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 3, with vectors and norming functionals interchanged, shows that \((y_n)\) is basic and \(|a_n|\le4\|\sum_i a_i y_i\|\). Therefore \[\left\|T\sum_n a_n y_n\right\| \le4\sum_n\|Ty_n\|\left\|\sum_n a_n y_n\right\| \le\varepsilon\left\|\sum_n a_n y_n\right\|.\] 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 \(b:\kappa\to\{0,1\}^{\mathbb N}\). 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 \[ \|a\|_2\le\Bigl\|\sum_{j=1}^{N_n}a_j u_{n,j}\Bigr\| \le2\|a\|_2\qquad(a\in\mathbb R^{N_n}). \tag{3}\] 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 6 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 \[\int_\kappa |y(u_{n,j_n(\alpha)})|\,d\mu(\alpha) \le \frac2{\sqrt{N_n}}.\] 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 3 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 3 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 7.

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 (Odell and Rosenthal 1975), each \(h_\alpha\) is the weak-star limit of a sequence from \(E\). Choose one such sequence \(c_\alpha\in E^{\mathbb N}\). Equip \(E^{\mathbb N}\) 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 \[\Phi:E^{\mathbb N}\times K\longrightarrow\mathbb R\] 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 \(E^{\mathbb N}\). The function \[ H(y)=\int_{E^{\mathbb N}}\mathbf1_{\{\Phi(c,y)\ne0\}}\,d\eta(c) \tag{6}\] 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: \[\lambda_n(A)=\int_I\cdots\int_I \mathbf1_A(\alpha_1,\ldots,\alpha_n) \,d\mu(\alpha_n)\cdots d\mu(\alpha_1).\] 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 \[ \left|\sum_{j\in s}a_j f_{\alpha_j}(x_s)\right| \ge\frac12\left\|\sum_{j\in s}a_j f_{\alpha_j}\right\|. \tag{7}\] Choose \(x_s=0\) if that sum is zero. The choice depends only on the indices in \(s\). For \(i\notin s\) we claim \[ f_{\alpha_i}(x_s)=0\quad\text{for $\lambda_n$-almost every tuple.} \tag{8}\]

Fix any earlier prefix \((\alpha_1,\ldots,\alpha_{i-1})\). The map \[Y:I^{n-i}\longrightarrow K,\qquad Y(\alpha_{i+1},\ldots,\alpha_n)=R_Ex_s\] 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 \[\begin{align*} &\int_I\int_{I^{n-i}} \mathbf1_{\{\Phi(c_{\alpha_i},Y(t))\ne0\}} \,d\lambda_{n-i}(t)\,d\mu(\alpha_i)\\ &\qquad=\int_{E^{\mathbb N}}\int_K \mathbf1_{\{\Phi(c,y)\ne0\}}\,d\nu(y)\,d\eta(c) =\int_K H(y)\,d\nu(y)=0. \end{align*}\] 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 \[ \left\|\sum_{j\in s}a_j f_{\alpha_j}\right\| \le2\left\|\sum_{j=1}^n a_j f_{\alpha_j}\right\| \tag{9}\] 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 \[ \lambda_{n-m}((G_n)_p)=1\qquad\text{for every }n>m. \tag{10}\] 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 \[1=\lambda_{n-m}((G_n)_p) =\int_I\lambda_{n-m-1}((G_n)_{(p,\alpha)})\,d\mu(\alpha).\] 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 (Argyros et al. 2008, 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 7 and 8 give a seminormalized family \(f_\alpha=h_\alpha R_E\) with the pointwise null-support property. Lemma 9 then produces an unconditional basic sequence in \(X^*\), the desired contradiction. ◻

