Countable compactness closes in the bidual

If $A=\mathrm{\varnothing }$, it is norm bounded and $J_X(A)=\mathrm{\varnothing }$ has empty weak-star closure, so both conclusions hold. Hence assume $A\mathrm{\ne }\mathrm{\varnothing }$.

Put $K=B_{X^{\ast }}$ with its weak-star topology and define $\mathrm{\Phi }:X\to {\mathbb{K}}^K$ by $\mathrm{\Phi }(x)(u)=u(x)$. Each $\mathrm{\Phi }(x)$ is continuous on $K$ by [F2], so $\mathrm{\Phi }(X)\subseteq C(K)$. The space $K$ is compact by [F3] and Hausdorff because distinct members of $X^{\ast }$ differ at some $x\in X$, whose evaluation separates them in the Hausdorff scalar field. By [F4], $\mathrm{\parallel }\mathrm{\Phi }(x){\mathrm{\parallel }}_{\mathrm{\infty }}=\mathrm{\parallel }x\mathrm{\parallel }$ and $\mathrm{\Phi }$ is injective. Since every $u\in X^{\ast }$ is zero or a scalar multiple of a member of $K$, [F2] shows that the pointwise topology on $\mathrm{\Phi }(X)$ is exactly the weak topology transported from $X$.

For each $u\in X^{\ast }$ the scalar set $u(A)$ is bounded. Otherwise every set $E_n=\{a\in A:\mathrm{\mid }u(a)\mathrm{\mid }>n\}$, $n\ge 1$, is nonempty. Apply DC to finite valid histories, starting with the empty history and extending the $n$th stage by an element of $E_{n+1}$; this gives $(a_n)$ with $\mathrm{\mid }u(a_n)\mathrm{\mid }>n+1$. Let $a$ be a weak cluster. The weak neighborhood $\mathrm{\mid }u(x-a)\mathrm{\mid }<1$ contains arbitrarily late $a_n$, while [F12] lets us take such an $n$ with $n+1>\mathrm{\mid }u(a)\mathrm{\mid }+1$. Then $\mathrm{\mid }u(a_n)\mathrm{\mid }\le \mathrm{\mid }u(a)\mathrm{\mid }+1<n+1$, a contradiction.

We shall repeatedly use this choice-free consequence of compactness. If $(z_n)$ is a sequence in a compact space, then the closed sets $C_N=\overline{\{z_n:n\ge N\}}$ are nonempty, nested, and have the finite-intersection property. By [F8] some $z$ lies in every $C_N$; by the closure characterization, every neighborhood of $z$ contains terms with arbitrarily large indices. Thus $z$ is a cluster point of the sequence.

Every sequence in $M:=\mathrm{\Phi }(A)$ has a pointwise cluster in $\mathrm{\Phi }(X)$. Indeed, injectivity gives its unique lift $(a_n)$ in $A$; [F1] gives a weak cluster $a\in X$, and the topology identification in step 1.2 makes $\mathrm{\Phi }(a)$ a pointwise cluster.

The dual $X^{\ast }=\mathcal{B}(X,\mathbb{K})$ is Banach: the real and complex scalar fields are Banach and [F5] applies to the bounded-operator space. The family $\{J_X(a):a\in A\}\subseteq \mathcal{B}(X^{\ast },\mathbb{K})$ is pointwise bounded by step 1.3, so UBP under the assumed DC gives $R:={\sup }_{a\in A}\mathrm{\parallel }{J}_X(a)\mathrm{\parallel }<\mathrm{\infty }$.

The HB isometry [F4] gives $\mathrm{\parallel }a\mathrm{\parallel }=\mathrm{\parallel }{J}_X(a)\mathrm{\parallel }\le R$ for every $a\in A$. Thus $A$, and equivalently $M$ in the supremum norm, is norm bounded.

Let $B={\bar{M}}^p$ be the closure in the full product ${\mathbb{K}}^K$. For each $t\in K$, step 3.1 gives $\mathrm{\mid }h(t)\mathrm{\mid }\le R$ for $h\in M$. The scalar disk $D_R=\{z:\mathrm{\mid }z\mathrm{\mid }\le R\}$ is compact Hausdorff by [F9], including $R=0$; hence $D_R^K$ is compact by [F3]. It is closed in ${\mathbb{K}}^K$, so $B\subseteq D_R^K$, and $B$ is closed in that product. Therefore $B$ is compact by [F8].

We prove $B\subseteq C(K)$. Suppose instead that $g\in B$ is discontinuous at $y\in K$. Then for some $\epsilon >0$, every neighborhood of $y$ meets $Z=\{z\in K:\mathrm{\mid }g(z)-g(y)\mathrm{\mid }\ge \epsilon \}$. This is exactly the negation of continuity at $y$ into the metric scalar field, written with one failed positive tolerance. Notice $y\mathrm{\notin }Z$.

