BANACH SPACE THEORY: A CONCISE COURSE
CONTENTS 1 CLASSICAL BANACH SPACES FINITE-DIMENSIONAL SPACES THE L P SPACES THE C(K) SPACES HILBERT SPACE “NEOCLASSICAL” SPACES THE BIG QUESTIONS 2 PRELIMINARIES CONTINUOUS LINEAR OPERATORS CONTINUOUS LINEAR FUNCTIONALS ADJOINTS PROJECTIONS QUOTIENTS A CURIOUS APPLICATION 3 BASES IN BANACH SPACES SCHAUDER’S BASIS FOR C[0, 1] THE HAAR SYSTEM 4 BASES IN BANACH SPACES II A WEALTH OF BASIC SEQUENCES DISJOINTLY SUPPORTED SEQUENCES IN L P AND ?P EQUIVALENT BASES 5 BASES IN BANACH SPACES III BLOCK BASIC SEQUENCES 6 SPECIAL PROPERTIES OF C0, ?1, AND ?? THE SECRET LIFE OF ?? 7 BASES AND DUALITY L P SPACES BASIC INEQUALITIES CONVEX FUNCTIONS AND JENSEN’S INEQUALITY A TEST FOR DISJOINTNESS CONDITIONAL EXPECTATION L P SPACES II THE RADEMACHER FUNCTIONS KHINCHINE’S INEQUALITY THE KADEC–PELCZY´NSKI THEOREM L P SPACES III UNCONDITIONAL CONVERGENCE ORLICZ’S THEOREM 11 CONVEXITY STRICT CONVEXITY NEAREST POINTS SMOOTHNESS UNIFORM CONVEXITY CLARKSON’S INEQUALITIES AN ELEMENTARY PROOF THAT L?= LQ 12 C(K) SPACES THE CANTOR SET COMPLETELY REGULAR SPACES 14 THE DUNFORD–PETTIS PROPERTY 15 C(K) SPACES II THE STONE– ?CECH COMPACTI?CATION RETURN TO C(K) 16 C(K) SPACES III THE STONE– ?CECH COMPACTI?CATION OF A DISCRETE SPACE A FEW FACTS ABOUT ?N “TOPOLOGICAL” MEASURE THEORY THE DUAL OF ?? THE RIESZ REPRESENTATION THEOREM FOR C(?D) CHAPTER |XN|P 1/P AND TAKE ?P TO BE THE COLLECTION OF THOSE X ?S FOR WHICH ?X?P < ? THE INEQUALITIES OF H¨OLDER AND MINKOWSKI SHOW THAT ?P IS A NORMED SPACE; FROM THERE IT’S NOT HARD TO SEE THAT ?P IS ACTUALLY A BANACH SPACE THE SPACE ?P IS DE?NED IN EXACTLY THE SAME WAY FOR 0 < P < 1 BUT, IN THIS CASE, ? ?P DE?NES A COMPLETE QUASI-NORM THAT IS, THE TRIANGLE INEQUALITY NOW HOLDS WITH AN EXTRA CONSTANT; SPECI?CALLY, ?X + Y?P ?21/P(?X?P + ?Y?Y) P |XI|P 1/P FOR P < ?, AND ?X??= MAX 1?I?N |XI| FOR P = ? A SHORT COURSE ON BANACH SPACE THEORY FOR ALL X = (X1, , XN) IN RN THUS, CONVERGENCE IN ANY NORM ON RN IS THE SAME AS “COORDINATEWISE” CONVERGENCE AND, HENCE, EVERY NORM ON RN IS COM- PLETE BECAUSE EVERY ?NITE-DIMENSIONAL NORMED SPACE IS JUST “RN IN DISGUISE,” IT FOLLOWS THAT EVERY ?NITE-DIMENSIONAL NORMED SPACE IS COMPLETE THE L P SPACES WE ?RST DE?NE THE VECTOR SPACE L0[0, 1] TO BE THE COLLECTION OF ALL (EQUIVALENCE CLASSES, UNDER EQUALITY ALMOST EVERYWHERE [A E ], OF) LEBESGUE-MEASURABLE FUNCTIONS F : [0, 1] ??R FOR OUR PURPOSES, L0 WILL SERVE AS THE “MEASURABLE ANALOGUE” OF THE SEQUENCE SPACE S FOR 1 ?P < ?, THE BANACH SPACE L P[0, 1] CONSISTS OF THOSE F ?L0[0, 1] FOR WHICH ?F ?P = | F (X)| = INF {B : | F | ?B A E } (IN PRACTICE, THOUGH, WE OFTEN JUST WRITE “SUP” IN PLACE OF “ESS SUP”) AGAIN, THE INEQUALITIES OF H¨OLDER AND MINKOWSKI PLAY AN IMPORTANT ROLE HERE AS BEFORE, THE SPACES L P[0, 1] ARE ALSO DE?NED FOR 0 < P < 1, BUT ? ?P DE?NES ONLY A QUASI-NORM AGAIN, D( F, G) = ?F ?G?P P DE?NES A COMPLETE, TRANSLATION-INVARIANT METRIC ON L P FOR 0 < P < 1 THE SPACE L0[0, 1] IS GIVEN THE TOPOLOGY OF CONVERGENCE (LOCALLY) IN MEASURE FOR LEBESGUE MEASURE ON [0, 1], THIS TOPOLOGY IS KNOWN TO BE EQUIVALENT TO THAT GIVEN BY THE METRIC D( F, G) = | F (X) ?G(X)| 1 + | F (X) ?G(X)| DX (1 5) AS SUBSETS OF L0[0, 1], WE HAVE THE FOLLOWING INCLUSIONS: L1[0, 1] ?L P[0, 1] ?LQ[0, 1] ?L?[0, 1], (1 6) FOR ANY 1 < P < Q < ? MOREOVER, THE INCLUSION MAPS ARE ALL NORM ONE: ?F ?1 ??F ?P ??F ?Q ??F ?? (1 7)