April 20, 2018 - Shawn Zhong

$Theorem 3.55 • Statement ○ If Σa_n is a series of complex numbers which converges absolutely ○ Then every rearrangement of Σa_n converges to the same sum • Proof ○ Let Σa_n^′ be a rearrangement of Σa_n with partial sum s_n^′ ○ By the Cauchy Criterion, given ε0, ∃N∈N s.t. § ∑_(i=n)^m▒|a_i | ε,∀m,n≥N ○ Choose p s.t. 1,2,…,N are all contained in the set {k_1,k_2,…,k_p } ○ Where k_1,…,k_p are the indices of the rearranged series ○ Then if np, a_1,…,a_N will be cancelled in the difference s_n−s_n^′ ○ So, |s_n−s_n^′ |≤ε⇒{s_n^′ } converges to the same value as {s_n } Limit of Functions • Definition ○ Let X,Y be metric spaces, and E⊂X ○ Suppose f:E→Y and p is a limit point of E ○ We write § f(x)→q as x→p, or § (lim)_(x→p)⁡f(x)=q ○ If ∃q∈Y s.t. § Given ε0, there exists δ0 s.t. § If 0d_X (x,p)δ, then d_Y (f(x),q)ε • Note ○ 0d_X (x,p)δ is the deleted neighborhood about p of radius δ ○ d_X and d_Y refer to the distances in X and Y, respectively • Relationship with sequence ○ Theorem 4.2 relates this type of limit to the limit of a sequence ○ Consequently, if f has a limit at p, then its limit is unique Theorem 4.3 • If f,g are complex function on E, we have • (f+g)(x)=f(x)+g(x) • (f−g)(x)=f(x)−g(x) • (fg)(x)=f(x)g(x) • (f/g)(x)=f(x)/g(x) where g(x)≠0 on E Theorem 4.4 (Algebraic Limit Theorem) • Let X be a metric space, E⊂X • Suppose p be a limit point of E • Let f,g be complex functions on E where ○ lim_(x→p)⁡f(x)=A ○ lim_(x→p)⁡g(x)=B • Then ○ lim_(x→p)⁡(f(x)+g(x))=A+B ○ lim_(x→p)⁡(f(x)g(x))=AB ○ lim_(x→p)⁡(f(x)/g(x) )=A/B where B≠0$

$Proposition 64 • Statement ○ Let n 0 ○ Every nonzero element in Z\/nZ is either a unit or a zero-divisor • Note ○ We don’t have this property in Z ○ In Z, the units are ±1, there are no zero-divisor ○ 2∈Z is not 0 or unit or zero-divisor • Proof ○ Suppose a ̅∈Z\/nZ is nonzero and not a unit ○ Then d≔(a,n) 1 ○ Write cd=a,md=n ○ Then a ̅m ̅=c ̅d ̅m ̅=c ̅n ̅=0 ̅ ○ Moreover, m ̅≠0 ̅ § Since md=n,1≤m≤n, and d 1 § m cannot be a multiple of n Field • Definition ○ Communitive ring R is called a field if ○ Every nonzero element of R is a unit ○ i.e. Every nonzero element of R have a multiplicative inverse • Examples ○ Q,R ○ ℂ § But not true for R2 with (r_1,r_2 )(r_1^′,r_2^′ )=(r_1 r_1^′,r_2 r_2^′ ) ○ Z\/pZ (p prime) § 1≤a≤p−1,(a,p)=1⇒a ̅∈Z\/pZ § Note: Z\/nZ is a field ⟺ n is prime Product Ring • If R_1,R_2 are rings, R_1×R_2 has the following ring structure • For addition, it s just the product as groups • For multiplication, (r_1,r_2 )(r_1^′,r_2^′ )=(r_1 r_1^′,r_2 r_2^′ ) with identity (1_(R_1 ),1_(R_2 ) ) Integral Domain • Definition ○ A communicative ring R is an integral domain (or just domain) if ○ R contains no zero-divisors • Example ○ Unites are not zero-divisors, so fields are domains ○ Z is a domain ○ Z\/nZ is a domain ⟺ it is a field ○ R_1×R_2 is a domain ⟺ one of them is trivial, and the other is a domain$

Shawn Zhong

Shawn Zhong

Math 521 - 4/11

Math 541 - 4/18

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