For a <Hilbert space> $H$, the linear span of the matrix-coefficient functionals $T\mapsto\langle Tx,y\rangle$ is a predual of $\mathcal B(H)$. The unit ball is compact in the resulting <weak operator topology> by the <Tychonoff theorem>, and the <compact norming dual-pair criterion> identifies $\mathcal B(H)$ with the dual of that span.

Operator predual from matrix coefficients

The useful forms of the <Hahn-Banach separation theorem> in a real locally convex space are:

* if $C$ is nonempty open convex and $x\notin C$, there are a continuous linear functional $f$ and $\alpha\in\mathbb R$ with $f(c)<\alpha\leq f(x)$ for every $c\in C$;
* if $C$ is closed convex and $x\notin C$, there are $f$ and $\alpha$ with $\sup_Cf<\alpha<f(x)$;
* if $K$ is compact convex, $C$ is closed convex, and $K\cap C=\varnothing$, there are $f$ and $\alpha<\beta$ with $\sup_Cf<\alpha<\beta<\inf_Kf$, after changing the sign of $f$ if needed.

The <Banach-Alaoglu theorem> says that the closed unit ball of $E^*$ is compact in $\sigma(E^*,E)$. <Goldstine theorem> says that the canonical image of the closed unit ball of a normed space $E$ is weak-star dense in the closed unit ball of $E^{**}$.

Give $Z$ its norm inherited from $X^*$ and define
$$
J:X\longrightarrow Z^*,\qquad Jx(f)=f(x).
$$
It is linear and contractive, and it is injective because $Z$ separates points. Goldstine's theorem followed by restriction from $X^*$ to $Z$ shows that $J(B_X)$ is weak-star dense in $B_{Z^*}$: a functional on $Z$ first extends norm-preservingly to $X^*$, and elements of $B_X$ approximate that extension on every finite subset of $Z$.

The topology induced by $\sigma(Z^*,Z)$ on $J(B_X)$ is exactly the given topology $\sigma(X,Z)$. By hypothesis $B_X$ is compact, so $J(B_X)$ is weak-star compact and therefore closed in the Hausdorff space $B_{Z^*}$. Density now gives
$$
J(B_X)=B_{Z^*}.
$$
Thus $J$ is surjective and maps closed unit ball onto closed unit ball, so it is an isometric isomorphism. Hence $X$ is a dual space. This is the <compact norming dual-pair criterion>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 106 / 4 / a / Solution

Let $Z\subseteq X^*$ separate points of a Banach space $X$. If the closed unit ball of $X$ is compact for $\sigma(X,Z)$ and $Z$ has the norm inherited from $X^*$, then the evaluation map $X\to Z^*$ is an isometric isomorphism. <Goldstine theorem> makes its unit-ball image weak-star dense, while the assumed compactness makes that image closed.

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 Incoming links: Compact norming dual-pair criterion