Relative independence of the separable quotient problem

Proof. The vectors are linearly independent: a vector f_n in the preceding span would vanish on its norming set D_{n-1} and hence be zero. For v=\sum_{i=1}^N c_i f_i and m\le N, the values of v on D_m equal those of its prefix. Thus

The prefix projections extend to G and converge to the identity there, because they do so on the dense subspace of finite sums. Taking differences of two consecutive prefixes gives (2).

Write T=R_G:X\to Z. For h\in Z^*, set

If d\in D_m, then Td=\sum_{i=1}^m f_i(d)f_i^\#, so (T^*h)(d)=v_m(d). Consequently \|T^*h\|\ge\|v_m\|/2. For every finite linear combination u of the coefficient functionals, we have h(u)=u(v_m) for all sufficiently large m. Since these combinations are dense in Z,

This argument does not require the vectors v_m to converge.

We recall why an adjoint lower bound gives surjectivity. If \|T^*z^*\|\ge c\|z^*\| for every z^*\in Z^*, Hahn–Banach separation of the balanced convex set \overline{T(B_X)} gives cB_Z\subset\overline{T(B_X)}. For a residual z\ne0, choose x\in X with \|x\|\le\|z\|/c and \|z-Tx\|\le\|z\|/2. Repeating this step makes the residuals decrease geometrically and the chosen vectors form an absolutely convergent series in X. Its sum maps to the original z and has norm at most 2\|z\|/c. Apply this with c=1/2.□

Proof. Suppose that T:X\to Y is such a surjection. Choose a closed separable infinite-dimensional subspace U\subset Y^*. The open mapping theorem makes T^* bounded below, so F=T^*U is a closed separable infinite-dimensional subspace of X^*. Under the isomorphism T^*|_U:U\to F, evaluation at x becomes evaluation at Tx. The evaluation map Y\to U^* is bounded and Y is separable; hence R_F(X) is norm separable.

Conversely, assume the second assertion and choose x_j\in X so that \{R_Fx_j:j\ge1\} is norm dense in R_F(X). We construct f_n\in F of norm one. At stage n, require f_n to annihilate x_1,\ldots,x_n and all finite sets D_m already chosen, m<n. These conditions define a finite-codimensional subspace of F, so a unit vector satisfying them exists. After choosing f_n, choose D_n\subset B_X that is 1/2-norming for \operatorname{span}\{f_1,\ldots,f_n\}. Such a finite set exists by compactness of the finite-dimensional unit sphere and the definition of the dual norm.

Use the spaces G and Z from Lemma 2.1. For each j,

Restriction from F^* to G^* is contractive. Density of the evaluations at the x_j, and closedness of Z, therefore give R_G(X)\subset Z. Lemma 2.1 makes R_G:X\to Z onto. The coefficient functionals are linearly independent, so Z is separable and infinite-dimensional. The space G also supplies the separable witness in the second assertion.□

Proof. Inductively choose unit vectors y_n\in M and finite 1/2-norming sets in B_{Y^*} for their prefix spans. Require each new vector to annihilate all earlier norming sets and to satisfy \|Ty_n\|<\varepsilon 2^{-n}/4. The annihilation conditions leave an infinite-dimensional closed subspace; strict singularity says that T is not bounded below there, so the choice is possible. The prefix calculation in Lemma 2.1, with vectors and norming functionals interchanged, shows that (y_n) is basic and |a_n|\le4\|\sum_i a_i y_i\|. Therefore

Continuity gives the bound on N=\overline{\operatorname{span}}\{y_n:n\ge1\}. For finitely many operators, apply the result successively inside the subspace already obtained. Restrictions of strictly singular operators remain strictly singular, and previous bounds persist on smaller subspaces.□

Proof. Every set of cardinality less than \kappa is null. If \operatorname{cf}(\kappa)<\kappa, a cofinal family of fewer than \kappa initial segments would therefore express \kappa as a null set.

Fix an injection b:\kappa\to\{0,1\}^{\N}. If A were a positive atom, then for each successive bit one of the two cells would have full measure in the preceding cell of A. This gives decreasing subsets A_n\subset A of measure \mu(A) on which the first n bits are fixed. Their intersection has at most one point, whereas countable additivity gives it measure \mu(A)>0. This contradiction proves atomlessness.

We include the elementary divisibility argument. Every positive set has a positive subset of measure at most any given \varepsilon>0: repeatedly split a positive set into two positive sets and retain the smaller part. A maximal disjoint family of such small positive subsets of a measurable set A is countable, since the measure is finite, and covers A modulo a null set. For 0\le t\le\mu(A), successive unions of these pieces either attain measure t, converge upwards to t, or first cross t. In the last case the union before that crossing has measure in (t-\varepsilon,t). Thus one can approximate t from below with error at most \varepsilon. Apply this construction repeatedly in the remainder of A, with errors tending to zero, and take the union. It has measure exactly t. Removing sets of measure 1/N one at a time gives the required partition.□

Proof. Every member of E^* is determined by its values on a fixed countable dense subset of E, so |D|\le\mathfrak c=\kappa. Suppose that A\subset D is norm dense and |A|<\kappa.

For n\ge1 put N_n=16^n. Inductively choose vectors (u_{n,j})_{j=1}^{N_n}\subset E such that

After choosing row n, choose a finite set D_n\subset B_X that 1/2-norms the span of all rows up to n. Require every subsequent row to annihilate D_n. At each step the finitely many annihilation conditions leave an infinite-dimensional closed subspace of E, so Dvoretzky’s Theorem gives (3) there.

For each n, Lemma 3.1 gives a map j_n:\kappa\to\{1,\ldots,N_n\} whose fibers have measure 1/N_n. For y\in A, (3) implies \|(y(u_{n,j}))_{j=1}^{N_n}\|_2\le2, and hence

The sum of these expectations is finite. Monotone convergence shows that \sum_n|y(u_{n,j_n(\alpha)})|<\infty for almost every \alpha. Intersecting these conull sets over y\in A is valid because |A|<\kappa. Choose one \alpha in their intersection and set g_n=u_{n,j_n(\alpha)}.

The vectors g_n have norms between 1 and 2, and the sets D_n are actual half-normers of their prefix spans, annihilated by all later vectors. Lemma 2.1 therefore makes (g_n) basic, with coefficient functionals g_n^\#\in G^* of norm at most 4, where G=[g_n:n\ge1]. Set Z=[g_n^\#:n\ge1]\subset G^*. For y\in A, the series \sum_n y(g_n)g_n^\# converges in G^*, since its tail has norm at most 4\sum_{n>m}|y(g_n)|. On the dense span of the g_n its sum agrees with y|_G, so y|_G\in Z. Restriction is contractive and A is dense in D, whence R_G(B_X)\subset Z. Scaling gives R_G(X)\subset Z. Lemma 2.1 now says that R_G:X\to Z is onto, contradicting the hypothesis on X.□

Proof. Write D=R_E(B_X). There is a \delta-separated subset of D of cardinality \kappa for some \delta>0. Otherwise, choose a maximal 1/n-separated subset for each n\ge1. Each would have cardinality less than \kappa, and their union would be dense in D with cardinality less than \kappa, by regularity. This contradicts Lemma 3.2.

