Existence and Uniqueness of Adjoint/Lemma 3

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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.

$\blacksquare$

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