TOPICS IN BANACH SPACE THEORY
CONTENTS 1 BASES AND BASIC SEQUENCES 1 1 1 SCHAUDER BASES 1 1 2 EXAMPLES FOURIER SERIES 7 1 3 EQUIVALENCE OF BASES AND BASIC SEQUENCES 11 1 4 BASES AND BASIC SEQUENCES DISCUSSION 16 1 5 CONSTRUCTING BASIC SEQUENCES 20 1 6 THE EBERLEIN SMULIAN THEOREM 24 PROBLEMS 26 2 THE CLASSICAL SEQUENCE SPACES 29 2 1 THE ISOMORPHIC STRUCTURE OF THE P SPACES AND C0 29 2 2 COMPLEMENTED SUBSPACES OF P 1 P 1 AND C0 33 2 3 THE SPACE 1 36 2 4 CONVERGENCE OF SERIES AND OPERATORS ON C0 38 2 5 COMPLEMENTABILITY OF C0 44 PROBLEMS 48 3 SPECIAL TYPES OF BASES 51 3 1 UNCONDITIONAL BASES 51 3 2 BASES AND DUALITY BOUNDEDLY COMPLETE AND SHRINKING BASES 55 3 3 NONRE EXIVE SPACES WITH UNCONDITIONAL BASES 63 3 4 THE JAMES SPACE J 66 3 5 A LITMUS TEST FOR HAVING UNCONDITIONAL BASES 70 PROBLEMS 73 4 BANACH SPACES OF CONTINUOUS FUNCTIONS 77 4 1 BASIC FACTS ABOUT THE SPACES C K 78 4 2 AN INTRINSIC CHARACTERIZATION OF REAL C K SPACES 79 4 3 ISOMETRICALLY INJECTIVE SPACES 86 4 4 SPACES OF CONTINUOUS FUNCTIONS ON UNCOUNTABLE COMPACT METRIC SPACES 94 4 5 SPACES OF CONTINUOUS FUNCTIONS ON COUNTABLE COMPACT METRIC SPACES 102 PROBLEMS 105 5 L1 SPACES AND C K SPACES 109 5 1 GENERAL REMARKS ABOUT L1 SPACES 109 5 2 WEAKLY COMPACT SUBSETS OF L1 111 5 3 WEAK COMPACTNESS IN M K 121 5 4 THE DUNFORD PETTIS PROPERTY 123 5 5 THE EMERGENCE OF THE RADON NIKODYM PROPERTY 126 5 6 WEAKLY COMPACT OPERATORS ON C K SPACES 131 5 7 SUBSPACES OF L1 SPACES AND C K SPACES 133 PROBLEMS 135 6 THE SPACES LP FOR 1 P 1 137 6 1 THE HAAR BASIS IN LP 0 1 1 P 1 138 6 2 AVERAGING IN BANACH SPACES 145 6 3 PROPERTIES OF L1 157 6 4 SUBSPACES OF LP 163 PROBLEMS 174 7 FACTORIZATION THEORY 177 7 1 MAUREY NIKISHIN FACTORIZATION THEOREMS 178 7 2 SUBSPACES OF LP FOR 1 P 2 185 7 3 FACTORING THROUGH HILBERT SPACES 194 7 4 THE KWAPIE N MAUREY THEOREMS FOR TYPE 2 SPACES 201 PROBLEMS 206 8 ABSOLUTELY SUMMING OPERATORS 209 8 1 GROTHENDIECK S INEQUALITY 210 8 2 ABSOLUTELY SUMMING OPERATORS 219 8 3 ABSOLUTELY SUMMING OPERATORS ON L1 SPACES AND AN APPLICATION TO UNIQUENESS OF UNCONDITIONAL BASES 228 PROBLEMS 232 9 PERFECTLY HOMOGENEOUS BASES AND THEIR APPLICATIONS 235 9 1 PERFECTLY HOMOGENEOUS BASES 235 9 2 SYMMETRIC BASES 242 9 3 UNIQUENESS OF UNCONDITIONAL BASIS 245 9 4 COMPLEMENTATION OF BLOCK BASIC SEQUENCES 246 9 5 THE EXISTENCE OF CONDITIONAL BASES 250 PROBLEMS 255 10 GREEDY TYPE BASES 259 10 1 GENERAL FRAMEWORK 260 10 2 QUASI GREEDY BASES 263 10 3 DEMOCRATIC BASES 273 10 4 GREEDY BASES 275 10 5 ALMOST GREEDY BASES 280 10 6 GREEDY BASES AND DUALITY 284 10 7 THE ZOO OF GREEDY LIKE BASES IN A BANACH SPACE 288 PROBLEMS 290 11 P SUBSPACES OF BANACH SPACES 295 11 1 RAMSEY THEORY 295 11 2 ROSENTHAL S 1 THEOREM 300 11 3 TSIRELSON SPACE 303 PROBLEMS 307 12 FINITE REPRESENTABILITY OF P SPACES 311 12 1 FINITE REPRESENTABILITY 311 12 2 THE PRINCIPLE OF LOCAL RE EXIVITY 319 12 3 KRIVINE S THEOREM 323 PROBLEMS 334 