8 CHAPTER 1.PRELIMINARY The othtThererexpectd oction of Fourer 1.2 Cardinality In following sections,we establishsome foundations on the set theory and the topology and geometry the E t this miliar with infinite elements?This requires the concept of the cardinality of a set For two sets with finite number of elements,it is clear which set contains more elements.For two sets with in me e if V2 B.th f()=2.A mapf:ABis called a bijection if f is both injective and surjective.Clearly,a map f:AB has a well-defined inverse,if and only if f is a bijection B are called to tion The cardinal number of natural numbers N is denoted by o.(Countable) Erample 1.Each infinite set contains a countable subset Erample 2.Countable uion of countable sets is countable. Proof.Array this union as an infinite square,and enumerate in a zigzag way. 0 Erample 3.All rational numbers Q is countable. Erample 4.Finite cartesian product of countable sets is countable Proof.Visualize this union as an infinitek dimensional cube,and enumerate in a zigzag way. ◇ Erample 5.The set of all real numbers R is not countable. (0.1l is not countable.We accept each real n om position.That we write0.25as0.249999999,1as0.99999. ete Now suppos (0,1]is countabl then we have an en meration for all mbers in(0,，小say that y is in The cardinality ofR is called The decimal representation shows that countable product of finite sets has cardinal number Erample 6.R,(0,1],(0,1],R all have same cardinal number Theorem 1.1.There docs not erist marimal cardinal number Proof.Given any set A.consider its power set 24.namely the set of all subsets of A.We ca show they have different cardinality.Otherwise,there exists a bijectionf:A24,where f(a) corresponds to a subset of A.Define a subset of A as follows: B={xz度f(z.8 CHAPTER 1. PRELIMINARY The second part begins with the rudiment of the function spaces, followed by an introduction to Fourier analysis. We study both Fourier series and Fourier transform together with their applications. The connection with real analysis is intimacy. There are also many unexpected connections of Fourier analysis to wide-ranging mathematical topics such as Number theory, Discrete geometry, Probability theory. We convey to the reader only a small portion of this fascinating subject. 1.2 Cardinality In following sections, we establish some foundations on the set theory and the topology and geometry of the Euclidean space. We assume the reader is familiar with basic notions of sets, operations between sets, etc. In this section, we address the following question: how to compare two sets with infinite elements? This requires the concept of the cardinality of a set. For two sets with finite number of elements, it is clear which set contains more elements. For two sets with infinite elements, which contains ’more’ elements relies on the mappings between them. A map f : A → B is an assignment to each element of A a unique element in B. f is called injective, if f(x) 6= f(y), for x 6= y. f is called surjective if ∀z ∈ B, there exists x ∈ A such that f(x) = z. A map f : A → B is called a bijection if f is both injective and surjective. Clearly, a map f : A → B has a well-defined inverse, if and only if f is a bijection. A and B are called to have same cardinality if there exists a bijection f : A → B, denoted by A ∼ B. Sometimes, we shall refer to the cardinal number of a set A, denoted by A ¯¯. The cardinal number of natural numbers N is denoted by ℵ0. (Countable) Example 1. Each infinite set contains a countable subset. Example 2. Countable union of countable sets is countable. Proof. Array this union as an infinite square, and enumerate in a zigzag way. Example 3. All rational numbers Q is countable. Example 4. Finite cartesian product of countable sets is countable. Proof. Visualize this union as an infinite k dimensional cube, and enumerate in a zigzag way. Example 5. The set of all real numbers R is not countable. Proof. We prove (0, 1] is not countable. We accept each real number in (0, 1] has a decimal repre￾sentation, which is unique if we don’t allow the appearance of all zeros after some position. That is we write 0.25 as 0.249999999..., 1 as 0.99999...., etc. Now suppose (0, 1] is countable, then we have an enumeration for all numbers in (0, 1], say 0.a11a12a13...., 0.a21a22a23..., ... We can choose bii ∈ {0, 1, 2, ..., 9} \ aii, for each i. Let y = 0.b11b22b33..., a moment of thought shows that y is indeed not in the enumeration list. A contradic￾tion. The cardinality of R is called ℵ1. The decimal representation shows that countable product of finite sets has cardinal number ℵ1. Example 6. R, (0, 1], [0, 1], R n all have same cardinal number ℵ1. Theorem 1.1. There does not exist maximal cardinal number. Proof. Given any set A, consider its power set 2A, namely the set of all subsets of A. We can show they have different cardinality. Otherwise, there exists a bijection f : A → 2 A, where f(a) corresponds to a subset of A. Define a subset of A as follows: B = {x|x /∈ f(x)}