Review of Tranposes

Review of Tranposes

Let $(E, F)$ and $(G, H)$ be dual pairs.

• Recall from The Transpose of a Linear Operator page that if $t : E \to G$ is a linear operator, then the Transpose of $t$ is the linear operator $t' : H \to E^*$ defined by $h \mapsto t'(h)$ where for each $e \in E$ we have that $t'(h)(e) := \langle t(e), h \rangle$. We note that $t'$ has the special identity that for all $e \in E$ and for all $h \in H$:
(1)
\begin{align} \quad \langle e, t'(h) \rangle = \langle t(e), h \rangle \end{align}
• On the Weakly Continuous Linear Operators page we said that $t : E \to G$ is Weakly Continuous if it is continuous when $E$ is equipped with the $\sigma(E, F)$ topology and when $G$ is equipped with the $\sigma(G, H)$ topology.
• We then characterized weak continuity of linear operators:

If $(E, F)$ and $(G, H)$ are dual pairs and $t : E \to G$ is a linear operator then $t$ is weakly continuous if and only if $t'(H) \subseteq F$.

If $E$ and $F$ are Hausdorff locally convex topological vector spaces (so that $(E, E')$ and $(F, F')$ are dual pairs) and if $t : E \to F$ is a continuous linear operator then $t$ is weakly continuous.

• We remarked that the converse of the result above is not true in general.

If $(E, F)$ and $(G, H)$ are dual pairs, $A \subseteq E$, and $t : E \to G$ is weakly continuous, then:

(2)
\begin{align} \quad (t(A))^{\circ} = t'^{-1}(A^{\circ}) \end{align}

Where $(t(A))^{\circ}$ is the polar of $t(A)$ (in $H$), and $A^{\circ}$ is the polar of $A$ (in $F$).