13 AN INTRODUCTION TO LOCAL THEORY 337 13 1 THE JOHN ELLIPSOID 338 13 2 THE CONCENTRATION OF MEASURE PHENOMENON 342 13 3 DVORETZKY S THEOREM 346 13 4 THE COMPLEMENTED SUBSPACE PROBLEM 352 PROBLEMS 357 14 NONLINEAR GEOMETRY OF BANACH SPACES 361 14 1 VARIOUS CATEGORIES OF NONLINEAR MAPS 362 14 2 THE LIPSCHITZ EMBEDDING PROBLEM 374 14 2 1 EXISTENCE OF DERIVATIVES OF LIPSCHITZ MAPS 376 14 2 2 LINEARIZATION OF LIPSCHITZ EMBEDDINGS 384 14 2 3 INVARIANCE OF THE LOCAL STRUCTURE UNDER COARSE LIPSCHITZ EMBEDDINGS 392 14 3 LIPSCHITZ ISOMORPHISMS BETWEEN BANACH SPACES 393 14 3 1 LINEARIZATION OF LIPSCHITZ RETRACTIONS 397 14 3 2 BANACH SPACES DETERMINED BY THEIR LIPSCHITZ STRUCTURE 400 14 4 LINEAR INVERSES TO NONLINEAR ISOMETRIC EMBEDDINGS 402 14 4 1 DERIVATIVES OF CONVEX FUNCTIONS ON FINITE DIMENSIONAL SPACES 402 14 4 2 THE STRUCTURE OF INTO ISOMETRIES 407 14 5 UNIFORM HOMEOMORPHISMS BETWEEN BANACH SPACES 409 14 5 1 THE COARSE LIPSCHITZ AND UNIFORM STRUCTURES OF P 411 14 6 LIPSCHITZ INVARIANCE OF ASYMPTOTIC SMOOTHNESS 417 PROBLEMS 422 15 IMPORTANT EXAMPLES OF BANACH SPACES 427 15 1 A GENERALIZATION OF THE JAMES SPACE 427 15 2 CONSTRUCTING BANACH SPACES VIA TREES 432 15 3 PE CZY NSKI S UNIVERSAL BASIS SPACE 434 15 4 THE JAMES TREE SPACE 436 CHAPTER 1 BASES AND BASIC SEQUENCES IN THIS CHAPTER WE ARE GOING TO INTRODUCE THE FUNDAMENTAL NOTION OF A SCHAUDER BASIS OF A BANACH SPACE AND THE CORRESPONDING NOTION OF A BASIC SEQUENCE ONE OF THE KEY IDEAS IN THE ISOMORPHIC THEORY OF BANACH SPACES IS TO USE THE PROPERTIES OF BASES AND BASIC SEQUENCES AS A TOOL TO UNDERSTANDING THE DIFFERENCES AND SIMILARITIES BETWEEN SPACES THE SYSTEMATIC USE OF BASIC SEQUENCE ARGUMENTS ALSO TURNS OUT TO SIMPLIFY SOME CLASSICAL THEOREMS AND WE ILLUSTRATE THIS WITH THE EBERLEIN SMULIAN THEOREM ON WEAKLY COMPACT SUBSETS OF A BANACH SPACE BEFORE PROCEEDING LET US REMIND THE READER THAT OUR CONVENTION WILL BE THAT ALL BANACH SPACES ARE REAL UNLESS OTHERWISE STATED IN FACT THERE IS VERY LITTLE CHANGE IN THE THEORY IN SWITCHING TO COMPLEX SCALARS BUT TO AVOID KEEPING TRACK OF MINOR NOTATIONAL CHANGES IT IS CONVENIENT TO RESTRICT OURSELVES TO THE REAL CASE OCCASIONALLY WE WILL GIVE PROOFS IN THE COMPLEX CASE WHEN IT APPEARS TO BE USEFUL TO DO SO IN OTHER CASES THE READER IS INVITED TO CONVINCE HIMSELF THAT HE CAN OBTAIN THE SAME RESULT IN THE COMPLEX CASE 1 1 SCHAUDER BASES THE BASIC IDEA OF FUNCTIONAL ANALYSIS IS TO COMBINE THE TECHNIQUES OF LINEAR ALGEBRA WITH TOPOLOGICAL CONSIDERATIONS OF CONVERGENCE IT IS THEREFORE VERY NATURAL TO LOOK FOR A CONCEPT TO EXTEND THE NOTION OF A BASIS OF A NITE DIMENSIONAL VECTOR SPACE IN THE CONTEXT OF HILBERT SPACES ORTHONORMAL BASES HAVE PROVED A VERY USEFUL TOOL IN MANY AREAS OF ANALYSIS WE RECALL THAT IF EN 1 ND1 IS AN ORTHONORMAL BASIS OF A HILBERT SPACE H THEN FOR EVERY X 2 H THERE IS A UNIQUE SEQUENCE OF SCALARS AN 1 ND1 GIVEN BY AN D HX ENI SUCH THAT 1