# Download e-book for iPad: Non-Archimedean Analysis: A Systematic Approach to Rigid by S. Bosch, U. Güntzer, R. Remmert

By S. Bosch, U. Güntzer, R. Remmert

ISBN-10: 0387125469

ISBN-13: 9780387125466

ISBN-10: 3540125469

ISBN-13: 9783540125464

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**Read e-book online Non-Archimedean Analysis: A Systematic Approach to Rigid PDF**

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By (i), m # I. Then for all positive integers x < m, x E S. But then by (ii), m E S, a contradiction. Therefore T = 0. 3 Theorem (fundamental theorem of arithmetic). Every positive integer is either I or it can be written in one and only one way as a product of positive primes. Proof The theorem is true ifn = Else, let n=n,n2, where 1. So let n> I. If n is prime, we are done. I, so n1, n2

For any set S. the power set Here A vB=AuB and A A ordered by set inclusion, is a lattice. for all A, Definition. Let E be an equivalence relation on a set X. and let a E I. The set of all elements in X that are in relation E to a is called the equivalence class of a under E and is denoted by E(a). Thai is, E(a)=(xEXIxEa). A subset C of X is called an equivalence class of E (or an E-class) in X if C = E(a) for some a in X. The set of all equivalence classes of E in X is called the quotient set of X by E and is writien X/E.

G. B —. C, and I:: C D. Then h(gf) = (hg)f Proof Clearly, h(gf) and (hg)f have the same domain A and the same codomain D. Let x E A. Then, by definition of the composite, h(gfXx) = h(gf(x)) = and (hg)f(x) = hg(f(x)) = h(g(f(x))). Hence, h( gf Xx) = (gf)f(x) for every x in A; therefore, h(gf) = (hg)f Definition. A mappingf A (a) B is injective (or one-to-one or I - I) ,ifor all x1 ,x2 E A. x1 (b) 0 x2 #f(x2) or. equivalent/v. f(x1) =f(x2) x1 = x2; surjective (or onto) '1 for every y E B, y =f(x) for some x E A.

### Non-Archimedean Analysis: A Systematic Approach to Rigid Analytic Geometry (Grundlehren der mathematischen Wissenschaften) by S. Bosch, U. Güntzer, R. Remmert

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