Home Analysis • A p-adic Property of Fourier Coefficients of Modular Forms by Guerzhoy P.

A p-adic Property of Fourier Coefficients of Modular Forms by Guerzhoy P.

By Guerzhoy P.

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18. Among other things, it contains some numbers that are infinitely large and some numbers (besides zero) that are infinitely small. " Yet another kind of infinity is the "number" of elements in an infinite set such as N or R; that "number" is called the c a r d i n a l i t y of the set. Some older mathematics books refer to it as an a c t u a l infinity, in contrast to the potential infinity mentioned above. f). 1. Some arithmetic of cardinalities is possible for instance, in later chapters we shall see that card(S x T) = max{card(S), card(T)} when S and T are infinite sets, and card(2 X) > card(X) for any set X.

If S is any set, then S ~ = {~}, since the only rule assigning to each element of O a corresponding element of S is the empty function. The set S ~ is also denoted S ~ If S is any nonempty set, then ~ s = ~, since there is no rule that assigns to each element of S a corresponding element of o. We emphasize that 2~ ~ {2~}. 38. An example using products. ~ : N U f E I I . ~ c A~ "T E C Note t h a t if C and the A~'s are finite sets, then I-I~Ec A~ is also a finite set; hence any finite intersection of finite unions can be represented as a finite union of finite intersections, and conversely.

36 we give an alternate definition t h a t is less intuitive but more precise, in terms of subsets of products of sets. We write f : X -~ Y to abbreviate the statement t h a t f is a function from X into Y. The set X is the d o m a i n of f , often abbreviated D o m a i n ( f ) or Dora(f). We say f is a function on X , or f is defined on X . The set Y is the c o d o m a i n of f. It should not be confused with the r a n g e of f , which is the set { f ( x ) : x E X}, often abbreviated R a n g e ( f ) or R a n ( f ) .

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