Preface ....................................................... VII
Standard Definitions, Notation, and Conventions .............. XVII
1. Separable Banach Spaces ...................................... 1
1.1. Basics .................................................. 2
Minimal systems; basic facts on biorthogonal systems
(b.o.s.); fundamental minimal systems; Markushevich bases
(M-bases); norming M-bases; shrinking b.o.s.;
boundedly complete b.o.s.
1.2. Auerbach Bases .......................................... 5
Every finite-dimensional space has an Auerbach basis;
Day's construction of a countable infinite Auerbach system
1.3. Existence of M-bases in Separable Spaces ................ 8
Markushevich theorem on the existence of M-bases in
separable spaces; every space with a ω*-separable dual
has a bounded total b.o.s.
1.4. Bounded Minimal Systems ................................. 9
Pelczyiiski-Plichko (1 + ε)-bounded M-basis theorem;
no shrinking Auerbach systems in C(K) (Plichko)
1.5. Strong M-bases ......................................... 21
Terenzi's theorem on the existence of a strong M-basis in
every separable space; flattened perturbations;
Vershynin's proof of Terenzi's theorem; strong and norming
M-bases; strong and shrinking M-bases
1.6. Extensions of M-bases .................................. 29
Gurarii-Kadets extension theorem; extension of bounded
M-bases to bounded M-bases (Terenzi); extension of bounded
M-bases and quotients; a negative result for extending
Schauder bases; extension in the direction of
quasicomplements (Milman)
1.7. ω-independence ......................................... 38
Relation to biorthogonal systems; every ω-independent
system in a separable space is countable
(Fremlin-Kalton-Sersouri)
1.8. Exercises .............................................. 42
2. Universality and the Szlenk Index ........................... 45
2.1. Trees in Polish Spaces ................................. 46
Review of some techniques on trees in Polish spaces;
Kunen-Martin result on well-founded trees in Polish spaces
2.2. Universality for Separable Spaces ...................... 49
Complementable universality of (unconditional) Schauder
bases (Kadets-Pelczynski); no complementable universality
for superreflexive spaces (Johnson-Szankowski);
if a separable space is isomorphically universal for all
reflexive separable spaces, then it is isomorphically
universal for all separable spaces (Bourgain);
in particular, there is no separable As-plund space
universal for all separable reflexive spaces (Szlenk);
there is no superreflexive space universal for all
superreflexive separable spaces (Bourgain);
there is a reflexive space with Schauder basis
complementable universal for all superreflexive spaces
with Schauder bases (Prus); there is a separable reflexive
space universal for all separable superreflexive spaces
(Odell-Schlumprecht)
2.3. Universality of M-bases ................................ 57
Iterated ω*-sequential closures of subspaces of dual
spaces (Banach-Godun-Ostrovskij); no universality for
countable M-bases (Plichko)
2.4. Szlenk Index ........................................... 62
Szlenk derivation; properties of the Szlenk index (Sz(X));
for separable spaces, Sz(X) = ω if and only if X admits
a UKK* renorming (Knaust-Odell-Schlumprecht)
2.5. Szlenk Index Applications to Universality .............. 70
Under GCH, if τ is an uncountable cardinal, there exists
a compact space К of weight τ such that every space of
density τ is isometrically isomorphic to a subspace of C(K)
(Yesenin-Volpin); for infinite cardinality τ, there is no
Asplund space of density τ universal for all reflexive
spaces of density τ; there is no WCG space
of density ω1 universal for all WCG spaces of
density ω1 (Argyros-Benyamini)
2.6. Classification of C[0, α] Spaces ....................... 73
Mazurkiewicz-Sierpiriski representation of countable
compacta; Bessaga-Pelczynski isomorphic classification
of C(K) spaces for К countable compacta
as spaces С[0,ωωα ] for α < ω1; Samuel's evaluation of the
Szlenk index of those spaces
2.7. Szlenk Index and Renormings ............................ 77
ω*-dentability index Δ(X) of a space X; Asplund spaces
with Δ(X) < ω1 have a dual LUR norm; characterization of
superreflexivity by Δ(X) ≤ ω; estimating Δ(X) from
above by Ψ(Sz(X))
2.8. Exercises .............................................. 82
3. Review of Weak Topology and Renormings ...................... 87
3.1. The Dual Mackey Topology ............................... 88
Grothendieck's results on the Mackey topology on dual
spaces
3.2. Sequential Agreement of Topologies in X* ............... 92
Characterizations of spaces not containing l1 by using the
Mackey topology τ(Х*, X) in the dual; results of Emmanuele,
Ørno and Valdivia
3.3. Weak Compactness in ca (Σ)and L1(λ) .................... 95