Enumerate D=(y_\beta)_{\beta<\kappa}. For \alpha<\kappa, let M_\alpha=[y_\beta:\beta<\alpha]\subset E^*. Its density is less than \kappa. Some y\in D has distance greater than \delta/4 from M_\alpha: if every point of the separated set had distance at most \delta/4, a dense subset of M_\alpha of cardinality less than \kappa would provide centers at distance less than 3\delta/8 from all those points. Two points cannot share such a center, a contradiction. Hahn–Banach gives h_\alpha\in S_{E^{**}} annihilating M_\alpha and satisfying |h_\alpha(y)|>\delta/4. Since y=R_Ex for some x\in B_X, the functional f_\alpha=h_\alpha R_E satisfies (4) with \varepsilon=\delta/4. For y=y_\beta, its nonzero evaluations occur only at indices \alpha\le\beta, a set of cardinality less than \kappa and therefore \mu-null.□

Proof. By the Odell–Rosenthal Theorem [18], each h_\alpha is the weak-star limit of a sequence from E. Choose one such sequence c_\alpha\in E^{\N}. Equip E^{\N} with the product of the norm topologies and K=B_{E^*} with the weak-star topology; both are Polish spaces, and K is compact. There is a Borel function

whose value at (c,y) is \lim_n y(c_n) if this limit exists and is zero otherwise. Indeed each map (c,y)\mapsto y(c_n) is continuous, and the Cauchy criterion makes its convergence set Borel. To check the joint continuity, use |y'(e')-y(e)|\le\|e'-e\|+|y'(e)-y(e)| for y,y'\in K.

The map \alpha\mapsto c_\alpha is measurable because its domain has the full power-set sigma algebra. Let \eta be its Borel pushforward probability on E^{\N}. The function

is Borel on K, by parameter integration of a bounded Borel function. Our chosen codes give \Phi(c_\alpha,y)=h_\alpha(y), so H(y)=0 for every y\in D.

For n\ge1 define a probability \lambda_n on all subsets of I^n by ordered iteration, with the first variable outermost:

Put \lambda_0 equal to the unit mass on the one empty tuple. Every section and every intermediate integrand is measurable. Induction and monotone convergence prove countable additivity of \lambda_n; approximation by simple functions proves the same ordered integral formula for bounded functions. This construction does not assert that changing the order of these power-set integrals preserves their value.

Fix n, rational coefficients a_1,\ldots,a_n, and a subset s\subset\{1,\ldots,n\}. For each tuple (\alpha_j)_{j\in s} choose x_s\in B_X such that

Choose x_s=0 if that sum is zero. The choice depends only on the indices in s. For i\notin s we claim

Fix any earlier prefix (\alpha_1,\ldots,\alpha_{i-1}). The map

is independent of \alpha_i and takes its values in D. It is measurable on the full power-set domain. Let \nu=Y_*\lambda_{n-i}, with the same interpretation when i=n. The ordered integral, starting at variable i, of the indicator of a nonzero evaluation in (8) is

The change of order in the last line is ordinary Fubini for a Borel function on the product of two Polish spaces with Borel probability measures. The final equality does not require D to be Borel: the Borel set \{H\ne0\} has empty inverse image under Y, so is \nu-null. Since the earlier prefix was arbitrary, integrating over those variables proves (8).

Combining (7) and (8) gives

almost surely. Also the coordinates are pairwise distinct almost surely: for r<t, the event \alpha_t=\alpha_r constrains variable t to a singleton after its preceding variables have been fixed. There are countably many rational coefficient tuples and finitely many subsets s. Consequently, for each n there is a set G_n\subset I^n of \lambda_n-measure one on which distinctness and all these inequalities hold.

We must choose one sequence whose prefixes all lie in these sets. Suppose a prefix p=(\alpha_1,\ldots,\alpha_m) has been chosen with all its nonempty prefixes in their respective G_j, and with

Here (G_n)_p denotes the section obtained by fixing the prefix p. The invariant holds for the empty prefix. For n=m+1, it says that (p,\alpha)\in G_{m+1} for almost every \alpha. For every n>m+1, the definition of the ordered measures gives

The integrand lies in [0,1], so it equals 1 for almost every \alpha. The intersection of these countably many conull sets is nonempty. Choose the next index there. This constructs the required infinite sequence without an infinite product measure.

Continuity extends (9) to real coefficients on every finite prefix. Singleton suppressions and \|f_{\alpha_j}\|\ge\varepsilon give linear independence. The initial-segment projections have norm at most 2 on the algebraic span and extend to its closure. Approximation by finite sums shows that these projections converge to the identity, so the selected sequence is a Schauder basis for its closed span. Every finite coordinate suppression also has norm at most 2. For a vector approximated within \delta by a finite sum, every finite coordinate sum outside that sum’s support has norm at most 2\delta. It follows that the basis expansion converges unconditionally.□

Proof. Suppose that X is a counterexample. A consequence of the Hagler–Johnson Theorem, in the precise form of [2, Proposition 16], says that an unconditional basic sequence in X^* gives a separable infinite-dimensional quotient of X. Thus X^* contains no such sequence. Choose an infinite-dimensional separable closed subspace E\subset X^*. It contains no isomorphic copy of \ell_1, whose usual basis would give an unconditional basic sequence in X^*.

Lemmas 3.2 and 3.3 give a seminormalized family f_\alpha=h_\alpha R_E with the pointwise null-support property. Lemma 3.4 then produces an unconditional basic sequence in X^*, the desired contradiction.□

Proof. The map

is a real-linear isometry with inverse f\mapsto\operatorname{Re}f. Indeed Ag is complex-linear, and phase rotation in B_X proves \|Ag\|=\|g\|. Apply Proposition 2.2 to X_{\R}, and transport its witness under A to a real closed infinite-dimensional subspace E\subset (X^*)_{\R}. The complex-valued real-linear evaluations on E have norm-separable image: under A, evaluation at x is the difference of real evaluation at x and i times real evaluation at ix. Evaluation on iE also has norm-separable image.