Put $U_0=K$ and ${\eta }_n=\epsilon \mathrm{/}(n+1)$ for $n\ge 1$. DC on finite valid histories constructs $h_n\in M$, open neighborhoods $U_n$ of $y$, and $x_n\in K$ such that $\mathrm{\mid }{h}_n(y)-g(y)\mathrm{\mid }<{\eta }_n\mathrm{/}2$ and $\mathrm{\mid }{h}_n(x_m)-g(x_m)\mathrm{\mid }<{\eta }_n\mathrm{/}2$ for $m<n$, while $y\in U_n\subseteq \overline{U_n}\subseteq U_{n-1}\cap \{z:\mathrm{\mid }{h}_n(z)-h_n(y)\mathrm{\mid }<{\eta }_n\}$ and $x_n\in U_n\cap Z$. At stage $n$, the approximation is possible because $g$ is in the pointwise closure of $M$; the set to be shrunk is an open neighborhood of $y$ because $h_n$ is continuous; [F7] supplies $U_n$; and step 5.1 makes $U_n\cap Z$ nonempty. Thus the relation extending a finite valid history is entire, exactly the hypothesis of DC.

By step 2.1, $(h_n)$ has a pointwise cluster $h=\mathrm{\Phi }(a)\in \mathrm{\Phi }(X)$. By step 1.4, $(x_n)$ has a cluster $x\in K$. The nesting in step 6.1 gives $x_m\in U_n$ whenever $m\ge n$, so the closure characterization gives $x\in \overline{U_n}$ for every $n$.

Hence $\mathrm{\mid }{h}_n(x)-h_n(y)\mathrm{\mid }<{\eta }_n$ and $\mathrm{\mid }{h}_n(y)-g(y)\mathrm{\mid }<{\eta }_n\mathrm{/}2$. Since ${\eta }_n\to 0$ by [F12], $h_n(x)\to g(y)$. But $h$ is a pointwise cluster of $(h_n)$, so $h(x)$ is a cluster of the convergent scalar sequence $(h_n(x))$; scalar Hausdorffness forces $h(x)=g(y)$.

For fixed $m$, step 6.1 gives $h_n(x_m)\to g(x_m)$ as $n\to \mathrm{\infty }$. The same cluster-and-uniqueness argument gives $h(x_m)=g(x_m)$. Since $x_m\in Z$, we therefore have $\mathrm{\mid }h(x_m)-h(x)\mathrm{\mid }=\mathrm{\mid }g(x_m)-g(y)\mathrm{\mid }\ge \epsilon$ for every $m$.

The function $h=\mathrm{\Phi }(a)$ is continuous on $K$. Thus $\{z:\mathrm{\mid }h(z)-h(x)\mathrm{\mid }<\epsilon \mathrm{/}2\}$ is a neighborhood of the cluster $x$ and must contain arbitrarily late $x_m$, contradicting step 9.1. Therefore every $g\in B$ is continuous and $B\subseteq C(K)$.

Fix $g\in B$. For positive integers $r,s$ and $h\in M$, define $V_h^{r,s}=\{(t_1,\dots ,t_r)\in K^r:\mathrm{\mid }h(t_j)-g(t_j)\mathrm{\mid }<1\mathrm{/}s\text{ for }1\le j\le r\}$. These sets are open because $g,h\in C(K)$ by step 10.1, and they cover $K^r$ because $g$ lies in the pointwise closure of $M$. The finite power $K^r$ is compact by [F3], so some nonempty finite list of members of $M$ has the corresponding $V_h^{r,s}$ covering $K^r$.

Pair the positive integer indices $(r,s)$ using [F12]. Apply DC to finite histories of choices of the finite subcovers from step 11.1; the extension relation is entire. Thus obtain one finite list $F_{r,s}\subseteq M$ for every pair. Their union $M_0$ is at most countable: retain the finite-list order, pad each nonempty list by its first term, and enumerate the pairs of natural indices using [F12]. No member of an uncountable family has been selected.

The point $g$ lies in the pointwise closure of $M_0$: for finitely many points and tolerance $\delta >0$, repeat points if needed to form a positive-length tuple and choose $s$ with $1\mathrm{/}s<\delta$; a member of $F_{r,s}$ gives all the inequalities. Let $P={\overline{M_0}}^p$. Then $g\in P\subseteq B$; $P$ is closed in compact $B$, hence compact by [F8], and it is separable because the at-most-countable $M_0$ is dense in it.

