The Transpose of a Linear Operator

Let $(E, F)$ and $(G, H)$ be dual pairs and let $t : E \to G$ be a linear operator. Observe that for each $h \in H$, the map $\langle t(\cdot), h \rangle : E \to \mathbf{F}$ defined by $\langle t(e), h \rangle$ is a linear operator on $E$.

So for each $h \in H$, let $t'(h) \in E^*$ be the linear operator defined for all $e \in E$ by:

(1)

\begin{align} \quad t'(h)(e) := \langle t(e), h \rangle \end{align}

Definition: Let $(E, F)$ and $(G, H)$ be dual pairs and let $t : E \to G$ be a linear operator. The Transpose of $t$ is the linear mapping $t' : H \to E^*$ defined above.

Note that the transpose $t'$ of $t$ has the following identify which holds for all $e \in E$ and for all $h \in H$:

(2)

\begin{align} \quad \langle e, t'(h) \rangle = \langle t(e), h \rangle \end{align}