Consider the real operator

First suppose that q is not strictly singular. There are a closed infinite-dimensional W\subset iE and c>0 such that \operatorname{dist}(w,E)\ge c\|w\| for w\in W. Since iW\subset E, the space H=W+iW is a complex subspace whose displayed sum is direct over \R, and

These estimates make H closed and its real coordinate projections bounded. The two coordinate projections combine the norm-separable evaluation images on W and iW into a separable set of real-linear maps H\to\C. The original evaluations are complex linear, with the same norm, so R_H(X) is norm separable in H^*. Proposition 2.2, over \C, gives the desired quotient.

Now suppose that q is strictly singular. Fix positive numbers \varepsilon_n with \sum_n\varepsilon_n<\infty. Choose f_n\in E of norm one inductively, requiring that all later vectors vanish on finite actual 1/2-norming sets D_m\subset B_X for the complex spans of the earlier vectors. Each such vanishing condition consists of two real linear equations, so their common kernel in E, say M, is closed and infinite dimensional. Thus iM is a closed infinite-dimensional subspace of iE. Strict singularity of q|_{iM} lets us choose f_n with \operatorname{dist}(if_n,E)<\varepsilon_n/2. Choose g_n\in E with \|g_n-if_n\|<\varepsilon_n. Lemma 2.1 shows that (f_n) is complex basic, with coefficient bound 4.

Put H=\overline{\operatorname{span}}_{\C}\{f_n:n\ge1\}. For finite sums, define a real-linear map by

The operator K=L-I extends to H, because

where K_N is the finite partial sum. In particular, K is a norm limit of finite-rank real-linear operators. Since E is closed and L maps finite sums into E, we have L(H)\subset E.

For x\in X, write e_x(h)=h(x) on H and also use e_x for evaluation on X^*. Work temporarily in the normed space of bounded real-linear maps from H to \C. The family \{e_x\circ L:x\in X\} is norm separable, by the evaluation property of E and boundedness of L. For x\in B_X, the functionals e_x\circ K_N form a bounded subset of a finite-dimensional space, and

Thus \{e_x\circ K:x\in B_X\} is totally bounded. The identity e_x=e_x\circ L-e_x\circ K proves norm separability of the evaluations on B_X, and their countably many dilates give the same conclusion on X. The complex-linear functionals on H embed isometrically in the space of real-linear maps just used. Hence R_H(X)\subset H^* is norm separable, and Proposition 2.2 applies again.□

Proof. Use the norm \|(x_1,x_2)\|=\max(\|x_1\|,\|x_2\|) on Y=X_1\oplus X_2, so Y^*=X_1^*\oplus_1X_2^*. If Y had such a quotient, Proposition 2.2 would provide a closed infinite-dimensional F\subset Y^* with norm-separable evaluation image. Let p_j:F\to X_j^* be its coordinate operators.

If p_1 is not strictly singular, choose a closed infinite-dimensional G\subset F on which it is bounded below. Otherwise, Lemma 2.3 gives a closed infinite-dimensional G\subset F with \|p_1|_G\|\le1/2, and then

Thus some p_j identifies G with a closed infinite-dimensional H\subset X_j^*. For x\in X_j, evaluation on H is obtained by evaluating the corresponding coordinate vector of Y on F, restricting to G, and composing with (p_j|_G)^{-1}. Its image is norm separable. Proposition 2.2 gives the forbidden quotient of X_j.□

Proof. Proposition 4.1 proves the positive implication from real to complex spaces. For the other implication, suppose X is a real counterexample. Give X_{\C}=X+iX the norm

Phase rotation proves absolute homogeneity over \C, and the triangle inequality follows from that of X. Moreover,

Thus X_{\C} is complete and its underlying real space is isomorphic to X\oplus_\infty X. A separable infinite-dimensional complex quotient would give a separable infinite-dimensional real quotient of this sum, contrary to Lemma 4.2.□

Proof. Put \eta_j=(2^{-10-j})^r and choose L_j=\lfloor\eta_jm_j^r\rfloor. By taking m_j large, arrange

It follows that

These bounds imply the first two inequalities in (11), since 2^{1+1/r}<4. Choose d_j before choosing m_j, so that (1/8)^{d_j}\le2^{-1/r}2^{-10-j}. At step j, the numbers B_j and d_j are already fixed. Because

the second inequality in (13) holds for all sufficiently large choices of m_j. Increase m_j further to make both integer sequences strictly increasing, to ensure 1/m_j<2^{-j-4}, and to make |\log\theta_j|/\log L_j<1/j. The last condition gives (12); its positive limit also gives L_j/m_j\to\infty.□

Proof. Construct the sets by induction on \beta. At zero use \{0\}, and at a successor append \beta to every closure of its predecessor. Suppose \beta is a nonzero limit. Enumerate its predecessors as (a_i), and choose an increasing cofinal sequence (\alpha_i)_{i\ge0} below \beta such that \alpha_i\ge a_h for h\le i. Let p_0=1. Recursively choose strictly increasing p_i>1, i\ge1, so large that F_{p_i}(\alpha_i) contains \alpha_{i-1} and a_0,\ldots,a_i. For p_i\le p<p_{i+1} set

At a switch, coherence at \alpha_i gives

so the sets remain nested. The enumerated predecessors are eventually included, and coherence follows directly from coherence at \alpha_i. The last assertion follows by applying (14) to the common coordinate.□

Proof. First note that each K_k is symmetric and stable under ordinal crops, and K is symmetric and stable under rectangle crops. For Type I, distribute the crop to the children and discard zeros; the remaining children still have the required succession. For Type II, intersect the existing crops with the new crop. This changes neither the paths nor their codes and only removes active weights. These observations prove stability by induction on the construction stage. A rectangle crop of a pure terminal either gives zero or leaves an ordinal crop within the same color.

Every coefficient of every member of K_k or K has absolute value at most one. At a Type I node, disjointness of children gives the stronger bound 1/m_j. At a Type II node, a given coordinate meets at most one piece of each path. Contributions from different paths have distinct active weights, so their absolute sum is at most \sum_j1/m_j<1. Induction first establishes these bounds in the pure families and then in the mixed family. Consequently

Crop stability proves contractivity of the asserted projections. Each coordinate functional therefore extends continuously to E. A countable sequence of finite approximations shows that every vector vanishes outside a countable set. Moreover, K remains norming on the completion, by finite approximation. Since its members use only finitely many coordinates, those coordinates determine every vector.

