By P.N. Natarajan

Ultrametric research has emerged as an incredible department of arithmetic lately. This ebook provides, for the 1st time, a quick survey of the learn so far in ultrametric summability idea, that's a fusion of a classical department of arithmetic (summability idea) with a latest department of study (ultrametric analysis). a number of mathematicians have contributed to summability concept in addition to sensible research. The booklet will attract either younger researchers and more matured mathematicians who're trying to discover new components in analysis.

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We recall that completeness means that every nested sequence of closed balls, whose diameters tend to 0, have non-empty intersection. , spherical completeness means that every nested sequence of closed balls have non-empty intersection. It is clear that spherical completeness is stronger than completeness. However, the converse is not true, a counterexample may not be easy! 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).

18) holds and ∞ ank = 1, k = 0, 1, 2, . . n=0 Leaving out the trivial part of the theorem, suppose x ∈ c0 − α and A ∈ ( α , α ; P) transforms every rearrangement of x into a sequence in α . Choose k(1) = 1 and a positive integer n(1) such that 52 4 Ultrametric Summability Theory ∞ |an,1 |α < 2−1 , n=n(1)+1 so that ∞ n(1) |an,1 |α = n=0 ∞ |an,1 |α − n=0 |an,1 |α n=n(1)+1 1 1 ≥1− = . 2 2 Having defined k( j), n( j), j ≤ m −1, choose a positive integer k(m) > k(m −1)+1 such that n(m−1) 1 |an,k(m) |α < 2−m , |xk(m) | < 2 mα n=0 and then choose a positive integer n(m) > n(m − 1) such that n(m) |an,k(m) |α ≥ n=n(m−1)+1 ∞ 1 , 2 |an,k(m) |α < 2−m .

75). Many results about classical and ultrametric Banach spaces have exactly same formal statements but the proofs are entirely different. For instance, consider the classical theorem that locally compact Banach spaces must be finite dimensional. The proof of this theorem involves use of unit vectors. In the ultrametric case, we cannot use this technique. We consider various cases and vectors whose norms are close to 1 and finally conclude that locally compact ultrametric Banach spaces must be finite dimensional and the underlying field K must be locally compact ([1], p.

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