$\def \AA{A^{**}}\newcommand{\cC}{\mathbf{C\mbox{-}Vct}}

\newcommand{\cW}{\mathbf{W\mbox{-}Vct}}

\newcommand{\BB}{B^{**}}\newcommand{\ob}[1]{\operatorname{Ob}{\mathbf{#1}}}\newcommand{\ad}{\operatorname{Ad^2}}

\newcommand{\id}{\operatorname{Id}}$

Lemma 1. Suppose $\pi :V\to W$ is a continuous linear map between two normed spaces, then

the adjoint of $\pi $ defined by \begin{align}

\pi^*:&W^*\to V^*\notag\\

& f\mapsto f\circ \pi\label{adjoint}

\end{align}

is a continous linear map respect to norms of two spaces or waek* topology of two

spaces. Moreover, the following equation always holds: \begin{align}

\label{adimbedding}

\pi^{**}\circ i=i\circ \pi,

\end{align}

i.e., the following diagram is always commutative: \begin{equation}

\label{icommutes}

\xymatrix{

V\ar[r]^\pi\ar[d]_i&W\ar[d]^i\\

V^{**}\ar[r]^{\pi^{**}}&W^{**}.

}

\end{equation}

Proof. Since \begin{equation}

\label{normdecrease}

\|f\circ \pi\|\leq\|f\|\|\pi\|,

\end{equation}

Suppose $(f_i)$ is a net converges to $f$ in $W^*$ with respect to the weak* topology,

i.e.,

$$f_i(w)\to f(w) \forall w\in W.$$

For any $v\in V$, $\pi (v)\in W$, so

$$f_i(\pi (v))\to f(\pi (v)),$$

so $\pi ^*(f_i)$ weak*-converges to $\pi ^*(f)$.

For any $v\in V$, \begin{align*}

&(\pi^{**}\circ i)(v)(w’)=\pi^{**}(i(v))(w’)=(i(v)\circ \pi^*)(w’)\\

=&i(v)(\pi^*(w’))=i(v)(w’\circ \pi)=(w’\circ \pi)(v)\\

=&w’(\pi(v))=i(\pi(v))(w’)=(i\circ \pi)(v)(w’)~(w’\in W^*),

\end{align*}

hence, $\pi ^{**}\circ i=i\circ \pi $. __

A normed space $M$ is said to be W-space if it admits a predual $^*M$, that is, $^*M$ is a normed

space and $M$ is isomorphic to $(^*M)^*$ as a normed space. Now Let $\cC $ and $\cW $ be two categories as

follows:

- $\cC $: objects, a collection of normed spaces; arrows, all bouned linear maps

between them.
- $\cW $: objects, a collection of objects of form $V^*$ for some normed space $V$; arrows,

all bouned linear maps between them that are also weak*-continuous.

Clearly, $\cW $ is a subcategory of $\cC $. Let $\ad $ assign to each bounded linear map $\pi :A\to B$ between two

normed space the double adjoint $\pi ^{**}: \AA \to \BB $, and $\id $ assign each bounded and weak*-continuous

map $\pi :M\to N$ between two W-spaces itself, then $\ad $ and $\id $ are two functors between $\cC $ and $\cW $.

Denotes the set of all arrows in $\cC $ from object $A$ to $B$ by $\cC (A,B)$, we have

Lemma 2. for each object $A\in \cC $ and each obeject $M\in \cW $, there is a bijection of sets

\begin{align}

\label{n}

n=n_{A,M}:&\cW(\ad(A),M)\to\cC(A,\id(M))\notag\\

&(\xymatrix{\ad(A)\ar[r]^\pi&M})\mapsto(\xymatrix{A\ar[r]^{\pi\circ i}&\id(M)}),

\end{align}

which is natural between two hom-functors

$$\hom _\cW (\ad (*),*), \hom _\cC (*,\id (*)): \cC ^{op}\times \cW \to \mathbf {Set}$$

as indicated by the following commutative diagram for each $\xymatrix {B\ar [r]^f & A}$ and $\xymatrix {M\ar [r]^g & N}$, \begin{align}

\label{naturalhom}\xymatrix{

\cW(\ad A,M)\ar[d]_{((\ad f)^*,g_*)}\ar[r]^{n_{A,M}}&\cC(A,\id M)\ar[d]^{(f^*,(\id g)_*)}\\

\cW(\ad B,N)\ar[r]^{n_{B,N}}&\cC(B,\id N),

}

\end{align}

where $f^*$ is the operation turning an arrow $\pi :A\to B$ in $\cC $ into $f\circ \pi $, adn $g_*$ turning $\pi :M\to N$ in $\cW $ into $\pi \circ g$. In one

sentence, the triple $\langle \ad ,\id , n\rangle $ is an ajunction from $\cC $ to $\cW $.

