Characterization of Dual Operator

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

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

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

Let $X^{\ast \ast}$ and $Y^{\ast \ast}$ be the second normed duals of $X$ and $Y$ respectively.

Let $T : Y^\ast \to X^\ast$ be a bounded linear transformation.

Let $\iota_X : X \to X^{\ast \ast}$ be the evaluation linear transformation for $X$.

Let $\iota_Y : Y \to Y^{\ast \ast}$ be the evaluation linear transformation for $Y$.


 * $(1) \quad$ $T$ is $\struct {w^\ast, w^\ast}$-continuous
 * $(2) \quad$ there exists a bounded linear transformation $S : X \to Y$ such that $S^\ast = T$, where $S^\ast$ denotes the dual operator of $S$
 * $(3) \quad$ $T^\ast \sqbrk {\iota_X X} \subseteq \iota_Y Y$

$(1)$ implies $(2)$
Suppose that $T$ is $\struct {w^\ast, w^\ast}$-continuous.

From Continuous Linear Functionals in Weak-* Topology, we have that:


 * $x^\wedge : \struct {X^\ast, w^\ast} \to \GF$ is continuous.

From Composite of Continuous Mappings is Continuous:


 * $x^\wedge \circ T : \struct {Y^\ast, w^\ast} \to \GF$ is continuous.

Applying Continuous Linear Functionals in Weak-* Topology, there exists $S x \in Y$ such that:


 * $x^\wedge \circ T = \paren {S x}^\wedge$

We first show that $S : X \to Y$ is linear.

We have:

We now show that $S$ is bounded.

We have, for each $x \in X$:

So $S$ is bounded.

Finally, for $f \in Y^\ast$ and $x \in X$ we have:

That is:


 * $f \circ S = T f$

for each $f \in Y^\ast$.

From the definition of the dual operator we have:


 * $S^\ast f = T f$

for each $f \in Y^\ast$.

So $S^\ast = T$.

$(2)$ implies $(3)$
Suppose that:


 * there exists a bounded linear transformation $S : X \to Y$ such that $S^\ast = T$, where $S^\ast$ denotes the dual operator of $S$.

We have $S^{\ast \ast} = T^\ast$, where $S^{\ast \ast}$ denotes the second dual operator.

Let $x \in X$ and $f \in Y^\ast$.

Then:

That is:


 * $\map {T^\ast} {\iota_X x} = \map {\iota_Y} {S x}$

$(3)$ implies $(1)$
Suppose that $T^\ast \sqbrk {\iota_X X} \subseteq \iota_Y Y$.

For each $x^\wedge \in \iota_X X$, we then have that:


 * $\map {T^\ast} {x^\wedge} \in \iota_Y Y$

That is, there exists $y \in Y$ such that:


 * $x^\wedge \circ T = y^\wedge$

From Continuous Linear Functionals in Weak-* Topology, we then have:


 * $x^\wedge \circ T : \struct {Y^\ast, w^\ast} \to \GF$ is continuous

for each $x \in X$.

Conversely from Continuous Linear Functionals in Weak-* Topology, we have that every continuous linear functional $\Phi : \struct {Y^\ast, w^\ast} \to \GF$ has the form $\Phi = x^\wedge$.

So from Continuity in Initial Topology, we have that:


 * $T : \struct {Y^\ast, w^\ast} \to \struct {X^\ast, w^\ast}$ is continuous.