Weak compactness in L1(μ); weak compactness in ca(Σ);
Grothendieck results on the Mackey topology
τ(X*,X) in (L1(μ))* and (C(K))*; Josefson-Nissenzweig
theorem
3.4. Decompositions of Nonseparable Banach Spaces .......... 102
Corson and Valdivia compacta; weakly Lindelöf-determined
spaces; pro-jectional resolutions of the identity (PRI);
projectional generators (PG); separable complementation
property; every nonseparable space with a PG has a PRI;
PG in WCG spaces
3.5. Some Renorming Techniques ............................. 107
LUR renorming of a space with strong M-basis (Troyanski);
LUR renorming of subspaces of l∞ closely related to
σ-shrinkable M-bases; weak 2-rotund property of Day's
norm on с0(Γ); Troyanski's results on uniform properties
of the Day norm
3.6. A Quantitative Version of Krein's Theorem ............. 119
Closed convex hulls of e-weakly relatively compact sets
3.7. Exercises ............................................. 125
4. Biorthogonal Systems in Nonseparable Spaces ................ 131
4.1. Long Schauder Bases ................................... 132
Uncountable version of Mazur's technique; bounded total
b.o.s. in every space
4.2. Fundamental Biorthogonal Systems ...................... 137
The existence of fundamental biorthogonal systems
(f.b.o.s.) implies the existence of bounded f.b.o.s;
lifting f.b.o.s. from quotients; f.b.o.s. in l∞;
f.b.o.s. in Johnson-Lindenstrauss spaces (JL);
lc∞(Γ) has an f.b.o.s. iff card Γ < с
4.3. Uncountable Biorthogonal Systems in ZFC ............... 143
Uncountable b.o.s. in C(K) when К contains a nonseparable
subset; b.o.s. in representable spaces; every nonseparable
dual space contains an uncountable b.o.s.
4.4. Nonexistence of Uncountable Biorthogonal Systems ...... 148
Under axiom ♣ there is a scattered nonmetrizable compact К
such that (C(K))* is hereditarily ω*-separable and C(K) is
hereditarily weakly Lindelof; in particular, C(K) does not
contain an uncountable b.o.s.
4.5. Fundamental Systems under Martin's Axiom .............. 152
Under Martin's axiom МАω1, any Banach space with
density ω1 with a ω*-countably tight dual unit ball has
an f.b.o.s.; under Martin's Maximum (MM), any Banach space
of density ω1 has an f.b.o.s. (Todorcevic)
4.6. Uncountable Auerbach Bases ............................ 158
Any nonseparable space with a ω*-separable dual ball has
a norm with no Auerbach basis (Godun-Lin-Troyanski)
4.7. Exercises ............................................. 161
5. Markushevich Bases ......................................... 165
5.1. Existence of Markushevich Bases ....................... 165
 -classes; every space in a -class has a strong M-basis;
spaces with M-bases and injections into с0(Γ); spaces with
strong M-bases failing the separable complementation
property; the space l∞(Γ) for infinite Γ has no
M-basis (Johnson)
5.2. M-bases with Additional Properties .................... 170
Every space with an M-basis has a bounded M-basis;
WCG spaces without a 1-norming M-basis and variations;
C[0,ω1] has no norming M-basis; countably norming M-bases
5.3. Σ-subsets of Compact Spaces ........................... 176
Topological properties of Valdivia compacta;
Deville-Godefroy-Kalenda-Valdivia results
5.4. WLD Banach Spaces and Plichko Spaces .................. 179
WLD spaces and full PG; characterizations of WLD spaces
in terms of M-bases; weakly Lindelöf property of WLD
spaces; property С of Corson; Plichko spaces;
Kalenda's characterization of WLD spaces by PRI
when dens X = ω1; spaces with ω*-angelic dual balls
without M-bases
5.5. C(K) Spaces that Are WLD .............................. 187
Corson compacta with property M; under CH, example of a
Corson compact L such that C(L) has a renorming without
PRI; on the other hand, under Martin's axiom МАω1, every
Corson compact has property M
5.6. Extending M-bases from Subspaces ...................... 191
Extensions from subspaces of WLD spaces; the bounded case
for dens X <  ω; c0(Γ) is complemented in every Plichko
overspace if card Γ < ω; on the other hand, under GCH,
со(ω) may not be complemented in a WCG overspace;
extensions of M-bases when the quotient is separable;
under ♣, example of a space with M-basis having a
complemented subspace with M-basis that cannot be extended
to the full space
5.7. Quasicomplements ...................................... 197
M-bases and quasicomplements;
the Johnson-Lindenstrauss-Rosenthal theorem that Y is
quasicomplemented in X whenever Y* is ω* -separable and X/Y
has a separable quotient; every subspace of l∞ is
quasicomplemented; Godun's results on quasicomplements
in l∞; quasicomplements in Grothendieck spaces;
Josefson's theorems on limited sets; quasicomple-mentation
of Asplund subspaces in l∞(Γ) iff X* is ω*-separable; in
particular, c0(Γ) is not quasicomplemented inl∞(Γ)
whenever Γ is uncountable (Lindenstrauss); the separable
