Theorem: Alaoglu-Bourbaki Theorem

Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $\map \sigma {E, F}$ be the weak topology on $E$ induced by $\innerprod E F$.
Let $V \subseteq E$ be a neighborhood of ${\mathbf 0}_X$ in $\map \sigma {E, F}$.
Let $V^\circ$ be the absolute polar of $V$.
Let $\map \sigma {F, E}$ be the weak-$\ast$ topology on $F$ induced by $\innerprod E F$.
Then $V^\circ$ is $\map \sigma {F, E}$-compact.

Proof:
Let:
:$B = \set {z \in \C : \cmod z \le 1}$
By the Heine-Borel Theorem: Normed Vector Space, we have that $B$ is compact. 
Hence by Tychonoff's Theorem, we have that:
:$\ds X = \prod_{x \mathop \in V} B$, equipped with the product topology $\tau_X$, is compact.
For each $f \in V^\circ$, define $\pi_f : E \to \GF$ by:
:$\map {\pi_f} x = \innerprod x f$ for each $x \in E$. 
Define a mapping $\phi : V^\circ \to \GF^V$ by:
:$\map \phi f = \pi_f \restriction_V$ for each $f \in V^\circ$.
For $f \in V^\circ$, we have:
:$\cmod {\map {\pi_f} x} = \cmod {\innerprod x f} \le 1$ for each $x \in V$.
Hence we have:
:$\map \phi f \in X$
for each $f \in V^\circ$.
Hence $\phi$ can be understood as a mapping $\phi : \struct {V^\circ, \map \sigma {F, E} } \to \struct {X, \tau_X}$. 
We show that $\phi$ is a homeomorphism onto $\phi \sqbrk {V^\circ} = \set {\pi_f \restriction_V : f \in V^\circ}$.
We first show that $\phi$ is an injection.
Suppose that:
:$\map \phi {f_1} = \map \phi {f_2}$ for each $f_1, f_2 \in V^\circ$.
That is:
:$\pi_{f_1} \restriction_V = \pi_{f_2} \restriction_V$
Let $x \in E$.
Let $U \subseteq V$ be an open neighborhood of ${\mathbf 0}_X$.
From Topological Vector Space as Union of Dilations of Open Neighborhood of Origin, there exists $n \in \N$ and $y \in U$ such that $x = n y$.
We then have:
{{begin-eqn}}
{{eqn	| l = \map {\pi_{f_1} } x
		| r = \map {\pi_{f_1} } {n y}
}}
{{eqn	| r = n \map {\pi_{f_1} } y
		| c = {{Defof|Linear Functional}}
}}
{{eqn	| r = n \map {\pi_{f_2} } y
		| c = $f_1 \restriction_V = f_2 \restriction_V$, $y \in V$
}}
{{eqn	| r = \map {\pi_{f_2} } {n y}
		| c = {{Defof|Linear Functional}}
}}
{{eqn	| r = \map {\pi_{f_2} } x
}}
{{end-eqn}}
That is, $\pi_{f_1} = \pi_{f_2}$.
Hence, we have:
:$\innerprod x {f_1} = \innerprod x {f_2}$ for each $x \in E$.
Since $\innerprod \cdot \cdot$ is bilinear, we have:
:$\innerprod x {f_1 - f_2} = 0$ for each $x \in X$.
By the definition of a dual system we have $f_1 - f_2 = {\mathbf 0}_F$ and hence $f_1 = f_2$. 
So $\phi$ is an injection.
We show that $\phi$ is continuous.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {f_\lambda}_{\lambda \mathop \in \Lambda}$ be a net in $V^\circ$ converging to $f \in V^\circ$ in $\struct {V^\circ, \map \sigma {F, E} }$.
By Characterization of Convergence of Net in Weak-* Topology Induced by Dual System, we have:
:$\innerprod x {f_\lambda} \to \innerprod x f$ for each $x \in E$.
In particular:
:$\map {\pi_{f_\lambda} } x = \map {\pi_f} x$ for each $x \in V$.
From Characterization of Convergence of Net in Product Topology, we have:
:$\pi_{f_\lambda} \restriction_V \to \pi_f \restriction_V$ in $\struct {X, \tau_X}$.
Hence from Characterization of Continuity in terms of Nets, $\phi$ is continuous.
We now show that $\phi^{-1}$ is continuous.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {\pi_{f_\lambda} \restriction_V}_{\lambda \mathop \in \Lambda}$ be a net in $\phi \sqbrk {V^\circ}$ converging to $\pi_f \restriction_V \in \phi \sqbrk {V^\circ}$ in $\struct {X, \tau_X}$.
From Characterization of Convergence of Net in Product Topology:
:$\map {\pi_{f_\lambda} } x \to \map {\pi_f} x$ for each $x \in V$. 