It remains to compare the pure and full norms. We claim that

where zero is allowed in the absolutely convex hull. It holds for terminals. At a mixed Type I node at most one child meets C_k. At a mixed Type II node at most one piece of each path meets C_k, and at most one child of that piece does so. By induction the restriction is an absolutely convex combination of pure functionals whose total coefficient mass is at most \sum_j1/m_j<1. This proves (18). The opposite norm inequality follows from K_k\subset K.□

Proof. Fix a pure family and show by induction that each member of K_k has coefficient \ell_r norm at most one. This holds at coordinates. A Type I piece of weight j has coefficient norm at most L_j^{1/r}/m_j=\theta_j, by the disjoint supports of its children. A cropped path with active weight set A_b therefore has coefficient norm at most

For a Type II expression, the triangle inequality and Hölder’s inequality give

We used q'>r and disjointness of the active weight sets. Now apply Hölder’s inequality to vectors on C_k and use Lemma 5.3. Applying (19) to finite tails proves convergence of every asserted infinite series.□

Proof. Each f\in K is a finite linear combination of coordinate functionals and has norm at most one on E. Thus K\subset B_{X_0} and the evaluation map is isometric.

For surjectivity fix g\in X_0^* and put a_\alpha=g(e_\alpha^*). We construct, by transfinite induction on \gamma\le\omega_1, a vector x_\gamma\in E whose coordinates are a_\alpha for \alpha<\gamma and zero otherwise. Each constructed prefix satisfies

Indeed, for finite f\in K,

using crop stability. This proves the bound whenever the prefix has been constructed.

Start with x_0=0, and at a successor append a_\gamma e_\gamma. Suppose \lambda is a limit ordinal and all earlier prefixes exist. If (x_\gamma)_{\gamma<\lambda} were not norm Cauchy, there would be \delta>0 and successive intervals [\alpha_i,\beta_i) below \lambda for which

The intervals can be chosen recursively from the failure of the Cauchy condition. Consider first the case that for some color k and some \delta'>0, infinitely many v_i satisfy \|P_{C_k}v_i\|\ge\delta'. For any j, select L_j of them. By the pure/full norm equality and symmetry, choose pure normers supported in their respective intervals, with evaluation at least \delta'/2. Their weight-j Type I average belongs to K_k. Evaluating on a prefix that covers all the selected intervals gives

contradicting L_j/m_j\to\infty.

In the remaining case, \|P_{C_k}v_i\|\to0 for every fixed k. Pass to a subsequence and choose successive finite color intervals D_i so that

Here is the recursive choice. After the preceding upper color cutoff has been fixed, its finitely many color projections tend to zero, so choose the next v_i with their sum of norms less than \delta/4. For this individual vector the projection onto sufficiently high colors has norm less than \delta/4, by finite approximation and contractivity. The intervening color interval has the desired norm. Take rectangle-cropped normers of evaluations at least \delta/4. Their supports are mixed-successive. Selecting L_j of them and using a mixed Type I average again contradicts (20), now with lower bound \delta L_j/(4m_j).

Thus the prefixes are Cauchy at every limit, including \omega_1. Completeness supplies their limit, with the required coordinates and earlier projections, and (20) is preserved. The terminal vector x_{\omega_1} agrees with g on every coordinate functional, hence on X_0 by density. This proves surjectivity.□

Proof. Every finite partial sum is a Type II functional, with one path and coefficient one. These sums have norm at most one. Their evaluations eventually stabilize on every finitely supported vector, since the pieces have successive disjoint supports. Uniform boundedness of the partial sums and finite approximation therefore give convergence on every vector in E, with limit norm at most one. Cropping follows either by the same argument or by composing with a contractive projection.□

Proof. Suppose two infinite paths have infinitely many common weights. Except possibly for their first weights, equality of weights implies equality of the preceding codes, by injectivity of \sigma. In particular, the preceding prefixes have the same length, the same metadata and weights, and the same arrays in their respective ordered closures. The lengths of these matching prefixes tend to infinity.

Interchange the two paths if necessary so that the supremum of the support of the first does not exceed that of the second. Fix a coordinate \alpha in the first path. There is a coordinate \xi\ge\alpha in the second path: an infinite successive path has no greatest support coordinate. For sufficiently long matching prefixes, both \alpha and \xi have appeared, and their common final metadata p is at least \rho(\alpha,\xi). The closure for the second prefix contains \xi, hence F_p(\xi), hence \alpha; the closure for the first prefix also contains \alpha. Their ordered initial segments through \alpha coincide by Lemma 5.2.

Every piece preceding the first path’s piece that contains \alpha is supported strictly before \alpha. Position alignment therefore makes this entire piece exactly equal to the corresponding piece in the second path, not merely equal on an initial interval. Its weight and metadata agree as well. Taking \alpha in arbitrarily late pieces identifies all the path data. Thus the paths were identical.

For finitely many distinct paths, discard initial pieces beyond all pairwise common weights. Their remaining weight sets are disjoint. Each finite sum of tail pieces is the crop of a finite prefix of the original path; this retains the required uncropped coding. Type II therefore gives (21) first for rational coefficients and finite tails. Pass to the path limits on each vector, use the norm bound, and approximate real coefficients. The discarded pieces have finite support and hence belong to X_0.□

Proof. Write M=\sup_i\|x_i\|. The proof has two steps. First we pass to a subsequence on which every actual infinite path in the family converges. Then we show that every pointwise limit of norming functionals converges on the same sequence.

Step 1: convergence on all actual paths. Fix \delta>0. If an as yet untreated path satisfies \limsup_i|P(x_i)|>\delta, pass to an infinite subsequence on which |P(x_i)|>\delta and P(x_i) converges. Continue this procedure, retaining all previously obtained convergence.

Only finitely many new paths can be selected at this threshold. Indeed, if h distinct paths have been selected, remove their finite prefixes as in Lemma 6.2. Those prefixes vanish on all sufficiently late blocks. By (21) and finite dimensional \ell_q duality, at each such block

Since every selected absolute value exceeds \delta on the retained subsequence, h\le(M/\delta)^{q'}. The selection must therefore stop. At that point every unselected path has limsup of its absolute evaluations at most \delta.

Apply this finite procedure successively to thresholds decreasing to zero and nested subsequences. Take a diagonal subsequence with the usual tail containment in each preceding subsequence. Every selected path converges on it. An unselected path has absolute limsup at most every threshold and hence tends to zero. Relabel this subsequence as (x_i).

Step 2: convergence on pointwise limits of norming functionals. Each individual norming functional is finitely supported and already vanishes on all sufficiently late blocks. This does not control every functional on the block span. We must instead control the compact pointwise closure of the norming set; at the end of the proof a measure representation will pass from this closure to the whole dual.

Let S=\bigcup_i\operatorname{supp}x_i, a countable set. Use K_k in the pure case and K in the mixed case, and let \mathcal C be the pointwise closure of its restrictions in [-1,1]^S. This is a compact metrizable space. Each u\in\mathcal C acts with norm at most one on the closed block span, because this holds on finite vectors by pointwise approximation. Define its oscillation on the blocks by

We will prove that D=0.

