Existence and Uniqueness of Adjoint/Lemma 3
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.
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:
\(\ds \forall y \in \KK: \, \) | \(\ds \norm {B y}_\HH^2\) | \(=\) | \(\ds \innerprod {B y} {B y}_\HH\) | Definition of Inner Product Norm | ||||||||||
\(\ds \) | \(=\) | \(\ds \innerprod {A B y} y_\KK\) | Definition of $B$ | |||||||||||
\(\ds \) | \(\le\) | \(\ds \norm {A B y}_\KK \norm y_\KK\) | Cauchy-Bunyakovsky-Schwarz Inequality for Inner Product Spaces | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm A \norm {B y}_\HH \norm y_\KK\) | Fundamental Property of Norm on Bounded Linear Transformation |
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.
$\blacksquare$