By P.N. Natarajan

ISBN-10: 8132225589

ISBN-13: 9788132225584

ISBN-10: 8132225597

ISBN-13: 9788132225591

This is the second one, thoroughly revised and extended variation of the author’s first publication, overlaying a variety of new subject matters and up to date advancements in ultrametric summability thought. Ultrametric research has emerged as an immense department of arithmetic in recent times. This e-book provides a short survey of the study to this point in ultrametric summability idea, that is a fusion of a classical department of arithmetic (summability thought) with a contemporary department of study (ultrametric analysis). a number of mathematicians have contributed to summability thought in addition to practical research. The e-book will attract either younger researchers and more matured mathematicians who're trying to discover new parts in research. The publication is additionally valuable as a textual content when you desire to concentrate on ultrametric summability theory.

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

**Example text**

There are equivalent ways of describing spherical completeness ([1], Chap. 2). The idea that spherical completeness should be substituted in the ultrametric set up whenever completeness appears in the classical case is not, however, true. In fact, spherical completeness plays a very little role in the case of ultrametric Banach algebras and ultrametric measure theory. In fact, in some ways, it is disadvantageous: For instance, if K is spherically complete, then no infinite-dimensional ultrametric Banach space over K is reflexive ([2], Chap.

Sup |tk | = ∞. In view of Definition k∈W ∩S0 ϕ. Thus there is no minimal ultrametric full We define χϕ = {χ S /S ∈ ϕ}, where χ S denotes the characteristic function of the subset S of N. C A denotes the set of all sequences x = {xk } which are A-summable. As an application to matrix summability, we have the following result. 7 Let ϕ be an ultrametric full class and A = (ank ) be any infinite matrix. Then C A ⊇ χϕ if and only if (i) lim ank = 0, n = 1, 2, . . ; k→∞ and (ii) lim sup |an+1,k − ank | = 0 for every S ∈ ϕ.

K=0 it being assumed that the series on the right converge. If lim (Ax)n = , we say that n→∞ the sequence x = {xk } is A-summable or summable A to . 2 Let X and Y be sequence spaces with elements whose entries are in K . The infinite matrix A = (ank ), ank ∈ K , n, k = 0, 1, 2, . . is said to transform X to Y if whenever the sequence x = {xk } ∈ X , (Ax)n is defined, n = 0, 1, 2, . . and the sequence {(Ax)n } ∈ Y . In this case, we write A ∈ (X, Y ). 3 If A ∈ (c, c) (where c is the ultrametric Banach space consisting of all convergent sequences in K with respect to the norm defined by ||x|| = sup |xk |, k≥0 x = {xk } ∈ c), A is said to be convergence preserving or conservative.

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

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