infinite-dimensional quotient problem
5.8. Exercises ............................................. 203
6. Weak Compact Generating .................................... 207
6.1. Reflexive and WCG Asplund Spaces ...................... 207
Characterization of reflexivity by M-bases and by
2R norms; characterization of WCG Asplund spaces
6.2. Reflexive Generated and Vasak Spaces .................. 212
Reflexive- and Hilbert-generated spaces; Frechet M-smooth
norms; weakly compact and σ-compact M-bases; M-basis
characterization of WCG spaces; weak dual 2-rotund
characterization of WCG; σ-shrinkable M-bases; M-basis
characterization of subspaces of WCG spaces; continuous
images of Eberlein compacts (Benyamini, Rudin,Wage);
weakly σ-shrinkable M-bases; M-basis characterization
of Vasak spaces; M-basis characterization of WLD spaces
6.3. Hilbert Generated Spaces .............................. 225
M-basis characterization of Hilbert-generated spaces;
uniformly Gateaux (UG) smooth norms; uniform Eberlein
compacts; M-basis characterization of spaces with UG
norms; characterizations of Eberlein, uniform Eberlein,
and Gul'ko compacts (Farmaki); continuous images of
uniform Eberlein compacts (Benyamini, Rudin, Wage)
6.4. Strongly Reflexive and Superreflexive Generated
Spaces ................................................ 233
Metrizability of Bx* in the Mackey topology τ(Х*,Х)\ weak
sequential completeness of strongly reflexive generated
spaces (Edgar-Wheeler); applications of strongly
superreflexive generated spaces to L1(μ);
super-reflexivity of reflexive subspaces of L1(μ)
(Rosenthal); uniform Eberlein compacta in L1(μ);
almost shrinking M-bases (Kalton)
6.5. Exercises ............................................. 239
7. Transfinite Sequence Spaces ................................ 241
7.1. Disjointization of Measures and Applications .......... 241
Rosenthal disjointization of measures; applications to
fixing c0(Г) and l∞(Г) in C(K) (Pelczynski, Rosenthal);
applications to weakly compact operators; subspaces
of c0(Г) and containment of c0(Г) in C(K) spaces
7.2. Banach Spaces Containing l1(Г) ........................ 252
Characterization of spaces containing l1(Г) by properties
of dual unit balls and quotients (Pelczynski, Argyros,
Talagrand, Haydon)
7.3. Long Unconditional Bases .............................. 259
Characterization of Asplund, WCG (Johnson), WLD
(Argyros-Mercourakis), strongly reflexive-generated
(Mercourakis-Stamati), UG or URED renormable (Troyanski)
spaces with unconditional bases
7.4. Long Symmetric Bases .................................. 266
Troyanski's characterizations of spaces with symmetric
basis having a UG or URED norm; applications to separable
spaces
7.5. Exercises ............................................. 270
8. More Applications .......................................... 273
8.1. Biorthogonal Systems and Support Sets ................. 273
There are no separable support sets (Rolewicz); X has
a bounded support set iff it has an uncountable
semibiorthogonal system; if К is a non-metrizable
scattered compact, then C(K) has a support set; under MM,
a space is separable iff there is no support set
(Todorcevic); consistency of the existence of nonseparable
spaces without support sets
8.2. Kunen-Shelah Properties in Banach Spaces .............. 276
Representation of closed convex sets as kernels of
nonnegative C∞ functions; representation of closed convex
sets as intersections of countably many half-spaces
and ω*-separability; ω1 -polyhedrons; for a separable
space X, X* is separable iff every dual ball
in X** is ω*-separable; if X* is ω*-hereditarily
separable (e.g., X = C(L) for L in Theorem 4.41),
then X has no uncountable ω-independent system
8.3. Norm-Attaining Operators .............................. 284
Lindenstrauss' properties α and β; renormings and
norm-attaining operators; renorming spaces to have
property α (Godun, Troyanski)
8.4. Mazur Intersection Properties ......................... 289
Characterizations of spaces with the Mazur intersection
property (MIP); Asplund spaces without MIP renormings
(Jimenez-Sevilla, Moreno); equivalence of Asplundness and
MIP in separable spaces; if X admits a b.o.s. {хγ;х*γ}γ Γ
such that  {x*γ; γ  Г} = X*, then X has an MIP
renorming; the non-Asplund space l1 × l2(c) has an MIP
renorming (Jimenez-Sevilla, Moreno)
8.5. Banach Spaces with only Trivial Isometries ............ 297
Every space can be renormed to have only ±identity
as isometries
8.6. Exercises ............................................. 300
References .................................................... 303
Symbol Index .................................................. 323
Subject Index ................................................. 327
Author Index .................................................. 335
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