if $B y = 0$.

So:


 * $\forall y \in \KK: \norm {B y}_\HH \le \norm A \norm y_\KK$

That is:


 * $B$ is a bounded linear transformation.