Existence and Uniqueness of Adjoint/Lemma 1

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

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

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

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

Then T  t here exists a unique mapping $B : \KK \to \HH$ such that:


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

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

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

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

For each $y \in \KK$, define the linear functional $f_y : \HH \to \mathbb F$ by:


 * $\map {f_y} x = \innerprod {A x} y_\KK$

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.

We therefore have:

Taking $M = \norm A \norm y_\KK$, we have:


 * $\size {\map {f_y} x} \le M \norm x_\HH$

for each $x \in \HH$, with $M$ independent of $x$.

So, $f_y$ is bounded.

So, by the Riesz Representation Theorem (Hilbert Spaces), there exists a unique $\map z y \in \HH$ such that:


 * $\map {f_y} x = \innerprod x {\map z y}_\HH$

for each $x \in \HH$.

That is, for each $y \in \KK$ there exists precisely one $\map z y \in \HH$ such that:


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

for all $x \in \HH$.

Define the mapping $B : \KK \to \HH$ by:


 * $B y = \map z y$

for each $y \in \KK$.

This map has:


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

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

Since the choice of $\map z y$ was unique, the map $B$ must also be unique, so $B$ is the unique map with the required properties.