Fix u\in\mathcal C and choose norming functionals converging pointwise on S to u. Subsequences may be taken without changing this limit. Choose one valid expression for each functional. After subselection the top expressions are all terminals, all Type I, or all Type II. The following analysis does not require their construction depths to be bounded: all child limits will lie in the same compact set \mathcal C.

Terminals and Type I limits. In a pure family, a sequence of coordinate terminals has either a fixed coordinate on a subsequence or the zero pointwise limit; its oscillation is zero. In the mixed family, terminal functionals belong to individual pure colors. If those colors drift, the pointwise limit is zero; if the color is fixed, it meets at most one of the mixed-successive blocks, again giving zero oscillation.

For Type I expressions whose weights tend to infinity, the coordinate bound 1/m_j gives pointwise limit zero. Otherwise fix the weight j, the number of children, and the pointwise limits of all children. Thus

Nonzero coordinates of different child limits remain in their original order. At most one child limit can meet infinitely many blocks. For if an earlier child met infinitely many, its hits would eventually occur beyond any fixed hit of a later child, contradicting the order. Every other child is zero on all sufficiently late blocks. Consequently

This conclusion also applies to a pointwise limit of cropped Type I pieces: distribute the crops to the children and discard zeros before taking the limits.

Individual terms in a Type II limit. The key issue is the limit of one cropped path. We shall show that, on all sufficiently late blocks, such a limit agrees either with zero, with a limit of a single Type I piece, or with an actual infinite path. The first and third alternatives have zero oscillation; the second has the small factor 1/m_j from (22). After classifying the individual terms, we will use their disjoint active weights to sum these factors.

Index each cropped path term by its smallest active weight b. These indices are distinct within each expression. A diagonal subsequence makes presence or absence eventually constant for each index, makes its coefficient converge to a number a_b, and makes its crop membership converge at every coordinate of S. Put a_b=0 for absent terms. Fatou’s inequality on finite sets gives

For each present term, also take pointwise limits of its path crop. Call the resulting functional on the block span u_b. All subsequent choices impose only countably many conditions, indexed by a term, a finite chain position and a coordinate of S. Nested subsequences and a diagonal retain every finite collection of these conditions simultaneously for all sufficiently late approximants.

The original uncropped weight list through b has only finitely many possibilities, since weights strictly increase. Fix this initial list by subselection. Examine its successive extensions one position at a time. At each position the path either ends, the next weight tends to infinity, or the next weight can be fixed on a subsequence. Diagonalizing produces a finite fixed chain, possibly followed by a divergent-weight suffix, or an infinite fixed chain. Take pointwise limits of all cropped pieces in this chain as well.

A suffix whose weights tend to infinity contributes zero in the pointwise limit. Each coordinate meets at most one piece of the suffix, and its coefficient is bounded by the reciprocal of its least weight’s m-parameter. Whenever a fixed piece has a fixed successor weight, that successor fixes its entire preceding code. In particular, the support size of the piece is uniformly bounded by the length of the finite array in that code. Its pointwise limit therefore has finite support.

For a finite fixed chain, only its last piece can have a limit with infinite support. If its weight is j, (22) gives

If the last cropped piece has zero pointwise limit, then u_b is finitely supported and its oscillation is zero. Otherwise the last weight is active in all sufficiently late approximating terms. The last weights for these potentially oscillating terms are therefore distinct: two of them cannot be active in different terms of the same Type II approximant.

Infinite fixed chains and their limiting crops. It remains to transfer convergence from actual paths to infinite fixed chains, then sum the Type II terms in dual norm. In this case every individual piece has a finitely supported pointwise limit. The pointwise limit of the path crop is the sum of these piece limits. To see that no additional contribution escapes to high piece positions, note that the coordinate error after the first l pieces is at most 1/m_{j_{l+1}}, uniformly in sufficiently late approximants. If only finitely many cropped piece limits are nonzero, u_b is finitely supported and has zero oscillation.

Suppose instead that arbitrarily late pieces have nonzero cropped limits. Choose a coordinate \beta\in S with nonzero limit in one such piece. Eventually \beta belongs to the actual support of that piece and is retained by its crop. The fixed next weight fixes the prefix code through this piece, including its metadata p. All corresponding approximating closures contain \beta, and their initial segments through \beta are therefore exactly F_p(\beta). Every earlier uncropped piece is supported strictly before \beta. The position-alignment observation following Lemma 5.2 shows that these earlier pieces are eventually the same actual functions on \Gamma.

Choosing \beta in arbitrarily late effective pieces proves eventual stabilization of every uncropped prefix, with its weights and metadata. These stabilized prefixes define an actual infinite path P. Its pieces are valid Type I expressions because each stabilized piece equals a piece from an approximating path; every finite prefix satisfies the required coding rule for the same reason.

The limiting crop changes P on only finitely many coordinates of S. To prove this, fix one retained coordinate \beta in a piece. For a coordinate \alpha\in S belonging to any strictly later piece, choose a retained coordinate \gamma in a still later effective piece. In late approximants the stabilized coordinates \beta,\alpha,\gamma have their fixed ordinal order, and, in the mixed case, their fixed color order. Both \beta and \gamma lie in the crop. Interval convexity, in both orders when required, forces \alpha to lie in it too. Thus the limiting crop retains every path coordinate of S after the piece containing \beta. Only the finitely many finite pieces up to and including the piece containing \beta can differ. It follows that u_b(x_i)-P(x_i) is eventually zero. Step 1 proves convergence of P(x_i), so every infinite-chain term has zero oscillation.

Reassembling the Type II limit. We have

Indeed, terms with smallest active weight b>N have coordinate error at most

uniformly in all approximants. At a given coordinate there is at most one contributing piece per path, and the contributing active weights are globally distinct. This proves (25) after passing to the finitely many indices b\le N and then letting N\to\infty.

The series also converges in dual norm on the block span. For every finite set of present indices and every real coefficient vector (c_b) with \ell_q norm at most one, replace the coefficients of the corresponding approximating Type II terms by rational approximations to (c_b) inside the closed \ell_q unit ball. They remain admissible, since the paths and their pairwise disjoint active weights have not changed. Passing to the pointwise limits on finite vectors yields

By (23), finite truncations of \sum_ba_bu_b are therefore Cauchy in dual norm. Their norm limit agrees with (25) on the block vectors.

Oscillation is subadditive and changes by at most 2M\|v-w\| when one bounded functional is replaced by another. Using finite truncations, the distinct potentially oscillating last weights in (24), and |a_b|\le1, we obtain

All other terms have zero oscillation.