Proof. By lemma 1, the following map is well-defined: \begin{align}

\label{m}

m=m_{A,M}:&\cC(A,\id M)\to\cW(\ad A,M)\notag\\

&(\xymatrix{A\ar[r]^{\pi}&M})\mapsto(\xymatrix{\AA\ar[r]^{(\pi^*\circ i)^*}&M})

\end{align}

For each $A\in \cC $ and each $M\in \cW $, we prove that $m_{A,M}$ is the inverse of $n_{A,M}$.

Pick an arrow $\xymatrix {\AA \ar [r]^{\pi ”} & M}$ in $\cW $. For any $a\in A$, \begin{align*}

&((\pi’’\circ i)^*\circ i)^*(i(a))(’m)=i(a)(((\pi’’\circ i)^*(i(’m)))=i(a)(i(’m)\circ (\pi’’\circ i))\\

=&(i(’m)\circ (\pi’’\circ i))(a)=i(’m)(\pi’’(i(a)))=\pi’’(i(a))(’m)~(’m\in^*M),

\end{align*}

i.e.,

$$((\pi ”\circ i)^*\circ i)^*(i(a))=\pi ”(i(a)).$$

By Goldstine theorem (proposition V.4.1, [1]), $i(A)$ is weak* dense in $A^{**}$.

$$(\pi ”\circ i)^*\circ i)^* = \pi ”$$

since the maps of both sides are weak*-continuous.

Pick an arrow $\xymatrix {A\ar [r]^{\pi } & M}$ in $\cC $. For any $a\in A$, \begin{align*}

&((\pi^*\circ i)^*\circ i)(a)=(\pi^*\circ i)^*(i(a))=i(a)\circ (\pi^*\circ i)\\

=&(\pi^*\circ i)(a)=\pi^*(i(a))=i(a)\circ \pi=\pi(a).

\end{align*}

So,

$$(\pi ^*\circ i)^*\circ i=\pi .$$

Suppose $\pi \in \cW (\ad A, M)$. For each $\xymatrix {B\ar [r]^f & A}$ and $\xymatrix {M\ar [r]^g & N}$, \begin{align*}

\id(g)\circ n_{A,M}(\pi)\circ f=g\circ \pi\circ i\circ f,\\

g\circ n_{B,N}(\pi)\circ \ad(f)=g\circ \pi\circ f^{**}\circ i.

\end{align*}

By \eqref{ad imbedding}, $\id (g)\circ n_{A,M}(\pi )\circ f=g\circ n_{B,N}(\pi )\circ \ad (f)$, hence the diagram \eqref{natural hom} is

commutative. __

By theorem IV.1.1, [2],

$$\eta _A:=n_{A,A}(\operatorname {1}_{\ad A})=i:A\to A^{**}$$

together with $\ad A$ is a universal arrow from $A\in \cC $ to the functor $\id :\cW \to \cC $ in the sense for any $M\in \cW $ and

any $\xymatrix { A\ar [r]^\pi & \id M}$ in $\cC $ there is an arrow $\xymatrix @C=1.75cm{\ad M\ar [r]^{\pi ”=n_{A,M}^{-1}(\pi )} & N}$ in $\cW $ filled in the commutative digram

$$\xymatrix { A\ar [r]^i\ar [rd]_\pi & \id \ad A\ar @{–>}[d]^{\id \pi ”} & \ad A\ar @{–>}[d]^{\pi ”=n_{A,M}^{-1}(\pi )} \\ & \id M. & M } $$

In other words,

Theorem 3. Each linear map $\pi :A\to M$ from a normed space $A$ and a W-space $M$ can be

extended to a unique linear map $(\pi ^*\circ i)^*:A^{**}\to M$ that are both bouded and weak*-continuous.

### References

[1] John B Conway. A course in functional analysis, volume 96. Springer

Science; Business Media, 2013.

[2] Saunders MacLane. Categories for the Working Mathematician.

Springer-Verlag, New York, 1971. Graduate Texts in Mathematics, Vol. 5.