Existence and Uniqueness of Adjoint/Lemma 2
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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 we have:
- $\map B {\alpha x + \beta y} = \alpha B x + \beta B y$
for all $\alpha, \beta \in \mathbb F$ and $x, y \in \KK$.
That is, $B$ is a linear transformation.
Proof
Let $\alpha, \beta \in \mathbb F$.
Let $x, y \in \KK$.
We have:
\(\ds \innerprod x {\map B {\alpha x + \beta y} }_\KK\) | \(=\) | \(\ds \innerprod {A x} {\alpha x + \beta y}_\HH\) | Definition of $B$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \overline \alpha \innerprod {A x} x_\HH + \overline \beta \innerprod {A x} y_\HH\) | Inner Product is Sesquilinear | |||||||||||
\(\ds \) | \(=\) | \(\ds \overline \alpha \innerprod x {B x}_\KK + \overline \beta \innerprod x {B y}_\KK\) | Definition of $B$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \innerprod x {\alpha B x}_\KK + \innerprod x {\beta B y}_\KK\) | Inner Product is Sesquilinear | |||||||||||
\(\ds \) | \(=\) | \(\ds \innerprod x {\alpha B x + \beta B y}_\KK\) | Inner Product is Sesquilinear |
Recall that there exists exactly one $z \in \HH$ such that:
- $\innerprod {A x} {\alpha x + \beta y}_\HH = \innerprod x z_\KK$
So we must have:
- $\map B {\alpha x + \beta y} = \alpha B x + \beta B y$
So:
- $\map B {\alpha x + \beta y} = \alpha B x + \beta B y$
for each $\alpha, \beta \in \mathbb F$ and $x, y \in \KK$.
So:
- $B$ is a linear transformation.
$\blacksquare$