Proof. Put \(\eta_j=(2^{-10-j})^r\) and choose \(L_j=\lfloor\eta_jm_j^r\rfloor\). By taking \(m_j\) large, arrange \[\eta_j/2\le L_j/m_j^r\le\eta_j.\] It follows that \[2^{-1/r}2^{-10-j}\le\theta_j\le2^{-10-j}.\] 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 \[\frac{L_j}{m_j}=\theta_jL_j^{1/s} \quad\hbox{and}\quad \frac1{q'}<\frac1s,\] 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 \[F_p(\beta)=F_p(\alpha_i)\cup\{\beta\}.\] At a switch, coherence at \(\alpha_i\) gives \[F_{p_i}(\alpha_{i-1}) =F_{p_i}(\alpha_i)\cap[0,\alpha_{i-1}],\] 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 \[\|x\|_\infty\le\|x\|\le\|x\|_1 \qquad(x\in c_{00}(\Gamma)).\] 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 \[ \{f|C_k:f\in K\}\subset\operatorname{absconv}(K_k), \tag{18}\] 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 \[t_b=\left(\sum_{j\in A_b}\theta_j^r\right)^{1/r}.\] For a Type II expression, the triangle inequality and Hölder’s inequality give \[\left\|\sum_b a_b P_{I_b}\sum_i\phi_{b,i}\right\|_{\ell_r} \le \|(a_b)\|_{\ell_q}\|(t_b)\|_{\ell_{q'}} \le\|(t_b)\|_{\ell_r} \le\left(\sum_j\theta_j^r\right)^{1/r}<1.\] We used \(q'>r\) and disjointness of the active weight sets. Now apply Hölder’s inequality to vectors on \(C_k\) and use Lemma 16. 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 \[ \|x_\gamma\|\le\|g\|. \tag{20}\] Indeed, for finite \(f\in K\), \[f(x_\gamma)=g(P_{<\gamma}f),\qquad \|P_{<\gamma}f\|_{X_0}\le1,\] 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 \[v_i=x_{\beta_i}-x_{\alpha_i},\qquad \|v_i\|\ge\delta.\] 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 \[\|g\|\ge\frac{\delta' L_j}{2m_j},\] 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 \[\|P_{\operatorname{col}^{-1}(D_i)}v_i\|\ge\delta/2.\] 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. 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 \[a(L_h)\ge\frac{L_h}{2m_h}.\] 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 14 implies that \(a(k)/k^{1/t}\) is unbounded. Consequently there is a record index \(k\) such that \[a(\ell)\le a(k)(\ell/k)^{1/t}\quad(1\le\ell\le k), \qquad a(k)\ge N^{1/s}.\] 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 \[\begin{align*} \sum_{b=1}^d\|P_{I_b}u\| &\le2\sum_{b:\ell_b>0}(\ell_b/k)^{1/t}+\frac{4d}{a(k)}\\ &\le2d^{1/t'}+\frac{4d}{N^{1/s}} \le8d^{1/r}. \end{align*}\] ◻

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 15.

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 \[\big\|(P_b[k](x_k))_b\big\|_{q'}\le\|x_k\|\le2.\] 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 \[A\sum_{j\ge J_{w_0}}\theta_j<\delta/4.\] 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 \[|P_n(x_w)|\ge\delta/2\] 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. 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 \[ C\|w\|,\qquad C=\frac{11/10}{1-(11/10)b}. \tag{43}\] In particular \(L\) is onto and \(Cb<1\).

Restriction to \(Y\) gives the decomposition \[E^{**}=Y^\perp\oplus J_E(E).\] 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, \[X^\perp=L^*(W^*)=\{Q^*w^*-B^*w^*:w^*\in W^*\}.\] 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 \[A:W\longrightarrow V/G,\qquad A(Ly)=By+G.\] 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 20, 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 \[\Big\|\sum_P a_P QP\Big\|\le \Big(\sum_P|a_P|^{q_t}\Big)^{1/q_t}\] 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 \[v_t=\sum_{P\in\mathcal P_t}(S_t^*w^*)(P)e_{\iota_t(P)}\] 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 \[C_F=\overline{\{y|_F:y\in X_0\}}\subset F^*.\] 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 \[T_F:W\longrightarrow F^*/C_F,\qquad T_F(Qy)=y|_F+C_F.\] 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 26 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 4 therefore rules out a separable infinite-dimensional quotient of \(X\). ◻

Proof. Lemma 16 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 \(\mathop{\mathrm{dens}}E=\aleph_1\), and the isomorphism \(E\simeq X^*\) from Lemma 27 gives \(\mathop{\mathrm{dens}}X^*=\aleph_1\).

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 \(\mathop{\mathrm{dens}}X\le\aleph_1\). The space \(X\) cannot be separable, since it is infinite dimensional and would then be its own forbidden quotient. Thus \(\mathop{\mathrm{dens}}X=\aleph_1\).

The complexification in Corollary 13 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 of Corollary 2. Starting with a ground model of ZFC and GCH, the forcing construction in (Kunen 1980, 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 \[\operatorname{Con}(\mathrm{ZFC})\Longrightarrow \operatorname{Con}(\mathrm{ZFC}+\mathrm{MA}+\mathfrak c=\aleph_2).\] 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 11 gives the corresponding complex conclusion.

On the other side, consistency of ZFC gives a CH model, and Corollary 31 supplies real and complex counterexamples of norm density \(\aleph_1\) there. Consequently, for each field \(\mathbb F\in\{\mathbb R,\mathbb C\}\), \[\operatorname{Con}(\mathrm{ZFC})\Longrightarrow \begin{cases} \operatorname{Con}(\mathrm{ZFC}+\mathrm{SQ}^{\aleph_1}_{\mathbb F}),\\ \operatorname{Con}(\mathrm{ZFC}+\neg\mathrm{SQ}^{\aleph_1}_{\mathbb F}). \end{cases}\] This proves the asserted independence with no large-cardinal consistency hypothesis. ◻