Biorthogonality and its Applications to Numerical Analysis by Claude Brezinski

By Claude Brezinski

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Proof. Á that is, we show that ²³  i ²)³ and  i ²)  ³ are disjoint and ²³  i ²)³ complements  i ²)  ³ up to ? We start with () Suppose  i ²)³ and  i ²)  ³ have a common point %. Then there is &  ) such that  ²%³ ~ & and &  )  such that  ²%³ ~ & . Thus, & £ & and  is not a single-valued function. 1 24 CHAPTER 1. SET-THEORETIC AND ALGEBRAIC PRELIMINARIES () If  i ²)³ does not complement  i ()  ) up to ? Á there will be at least one point % that does not belong to either of these sets ²for they are disjoint as shown above³.

Define ´µ ~ ¸  {¢  –  ²mod ³¹ ²  {³. In other words, ´µ ~ ¸  {¢ E  {¢  ~  b ¹. Then any two integers  and  are related in terms of [ h µ if and only if   ´µ . This is an equivalence relation. ³ (#) Let ? be a nonempty set and R ‹ ? d ? be a binary relation. ¹ we have with (? Á +) the “smallest” (by the contents of elements of ? d ? ) equivalence relation on ? , where each element forms a singletonclass, and + partitions ? into the “number” of classes corresponding to the quantity of all elements of ?

5 Corollary. Á @ Á  ] be a function and let E denote its equivalence kernel. Then ¢  there is a unique reducer function [? Á ? |E Á E ] and [? |E Á @ Á  ]  if  is surjectiveÁ the reducer  is bijective. 6 Example. Á cos!. # is the quotient set of ? Ecos ~ " f % b  ¢   { # 34 CHAPTER 1. SET-THEORETIC AND ALGEBRAIC PRELIMINARIES ~ cosi "cos%# making sÁ sOEcos Á Ecos ! the projection of ? ~ s on its quotient by Ecos . Á cos! 5, it is bijective. 5, cos k Ecos % ~ cos f % b  ¢   { ~ cos%À … Now we turn to a discussion on the partial order relation and all relevant notions and theorems, which we apply throughout the rest of the book.

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