Reflexive space

Abstract: Let E1, E2 be two Banach spaces, and let f: E1 -* E2 be a mapping, that is "approximately linear". S. M. Ulam posed the problem: "Give conditions in order for a linear mapping near an approximately linear mapping to exist". The purpose of this paper is to give an answer to Ulam's problem. THEOREM. Consider E1, E2 to be two Banach spaces, and let f: E1 -> E2 be a mapping such that f (tx) is continuous in t for each fixed x. Assume that there exists 0 > 0 andp E [0, 1) such that IIf(x + y) f (x) f(A)lI 0. The verification of (3) follows by induction on n. Indeed the case n = 1 is clear because by the hypothesis we can find 0, that is greater or equal to zero, andp such that 0 < p < 1 with 11[f(2x)]/2 -f(x)ll (4) IIxIIp Assume now that (3) holds and we want to prove it for the case (n + 1). However this is true because by (3) we obtain II [f (2n 2x)]/2 n f(2x)llI nI I*2x)2P < m E 2m(P therefore Received by the editors December 1, 1977. AMS (MOS) subject classifications (1970). Primary 47H15; Secondary 39A15.

Abstract: The Concentration of Measure Phenomenon in the Theory of Normed Spaces.- Preliminaries.- The Isoperimetric Inequality on Sn?1 and Some Consequences.- Finite Dimensional Normed Spaces, Preliminaries.- Almost Euclidean Subspaces of A Normed Space.- Almost Euclidean Subspaces of ?{p}n Spaces, of General n-Dimensional Normed Spaces, and of Quotient of n-Dimensional Spaces.- Levy Families.- Martingales.- Embedding ?pm into ?1n.- Type and Cotype of Normed Spaces, and Some Simple Relations with Geometrical Properties.- Additional Applications of Levy Families in the Theory of Finite Dimensional Normed Spaces.- Type and Cotype of Normed Spaces.- Ramsey's Theorem with Some Applications to Normed Spaces.- Krivine's Theorem.- The Maurey-Pisier Theorem.- The Rademacher Projection.- Projections on Random Euclidean Subspaces of Finite Dimensional Normed Spaces.

Abstract: Introduction 1. Notation and preliminary background 2. Gaussian variables. K-convexity 3. Ellipsoids 4. Dvoretzky's theorem 5. Entropy, approximation numbers, and Gaussian processes 6. Volume ratio 7. Milman's ellipsoids 8. Another proof of the QS theorem 9. Volume numbers 10. Weak cotype 2 11. Weak type 2 12. Weak Hilbert spaces 13. Some examples: the Tsirelson spaces 14. Reflexivity of weak Hilbert spaces 15. Fredholm determinants Final remarks Bibliography Index.

Abstract: Support functionals for closed bounded convex subsets of a Banach space- Convexity and differentiability of norms- Uniformly convex and uniformly smooth Banach spaces- The classical renorming theorems- Weakly compactly generated banach spaces- The Radon-Nikodym theorem for vector measures