That is:
:$\innerprod x {f_\lambda} \to \innerprod x f$ for each $x \in V$.
Now let $x \in E$ be general.
From Topological Vector Space as Union of Dilations of Open Neighborhood of Origin, there exists $n \in \N$ and $y \in U \subseteq V$ such that $x = n y$.
Then we have:
:$\innerprod y {f_\lambda} \to \innerprod y f$ 
From Scalar Multiple of Convergent Net in Topological Vector Space is Convergent, we have:
:$n \innerprod y {f_\lambda} \to n \innerprod y f$
Since $\innerprod \cdot \cdot$ is bilinear, we have:
:$\innerprod {n y} {f_\lambda} \to \innerprod {n y} f$
Hence:
:$\innerprod x {f_\lambda} \to \innerprod x f$ for each $x \in E$.
Hence by Characterization of Convergence of Net in Weak-* Topology Induced by Dual System, we have:
:$f_\lambda \to f$ in $\struct {V^\circ, \map \sigma {F, E} }$
Hence by Characterization of Continuity in terms of Nets, $\phi^{-1}$ is continuous.
Hence $\phi$ is a homeomorphism.
Hence from Closed Subspace of Compact Space is Compact, it is enough to show that $\phi \sqbrk {V^\circ}$ is closed in $\struct {X, \tau_X}$.
We use Characterization of Closedness in terms of Nets.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {\pi_{f_\lambda} \restriction_V}_{\lambda \mathop \in \Lambda}$ be a net in $\phi \sqbrk {V^\circ}$ converging to $\pi \in X$.
We aim to show that there exists $f \in V^\circ$ such that $\pi = \pi_f \restriction_V$.
We first extend $\pi$ to a linear functional on $E$.
Let $x \in E$. 
By Topological Vector Space as Union of Dilations of Open Neighborhood of Origin, for each $x \in E$ there exists $n \in \N$ such that $x \in n V$.
We have:
{{begin-eqn}}
{{eqn	| l = \innerprod x {f_\lambda} 
		| r = n \innerprod {\frac x n} {f_\lambda}
}}
{{eqn	| o = \to
		| r = n \map \pi {\frac x n}
}}
{{end-eqn}}
Hence the limit:
:$\ds \lim_\lambda \innerprod x {f_\lambda}$ exists.
Define $\pi_\ast : E \to \GF$ by:
:$\ds \map {\pi_\ast} x = \lim_\lambda \innerprod x {f_\lambda}$
This extends the previously discovered $\pi$. 
We show that $\pi_\ast$ is linear.
Let $x, y \in E$ and $t \in \GF$.
Since $\innerprod \cdot \cdot$ is bilinear, we have:
:$\innerprod x {f_\lambda} + t \innerprod y {f_\lambda} = \innerprod {x + t y} {f_\lambda}$ 
Taking the limit in $\lambda$, we obtain:
:$\map {\pi_\ast} x + t \map {\pi_\ast} y = \map {\pi_\ast} {x + t y}$
Hence $\pi_\ast$ is linear.
We now show that $\pi_\ast$ is continuous mapping $\struct {E, \map \sigma {E, F} } \to \GF$.
Let $U \subseteq V$ be an open neighborhood of ${\mathbf 0}_X$.
We have:
:$\cmod {\innerprod x {f_\lambda} } \le 1$ for each $\lambda \in \Lambda$ and $x \in V$
since $f_\lambda \in V^\circ$ for each $\lambda \in \Lambda$.
Taking the limit in $\lambda$, we have:
:$\cmod {\map {\pi_\ast} x} \le 1$ for each $x \in V$.
Hence $\pi_\ast$ is bounded on an open neighborhood $U$ of ${\mathbf 0}_X$.
Hence from Characterization of Continuous Linear Functionals on Topological Vector Space, $\pi_\ast$ is a continuous linear functional.
From Continuity of Linear Functionals in Weak Topology Induced by Pair of Vector Spaces with Bilinear Mapping, there exists $f \in F$ such that $\pi_\ast = \pi_f$. 
Hence we have $\pi = \pi_f \restriction_V \in \phi \sqbrk {V^\circ}$.
Hence from Characterization of Closedness in terms of Nets, $\phi \sqbrk {V^\circ}$ is closed in $\struct {X, \tau_X}$.
Hence from Closed Subspace of Compact Space is Compact and the fact that $V^\circ$ is homeomorphic to $\phi \sqbrk {V^\circ}$, we have that $V^\circ$ is $\map \sigma {F, E}$-compact.
{{qed}}
{{Namedfor|Leonidas Alaoglu|name2 = Nicolas Bourbaki|cat = Alaoglu|cat2 = Bourbaki}}