Dual Operator of Composition

Theorem

Let $\GF \in \set {\R, \C}$.

Let $X$, $Y$ and $Z$ be normed vector spaces over $\GF$.

Let $X^\ast$, $Y^\ast$ and $Z^\ast$ be the normed dual spaces of $X$, $Y$ and $Z$ respectively.

Let $T : X \to Y$ and $S : Y \to Z$ be bounded linear transformations.

Let $T^\ast : Y^\ast \to X^\ast$ and $S^\ast : Z^\ast \to Y^\ast$ be the dual operators of $T$ and $S$ respectively.

Then:

$\paren {S T}^\ast = T^\ast S^\ast$

where $\paren {S T}^\ast$ is the dual operator of $S T : X \to Z$.

Let $f \in Z^\ast$.

Then, we have:

$\ds \map {\paren {T^\ast S^\ast} } f$

$=$

$\ds \map {T^\ast} {S^\ast f}$

$\ds $

$=$

$\ds \map {T^\ast} {f \circ S}$

Definition of Dual Operator

$\ds $

$=$

$\ds \paren {f \circ S} \circ T$

Definition of Dual Operator

$\ds $

$=$

$\ds f \circ \paren {S T}$

$\ds $

$=$

$\ds \map {\paren {S T}^\ast} f$

Definition of Dual Operator

$\blacksquare$