By P.N. Natarajan

ISBN-10: 8132216466

ISBN-13: 9788132216469

ISBN-10: 8132216474

ISBN-13: 9788132216476

Ultrametric research has emerged as an incredible department of arithmetic lately. This ebook offers, for the 1st time, a quick survey of the learn to this point in ultrametric summability conception, that is a fusion of a classical department of arithmetic (summability idea) with a contemporary department of study (ultrametric analysis). numerous mathematicians have contributed to summability conception in addition to useful research. The booklet will entice either younger researchers and more matured mathematicians who're trying to discover new parts in analysis.

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**Additional info for An Introduction to Ultrametric Summability Theory**

**Sample text**

K≥0 We now claim that sup |an+1,k − ank | < ∞. 1 Classes of Matrix Transformations 33 Suppose not. We can then choose a strictly increasing sequence {n(m)} of positive integers such that sup |an(m)+1,k − an(m),k | > m 2 , m = 1, 2, . . 6) k≥0 In particular, sup |an(1)+1,k − an(1),k | > 12 . 7) k≥0 Since an(1)+1,k − an(1),k → 0, k → ∞, given ε > 0 (we can suppose that ε < 1), we can choose a positive integer k(n(1)) such that sup |an(1)+1,k − an(1),k | < ε. 8), we have, sup |an(1)+1,k − an(1),k | > 12 , 0≤k

For a counterexample, see [1], Example 4, pp. 81–83. Ingleton proved the following result. 78) (The ultrametric Hahn-Banach theorem) Let X be a normed linear space and Y be an ultrametric normed linear space over K (K may even be trivially valued). 1), if and only if Y is spherically complete. We now prove the above theorem to exhibit the crucial role played by “Spherical Completeness” in proving the ultrametric version of the Hahn-Banach theorem. 1 Let A be a bounded linear transformation from M into Y , where M ⊆ X and let Y be spherically complete.

1 Classes of Matrix Transformations We now present for the first time a brief survey, though not exhaustive, of the work done so far on ultrametric summability theory. Divergent series have been the motivating factor for the introduction of summability theory both in classical as well as ultrametric analysis. Study of infinite matrix transformations in the classical case is quite an old one. Numerous authors have studied general matrix transformations or matrix P. N. 1007/978-81-322-1647-6_4, © The Author(s) 2014 29 30 4 Ultrametric Summability Theory transformations pertaining to some special classes for the past several decades.

### An Introduction to Ultrametric Summability Theory by P.N. Natarajan

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