Pure Mathematics

## Naive Set Theory by Paul R. Halmos

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By Paul R. Halmos

From the studies: "...He (the writer) makes use of the language and notation of normal casual arithmetic to country the elemental set-theoretic evidence which a starting pupil of complicated arithmetic must know...Because of the casual approach to presentation, the booklet is eminently fitted to use as a textbook or for self-study. The reader may still derive from this quantity a greatest of figuring out of the theorems of set thought and in their uncomplicated significance within the learn of mathematics." - "Philosophy and Phenomenological Research".

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Additional resources for Naive Set Theory

Sample text

T£ = Ind^(7_£, where (r_^(m) = |detm|"*^£ be a holomorphic section in TT^. We assume that £ takes values in W( the standard Whittaker model of dugji)rl{u)du . 1) JYt This integral converges absolutely for 3ft(C) large enough, it has a holomorphic extension to the whole plane, which defines for g = 1 a Whittaker functional for wc for all < G C. l]. 2. Assume that £>n. Let Wa and WT be elements of W((r, ip) and W(T, tp~~l) respectively. Let a be a Schwartz function on Fn (row vectors).

Jxl***) firtm (zfanitM) *a(*)dSdg O 6. A(W^Tf8) CAN BE MADE CONSTANT ( N O N A R C H I M E D E A N CASE) In this chapter F is a nonarchimedean local field. Assumptions are as in section 4. 1 Proposition. Assume I > n. There is a choice of W G W(7r, ip) and £T,S € VpTt8, such that A(w,tTfS) = i, Vsec. Proof. 1) f _ JAnxVn f JX(ntt) W (xjn,t(m(a)u)) «- 1 (m(a))| det a | * + ^ T ) , ( i l ( y ) ; az)dxdadu Here tJ = m(J)u(y), J € Zn. Choose £T>5 to have support in Qn • V, where V is a small open compact subgroup of Hn, and such that fTj, is fixed by the elements of V.

The proofs run in analogy with the proofs of the Euler product expansions in the global case. 1) the proof of the Euler product expansion in case £ > n. We do this in a little different manner than in [G], so that the "translation" to the local field case is transparent. This can be seen in the case £ > n as well. 3). Finally, we conclude from a theorem of Bernstein, that A(W,£T}s) is a rational function in #~ # , where q is the number of elements in the residue field of F. 1. We assume, (in this section only), that Jb is a number field, A - its ring of adeles, W,T - irreducible, automorphic, cuspidal representations of G/(A) and GL n (A) respectively, V~ nontrivial character of k\A.