The terminal, Type I and Type II cases are exhaustive. Equations (22) and (26), together with Lemma 5.1, imply

Hence D=0: every u\in\mathcal C gives a convergent sequence u(x_i).

Finally, evaluation embeds the block span isometrically into C(\mathcal C); for a finite vector, its evaluation is continuous, and finite approximation extends this assertion to the closed span. Given any functional on the block span, extend it by Hahn–Banach to C(\mathcal C). The Riesz–Markov representation theorem [20, Theorem 6.19] represents the extension by a finite signed Borel measure \nu, with |\nu|(\mathcal C)<\infty. The functions u\mapsto u(x_i) are uniformly bounded by M and converge at every u\in\mathcal C. Bounded convergence with respect to |\nu| proves convergence under the given functional. The selected subsequence is therefore weak-Cauchy.□

Proof. Work in that tail. Let a(k) be the supremum of the norms of sums of k successive finite unit blocks. Contractivity of interval projections gives a(k)\le k. Norm each of L_h successive unit blocks to within a factor 2, crop the normers to their block ranges, and combine them by a Type I operation of weight h. In the mixed case use the ordinal and color ranges. Thus

Choose t>s sufficiently close to s that d^{1/t'}\le2d^{1/r} for 1\le d\le N. The parameter limit in Lemma 5.1 implies that a(k)/k^{1/t} is unbounded. Consequently there is a record index k such that

Choose successive unit blocks v_1,\ldots,v_k with \|\sum v_i\|\ge a(k)/2, and normalize their sum to obtain u.

For each I_b, let \ell_b count the blocks whose ordinal ranges are contained in I_b. There are at most two further blocks whose ranges partly meet I_b. Their projected norms are at most 1 each. Since \sum_b\ell_b\le k, the record inequality and Hölder’s inequality give

□

Proof. Color a four-element set positively if one finite path hits all four indices. By the infinite Ramsey theorem, it suffices to rule out an infinite set M all of whose four-element subsets are positive.

Fix v<t<k in M. For each w\in M with w<v, choose a path hitting w,v,t,k. Partition these paths according to their actual histories of weights below J_v: this history records the uncropped pieces, their weights, and their metadata, with their actual ordinal coordinates. We claim that representatives of different classes have disjoint weight sets in [J_k,T_k].

Indeed, suppose two such paths share a weight in that window. It is a successor weight in both paths, because both hit t. Injectivity of the coding map identifies their preceding compressed codes. These codes have a common final metadata value p\ge J_t, so (29) gives p>\max\rho(\operatorname{supp}x_v). Choose coordinates a,b of \operatorname{supp}x_v at which the respective v-window pieces are nonzero, and suppose a\le b. The first closure contains a; the second contains b, and hence a, by coherence and the bound on p. The two closures have the same initial segment through a, by Lemma 5.2.

In the first path every piece of weight below J_v lies strictly before a. Its compressed array is zero after the position of a. Equality of the codes therefore identifies that entire uncropped piece in the second path: the positions through a have the same ordinal labels, and every later coefficient is zero. The metadata agree as well. Thus the two actual histories below J_v are equal, proving the claim.

Crop each class representative to its pieces in [J_k,T_k]. This is a legal ordinal crop because pieces and weights are successive. Their active weight sets are pairwise disjoint. Type II combinations and rational approximation therefore give

Every coordinate has absolute value greater than \delta. In particular, the number of classes is bounded by an integer D depending only on \delta, for example any integer D\ge(2/\delta)^{q'}.

Choose triples v_n<t_n<k_n from M with v_n\to\infty, and label the classes for each triple by elements of \{1,\ldots,D\}. Each fixed w\in M has a label once w<v_n. Successive subsequences followed by a diagonal subsequence make its label eventually constant, for every fixed w. One label occurs for an infinite set M_0 of fixed indices. For each sufficiently large n, let P_n be a representative of that label. Then, for every fixed w\in M_0, P_n hits w for all sufficiently large n: eventually T_w<J_{v_n}, and equality of actual histories below J_{v_n} preserves the whole w-window.

Choose w_0\in M_0 so large that

For fixed w\in M_0 with w>w_0, consider n large enough that P_n hits both w_0 and w. Every piece of weight below J_{w_0} precedes a coordinate in \operatorname{supp}x_{w_0}, and hence vanishes on x_w. For J_{w_0}\le j<J_w, the child intervals of a weight-j piece and (30) give the bound A\theta_j on its absolute evaluation at x_w. The total contribution of weights above T_w is at most the expression in (31), which is less than \delta/4 for all sufficiently large w. Consequently

for every sufficiently large fixed w\in M_0, once n is sufficiently large depending on w.

The finite path sums belong to B_{E^*}. A weak-star convergent subnet of (P_n) has a limit P satisfying |P(x_w)|\ge\delta/2 at every one of these fixed indices: each inequality is eventually true and defines a weak-star closed set. This contradicts weak nullity of (x_w).□

Proof of Proposition 7.1. Perform the preceding construction in the prescribed tail and put y=(m_J/L)\sum_{i=1}^Lx_i. Equation (33) gives \|y\|\le500. Choose a normer g_i in the family with g_i(x_i)\ge1/2 and crop it to the ordinal range of x_i, also to its color range in the mixed case. These nonzero normers are successive, so

is a Type I functional of weight J. Their cross evaluations vanish, and hence \phi(y)=L^{-1}\sum_i g_i(x_i)\ge1/2. The crops give the asserted support bounds.□

Proof. Choose \gamma\leq\omega_1 least such that P_{<\gamma}|_F is not strictly singular; it exists because P_{<\omega_1}=I_E. It is nonzero. It cannot be a successor \beta+1: on an infinite-dimensional closed subspace where P_{<\beta+1} is bounded below, intersect with the kernel of e_\beta^*. This intersection is still infinite dimensional, and P_{<\beta}=P_{<\beta+1} there, contradicting minimality.

Choose an infinite-dimensional closed F_0\subset F on which P_{<\gamma} is bounded below. Its image H_0=P_{<\gamma}F_0 is closed, and its inverse U_0:H_0\to F_0 is bounded. Every earlier projection is strictly singular on H_0. Indeed, if P_{<\beta} were bounded below by a>0 on an infinite-dimensional closed M\subset H_0, then

would contradict the choice of \gamma.

If all the color projections are strictly singular on H_0, take H=H_0 and U=U_0, giving the second alternative. Otherwise choose k and an infinite-dimensional closed H_1\subset H_0 such that C=P_{C_k} is bounded below by b>0 on H_1. Set

The space H is closed, and CP_{<\gamma}U=I_H. To check earlier projections, suppose P_{<\beta} is bounded below by a>0 on an infinite-dimensional closed M\subset H. On (C|_{H_1})^{-1}M, commutation and contractivity of the projections give

contradicting strict singularity on H_0.