We record the compact-cluster argument of Haase's Lemma E.1. Let $(q_n)\subseteq K$, $q\in K$, and let $(v_j)$ be pointwise dense in $P$. If $v_j(q_n)\to v_j(q)$ for every $j$, then $v(q_n)\to v(q)$ for every $v\in P$. Indeed, let $C$ be the intersection of the closures of all tails of $(q_n)$; it is nonempty by step 1.4. For $z\in C$, continuity and the assumed scalar convergence give $v_j(z)=v_j(q)$ for every $j$. Pointwise density then gives $v(z)=v(q)$ for every $v\in P$: otherwise the two-coordinate neighborhood of $v$ at $z,q$ with radius $\mathrm{\mid }v(z)-v(q)\mathrm{\mid }\mathrm{/}3$ would contain no $v_j$. If $v(q_n)$ failed to converge to $v(q)$, least-index recursion would give a subsequence staying some fixed positive distance away. Its closed tail closures have a common point $z\in C$ by [F8], while continuity of $v$ at $z$ contradicts both $v(z)=v(q)$ and that fixed separation.

The compact separable pointwise space $P$ is nonempty because it contains $g$. Enumerate a nonempty pointwise-dense subset as $(v_j)$ using [F12]. On the countable product of the bounded scalar metrics $\rho$ use the standard weighted product metric $D$ from [F10], and pull it back along $q\mapsto (v_j(q){)}_j$ to a continuous pseudometric $d$ on $K$.

For each positive integer $n$, the open $d$-balls of radius $1\mathrm{/}n$ cover $K$, so compactness gives a nonempty finite list of centers. DC, applied to finite histories of such lists, chooses one list for each $n$. Pad every list by its first center and use the countable pairing in [F12] to enumerate the union as $(q_m)$. For each $q\in K$ and each $n$, take the first center in the $n$th list whose ball contains $q$; the resulting sequence $q_{m_n}$ satisfies $d(q_{m_n},q)<1\mathrm{/}n$, hence $v_j(q_{m_n})\to v_j(q)$ for every $j$.

The evaluations at $(q_m)$ separate $P$. If $v,w\in P$ agree at every $q_m$, then for arbitrary $q\in K$ use the sequence from step 15.1. Step 14.1 gives $v(q_{m_n})\to v(q)$ and $w(q_{m_n})\to w(q)$; equality term by term and scalar Hausdorffness give $v(q)=w(q)$. Thus $v=w$.

The evaluation map $E:P\to {\mathbb{K}}^{\mathbb{N}}$, $E(v)=(v(q_m){)}_m$, is continuous for the pointwise and product topologies by [F2] and injective by step 16.1. Its corestriction to $E(P)$ is a continuous bijection from compact $P$ to a metric, hence Hausdorff, space. By [F11] it is a homeomorphism. Pulling back the restriction of the weighted metric built from $\rho$ therefore metrizes the pointwise topology of $P$.

Enumerate $M_0$ as $(w_k)$, with repetitions allowed. Since it is dense in the metric space $P$, for each $n\ge 1$ there is a $k$ with $d_P(w_k,g)<1\mathrm{/}n$; take the least such $k$. This defines, without choice, a sequence $(w_{k_n})$ in $M_0$ converging to $g$ pointwise.

Lift this sequence uniquely to $(a_n)$ in $A$. By [F1] it has a weak cluster $a\in X$, so $\mathrm{\Phi }(a)$ is a pointwise cluster of $(w_{k_n})$ by step 1.2. Every scalar coordinate of that sequence converges to the corresponding coordinate of $g$ by step 18.1; uniqueness of scalar cluster points gives $g=\mathrm{\Phi }(a)$. Since $g\in B$ was arbitrary, $B\subseteq \mathrm{\Phi }(X)$.

Let $x^{\ast \ast }\in {\overline{J_X(A)}}^{w^{\ast }}$ and restrict it to $K$: $g(t)=x^{\ast \ast }(t)$. Every finite pointwise neighborhood of $g$ on $K$ is the restriction of a basic weak-star neighborhood of $x^{\ast \ast }$ in $X^{\ast \ast }$, so it meets $J_X(A)$ by [F2]. Hence $g\in B$, and step 19.1 gives $a\in X$ with $g(t)=t(a)$ for every $t\in K$.

For arbitrary $u\in X^{\ast }$, the equality is immediate if $u=0$; otherwise $u\mathrm{/}\mathrm{\parallel }u\mathrm{\parallel }\in K$, and linearity gives $x^{\ast \ast }(u)=\mathrm{\parallel }u\mathrm{\parallel }g(u\mathrm{/}\mathrm{\parallel }u\mathrm{\parallel })=u(a)=J_X(a)(u)$. Thus $x^{\ast \ast }=J_X(a)\in J_X(X)$. Together with step 3.1 and the empty case of step 1.1 this proves both assertions. The ultrafilter lemma is spent in steps 1.2, 4.1 and 11.1 through compactness; HB is spent only in the isometry in steps 1.2 and 3.1; DC is spent in UBP at step 2.2 and in the explicit finite-history constructions of steps 1.3, 6.1, 12.1 and 15.1.