By P.N. Natarajan

This is the second one, thoroughly revised and elevated version of the author’s first ebook, masking various new themes and up to date advancements in ultrametric summability idea. Ultrametric research has emerged as an enormous department of arithmetic in recent times. This ebook offers a quick survey of the learn to this point in ultrametric summability concept, that is a fusion of a classical department of arithmetic (summability concept) with a latest department of study (ultrametric analysis). a number of mathematicians have contributed to summability idea in addition to sensible research. The booklet will entice either younger researchers and more matured mathematicians who're seeking to discover new parts in research. The e-book can be priceless as a textual content if you happen to desire to concentrate on ultrametric summability theory.

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**Example text**

E. closure of of “generalized semiperiodic sequences”. 7 x = {xk } is called a “generalized semiperiodic sequence”, if for any > 0, there exist n, k0 ∈ N such that |xk − xk+sn | < , k ≥ k0 , s = 0, 1, 2, . . 44 4 Ultrametric Summability Theory Let Q denote the set of all generalized semiperiodic sequences. One can prove that Q is a closed linear subspace of ∞ . Further, whatever be K , ∞ Q⊆ r . r =1 When K is a complete, non-trivially valued, ultrametric field, ∞ Q= r . r =1 Whatever be K , we shall define for α > 0, ∞ α |xk |α < ∞ .

Some results of the classical Banach space theory which hold in the ultrametric setting too are the closed graph theorem, Banach-Steinhaus theorem and open mapping theorem. However, the classical Hahn–Banach theorem does not carry over to the ultrametric setting—this makes the situation more interesting. In the case of ultrametric Banach spaces, it may not be possible to extend a given continuous linear functional from a subspace to the entire space. The fault is not with the linear space X but with the underlying field K .

Also, n,k ∞ ∞ α ∞ |(Ax)n | ≤ n=0 n=0 k=0 ∞ |xk |α ≤ k=0 < ∞, so that {(Ax)n } ∈ α. 2 Steinhaus-Type Theorems 45 Necessity. Suppose A ∈ ( α , α ). We first note that sup |ank |α = Bn < ∞, k≥0 n = 0, 1, 2, . . For, if for some m, sup |amk |α = Bm = ∞, then, we can choose k≥0 a strictly increasing sequence {k(i)} of positive integers such that |am,k(i) |α > i 2 , i = 1, 2, . . Define the sequence {xk }, where xk = 1 am,k , k = k(i) 0, k = k(i) Then ∞ {xk } ∈ α, |xk |α = for, k=0 ∞ , i = 1, 2, .