Finally, put R=P_{<\gamma} in the second alternative and R=P_{C_k}P_{<\gamma} in the first. Then RU=I_H. Each finite partial sum of a path with the stated support is fixed by composition with R. Passing to its pointwise limit on E gives P\circ R=P, and hence (38).□

Proof. Apply Lemma 8.1. We use the pure k family in its first alternative and the mixed family in its second. There are two steps: approximate vectors in H by successive finite blocks, and use exact pairs in those blocks to label a binary tree.

Successive blocks and a small perturbation. Choose positive numbers \varepsilon_i with

We construct h_i\in H and nonzero finite vectors z_i\in E satisfying

with the z_i successive on C_k in the pure case, and successive in both ordinal and color ranges in the mixed case.

Here is the inductive step. After z_1,\ldots,z_{i-1} have been chosen, let \beta<\gamma lie strictly above their supports; this is possible because \gamma is a limit ordinal. For i=1, take \beta=0. In the mixed case let l be the largest color previously used, with l=-1 at the first step. Apply Lemma 2.3 to P_{<\beta}|_H, and in the mixed case also to the finitely many maps P_{C_j}|_H, 0\leq j\leq l. We may therefore choose a unit vector h_i\in H for which

The second sum is zero when l=-1. In the pure case remove the initial ordinal segment from h_i; in the mixed case remove also all colors at most l. The resulting vector differs from h_i by less than 2\varepsilon_i/3, because the remaining ordinal projection is contractive. It is supported below \gamma and in the required ordinal and color tails. A finite-support approximation, cropped to those same sets, gives z_i satisfying (39). In the pure case the approximation is additionally cropped to C_k.

The ordinal interval projections show that (z_i) is a basic sequence with prefix projections of norm at most one. Since \|z_i\|\geq1/2, its coefficients satisfy

on finite sums. Consequently the map defined there by Vz_i=h_i extends to a bounded map V:[z_i]\to H with

Here I:[z_i]\to E is the inclusion, so the first norm is taken for maps into E; the image of V lies in H because H is closed.

A binary tree of exact pairs. As in the James tree construction [14], branches will separate dual functionals. Here the nodes carry the prescribed-weight pairs constructed in Proposition 7.1. List the finite binary sequences in breadth-first order, ordering each finite level lexicographically. For each node s we shall choose a finite block vector y_s\in\operatorname{span}\{z_i:i\in\N\}, a Type I functional \phi_s, its declared weight J_s, and its integer metadata p_s. Apply Proposition 7.1 successively in this order, always beyond every block used at earlier nodes. It gives

The support of \phi_s lies in the ordinal block range used for y_s, and in its color range as well in the mixed case. Thus the supporting ranges for different nodes are disjoint and successive in the chosen enumeration. In particular,

We now specify the weights and metadata, so that every branch is an actual path. Give the root any positive weight. Once \phi_s is chosen, choose p_s greater than the metadata at its predecessor, and at least J_s and the maximum of \rho on the finite union of supports along the branch from the root to s. At the root the predecessor condition is omitted. Form the prescribed order-position code of this whole prefix, and let both children of s have weight equal to its \sigma-value. Their weight is therefore already known when they are processed. Proposition 7.1 allows any target weight in any tail, so every step is possible. The siblings have the same weight; this causes no conflict, since they never belong to the same path. The disjoint-active-weight requirement for a Type II combination places no additional restriction on the individual paths constructed here.

For b\in\{0,1\}^{\N}, let

where b|0 is the root. Succession and the chosen codes make this an actual path in the selected family. Its finite partial sums have norm at most one, so it defines P_b\in B_{E^*}. Every piece is supported below \gamma, and on C_k in the pure case. Thus (38) applies to every P_b.

If b\ne c, choose a node s on b beyond their first divergence. Equations (41) and (42) give

Test their restrictions to F at UVy_s. By the lift identity and (40),

The conclusion follows with

□

Proof. Put L=Q-B and b=\|B\|. Every w\in W has a Q-lift of norm at most (11/10)\|w\|. Starting with such a lift y_0 of w, lift the error By_0, then the next error, and so on. The resulting series converges because (11/10)b<1. Its sum is an L-lift of w of norm at most

In particular L is onto and Cb<1.

Restriction to Y gives the decomposition

Indeed the restriction of z^{**} belongs to Y^*=E, and subtracting its canonical image leaves an element of Y^\perp. Moreover Q^* is an isometric isomorphism from W^* onto Y^\perp. Since L is onto,

For completeness, the first equality follows by factoring a bounded functional vanishing on \ker L through L; the lift bound makes the factor bounded. The displayed subspace is a graph over Y^\perp with values in J_E(E). Consequently J_E(E) maps bijectively onto E^{**}/X^\perp=X^*. This bijection is bounded with bounded inverse, by the open mapping theorem. Thus restriction identifies E with X^*.

Let G=\overline{B(X)}\subset V. Define

This is well-defined because two L-lifts differ by an element of X. The lift bound gives \|A\|\le Cb<1. For y\in Y one has Ly=-By, and hence A(By)=-By+G. By density, A|_V is minus the canonical quotient map V\to V/G. If G\ne V, that quotient map has norm one, contradicting \|A\|<1. Therefore G=V.□

Proof. Consider finitely many distinct paths. By Lemma 6.2, removing a finite prefix from each leaves pairwise disjoint weight sets. A removed prefix has finite support and belongs to X_0, so it does not change the path coset. Finite truncations of the remaining tails are allowed crops of the original path prefixes, with their uncropped metadata unchanged. The Type II rule bounds every rational \ell_{q_t}-unit combination of them by one in E^*. Their pointwise limits on E give bounded full-tail representatives in E^*. Applying Q to those representatives, and then approximating real coefficients, gives

for every finitely supported real family (a_P). Completion gives S_t.□

Proof. The first two claims follow from finite approximation in each \ell_{q_t}(\mathcal P_t) and the summability of the \tau_t. For the reverse inclusion in the first claim, use the assigned coordinate functional: disjointness of the label ranges gives B e_{\iota_t(P)}^*=\tau_t QP.

Fix w^*\in W^*. The coefficient family S_t^*w^* belongs to \ell_{q_t'}(\mathcal P_t) with norm at most \|w^*\|. Since q_t'\le s_{\rm src}(t), it also belongs to \ell_{s_{\rm src}(t)} with no larger norm. The pure-coordinate bound therefore makes

a norm-convergent vector series in E, of norm at most \|w^*\|. Here, as usual, an \ell_p family with finite p has countable support. The vector v=\sum_t\tau_t v_t converges in E. Testing the definition of B against w^*, first on finite sums and then by continuity, gives B^*w^*=J_Ev.□

Proof. Use the space X just constructed. Let F\subset E be any infinite-dimensional separable closed subspace, and put

This space is separable. Indeed a countable dense subset of F is supported on a common countable set of coordinates; all other coordinate functionals vanish on F. Since X_0 is the closed coordinate span, its restrictions belong to the closed span of a countable family.

