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I - Introduction

TATSUO KAWATA - No affiliation found

1972/01/01 Simple chapter DOI: 10.1016/B978-0-12-403650-5.50005-9 Journal:

Abstract:

Publisher SummaryThis chapter discusses the basic notions and results from real and complex function theories and probability theory. It discusses the Fubini–Tonelli theorem, Lebesgue decomposition theorem, Jensen's theorems, Carleman's theorems, maximum modulus theorem, and theorems of Phragmen–Lindelöf. The notion of a function of bounded variation is generalized to multidimensional functions. A nondecreasing function is equivalent to the assignment of a measure, and the total variation of a function of bounded variation is an additive interval function. A linear operator is bounded if and only if it is uniformly continuous, and it is uniformly continuous if and only if it is continuous at an element.