By P.N. Natarajan

Ultrametric research has emerged as an incredible department of arithmetic lately. This booklet offers, for the 1st time, a short survey of the examine thus far in ultrametric summability concept, that's a fusion of a classical department of arithmetic (summability concept) with a contemporary department of research (ultrametric analysis). a number of mathematicians have contributed to summability idea in addition to useful research. The booklet will entice either younger researchers and more matured mathematicians who're seeking to discover new parts in analysis.

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**Example text**

P≥0 Hence there exists p(n(i)) such that |an(i)+1, p(n(i)) − an(i), p(n(i)) | > ε, i = 1, 2, . . 26) Suppose { p(n(i))} is bounded, then there are only a finite number of distinct entries in that sequence. Consequently, there exists p = p(n(m)) which occurs in the sequence { p(n(i))} infinite number of times. 26) will then contradict the existence of lim anp , p = 0, 1, 2, . . established earlier. 25) holds. Consider now the sequence {y p } defined as follows: xk( p)+1 , p = p(n(i)), p = p(n(i)), i = 1, 2, .

1 A set S of vectors is said to be “absolutely K -convex” if ax + by ∈ S whenever |a|, |b| ≤ 1 and x, y ∈ S; translates w + S of such sets S are called “K convex”. A topological linear space X (which is defined as in the classical case) is said to be “locally K -convex” if its topology has a base of K -convex sets at 0. Using the above notion of K -convexity, a weaker form of compactness, called “ccompactness” could be defined. c-compactness demands that filterbases composed of K -convex sets possess adherence points rather than the usual requirement of compactness that all filterbases possess adherence points (see [4]).

Strange terrain—non-archimedean spaces. Amer. Math. Montly 88, 667–676 (1981) 7. : p-Adic Analysis and Mathematical Physics. World Scientific, Singapore (1994) Chapter 4 Ultrametric Summability Theory Abstract In this chapter, our survey of the literature on “ultrametric summability theory” starts with a paper of Andree and Petersen of 1956 (it was the earliest known paper on the topic) to the present. Most of the material discussed in the survey have not appeared in book form earlier. Silverman-Toeplitz theorem is proved using the “sliding-hump method”.