Restriction induces a bounded map

We claim that T_F(V) is nonseparable. Otherwise its closure M would be separable. The inverse image of M under the quotient map F^*\to F^*/C_F is then separable: its kernel C_F is separable, and the restriction of the quotient map to this inverse image still has the quotient norm. More explicitly, lift a countable dense set (m_n) in M to vectors (u_n) in F^*, and choose a countable dense set (c_j) in C_F. For any vector z in the inverse image, first make its quotient close to m_n. The quotient norm then makes z-u_n close to a vector of C_F, which can in turn be approximated by some c_j. Thus the countable set \{u_n+c_j:n,j\ge1\} is dense in the inverse image. But this inverse image contains every path restriction P|_F. Theorem 8.2 supplies an uncountable norm-separated family among those restrictions, a contradiction.

Because \overline{B(X)}=V, continuity gives \overline{T_F(BX)}=\overline{T_F(V)}; hence T_F(BX) is nonseparable. For x\in X we have Qx=Bx, so the evaluation image \{x|_F:x\in X\}\subset F^* has a nonseparable image in F^*/C_F. It is itself nonseparable. Given any infinite-dimensional separable closed F'\subset X^*, apply this conclusion to F=R^{-1}(F'). Composition with the bounded isomorphism R|_F:F\to F' identifies the two evaluation images and preserves norm separability. Thus the evaluations on every such F' are nonseparable. The necessary direction of Proposition 2.2 therefore rules out a separable infinite-dimensional quotient of X.□

Proof. Lemma 5.3 makes the finitely supported rational vectors a dense subset of E of cardinality \aleph_1, while the coordinate vectors e_\alpha, \alpha<\omega_1, are pairwise at distance at least one. Hence \dens E=\aleph_1, and the isomorphism E\simeq X^* from Lemma 9.1 gives \dens X^*=\aleph_1.

Choose a norm-dense set D\subset S_{X^*} of cardinality \aleph_1, and, for each f\in D, choose x_f\in S_X with |f(x_f)|>1/2. The vectors x_f have dense linear span: otherwise Hahn–Banach supplies g\in S_{X^*} vanishing on their closed span, and choosing f\in D with \|f-g\|<1/2 contradicts |f(x_f)|>1/2. Rational linear combinations of these vectors give \dens X\le\aleph_1. The space X cannot be separable, since it is infinite dimensional and would then be its own forbidden quotient. Thus \dens X=\aleph_1.

The complexification in Corollary 4.3 has underlying real space isomorphic to X\oplus_\infty X, so it has the same density. That corollary also gives the absence of separable infinite-dimensional complex quotients.□

Proof. Work in a ground model of ZFC with a measurable cardinal \kappa. A \kappa-complete nonprincipal ultrafilter on \kappa gives a \{0,1\}-valued probability measure witnessing its real-valued measurability.

We first record the cardinal arithmetic needed to control the continuum. The cardinal \kappa is regular: sets of size less than \kappa are null, so a cofinal union of fewer than \kappa such sets cannot cover \kappa. It is also a strong limit. Otherwise, for some \lambda<\kappa there would be an injection e:\kappa\to\mathcal P(\lambda). For each \xi<\lambda, the ultrafilter chooses one of the two sets on which the statement \xi\in e(\alpha) is constant. Their intersection belongs to the ultrafilter by \kappa-completeness, although injectivity makes that intersection have at most one element. This is impossible. Every countable sequence of ordinals below \kappa is therefore bounded, and

Here \lambda^{\aleph_0}\leq 2^{\max(\lambda,\aleph_0)}<\kappa for every \lambda<\kappa. The reverse inequality is immediate, so \kappa^{\aleph_0}=\kappa.

Let \mathbb B_\kappa be the measure algebra of the fair-bit product on 2^{\kappa\times\mathbb N}. We use the probability space generated by finite-coordinate cylinders, or its completion; the quotient algebra is the same. Every element of this algebra has a representative determined by countably many coordinates. Such a representative is the inverse image of a Borel set in a countable product. Thus

in the ground model. An antichain consists of disjoint nonzero elements modulo null sets and hence is countable: for each positive integer n, only finitely many of its elements can have measure greater than 1/n. Consequently the forcing has the countable chain condition and preserves cardinals [16, Chapter VII, Theorem 5.10].

The generic coordinate sequences give \kappa distinct reals. Indeed, for distinct \alpha,\beta<\kappa, the event that their coordinate sequences agree has product measure zero. For the reverse bound, every real has a name specified by countably many antichains deciding its bits. Each antichain is countable, so at most |\mathbb B_\kappa|^{\aleph_0}\leq\kappa such deciding-antichain descriptions suffice to name all reals [16, Chapter VII, Definition 5.11 and Lemmas 5.12–5.13]. Hence the extension satisfies \mathfrak c=\kappa.

Theorem 10.1 now gives the required measure on \mathcal P(\mathfrak c) in the extension. Set forcing preserves ZFC, including Choice [16, Chapter VII, Theorem 4.2]. The definable forcing relation and truth theorem [16, Chapter VII, Theorems 3.5–3.6] give the asserted relative-consistency implication in their syntactic, or finite-fragment, interpretation [16, Chapter VII, Sections 1 and 9]: a finite contradiction proof in the extension theory would yield one in the ground theory. This requires no transitive set model of all ZFC.□

Proof of Corollary 1.2. Starting with a ground model of ZFC and GCH, the forcing construction in [16, Chapter VIII, Theorem 6.3] gives a model of Martin’s axiom with \mathfrak c=\aleph_2. Gödel’s constructible-universe theorem and the relative-consistency interpretation of forcing therefore give

This model satisfies \mathrm{MA}_{\aleph_1}, hence \mathfrak b>\aleph_1 by the preceding argument. The small-density theorem gives a separable infinite-dimensional quotient for every real Banach space of norm density \aleph_1. A complex Banach space has the same norm density as its underlying real space, so Proposition 4.1 gives the corresponding complex conclusion.

On the other side, consistency of ZFC gives a CH model, and Corollary 9.5 supplies real and complex counterexamples of norm density \aleph_1 there. Consequently, for each field \mathbb F\in\{\R,\C\},

This proves the asserted independence with no large-cardinal consistency hypothesis.□