UA MATH523A 实分析1 度量空间 概念与定理总结
UA MATH523A 實分析1 集合論基礎 概念與定理總結
- 序關系
- 度量空間
limit superior and lim inferior
lim?sup?Fn=?k=1∞?n=k∞Fnlim?inf?Fn=?k=1∞?n=k∞Fn\limsup F_n = \bigcap_{k=1}^{\infty} \bigcup_{n=k}^{\infty} F_n \\ \liminf F_n = \bigcup_{k=1}^{\infty} \bigcap_{n=k}^{\infty} F_nlimsupFn?=k=1?∞?n=k?∞?Fn?liminfFn?=k=1?∞?n=k?∞?Fn?
de Morgan’s law
(?α∈AFα)C=?α∈AFαC(?α∈AFα)C=?α∈AFαC\left( \bigcup_{\alpha \in A} F_{\alpha} \right)^C = \bigcap_{\alpha \in A} F_{\alpha}^C \\ \left( \bigcap_{\alpha \in A} F_{\alpha} \right)^C = \bigcup_{\alpha \in A} F_{\alpha}^C(α∈A??Fα?)C=α∈A??FαC?(α∈A??Fα?)C=α∈A??FαC?
For preimage
f?1(?α∈AEα)=?α∈Af?1(Eα)f?1(?α∈AEα)=?α∈Af?1(Eα)f?1(EαC)=(f?1(Eα))Cf^{-1}\left(\bigcup_{\alpha \in A} E_{\alpha}\right) = \bigcup_{\alpha \in A}f^{-1}(E_{\alpha}) \\ f^{-1}\left(\bigcap_{\alpha \in A} E_{\alpha}\right) = \bigcap_{\alpha \in A}f^{-1}(E_{\alpha}) \\ f^{-1}(E_{\alpha}^C) = (f^{-1}(E_{\alpha}))^Cf?1(α∈A??Eα?)=α∈A??f?1(Eα?)f?1(α∈A??Eα?)=α∈A??f?1(Eα?)f?1(EαC?)=(f?1(Eα?))C
序關系
Partial order ?x,y∈X\forall x,y \in X?x,y∈X, RRR relation such that
Denoted as (X,≤)(X,\le)(X,≤), if one of x≤y,y≤xx\le y,y\le xx≤y,y≤x holds, it is total order (linear order).
Axiom of Choice（by Zermelo 1904）：一列非空集合的笛卡爾積也是非空集合
Zorn’s Lemma：如果偏序集的所有全序子集都有一個上界，那么這個偏序集有最大元
Hausdorff Maximal Principle：每個偏序集都有一個最大的全序子集
Well Ordering Principle (by Cantor 1883)：任意非空集合上都可以定義一個良序使之成為良序集
度量空間
Metric Space(X,ρ)(X,\rho)(X,ρ) ，ρ:X×X→[0,∞)\rho:X\times X \to [0,\infty)ρ:X×X→[0,∞) is metric, if
Product measure ρ((x1,y1),(x2,y2))=max?{ρ1(x1,x2),ρ2(y1,y2)}\rho((x_1,y_1),(x_2,y_2)) = \max\{\rho_1(x_1,x_2),\rho_2(y_1,y_2)\}ρ((x1?,y1?),(x2?,y2?))=max{ρ1?(x1?,x2?),ρ2?(y1?,y2?)}
Open balls B(r,x)={z∈X:ρ(x,z)<r}B(r,x) = \{z \in X:\rho(x,z)<r\}B(r,x)={z∈X:ρ(x,z)<r}
Interior point：?x∈X\forall x \in X?x∈X, if ?r>0\exists r>0?r>0, B(r,x)?AB(r,x) \subset AB(r,x)?A
Exterior point：?x∈X\forall x \in X?x∈X, if ?r>0\exists r>0?r>0, B(r,x)?ACB(r,x) \subset A^CB(r,x)?AC
Boundary point：?x∈X\forall x \in X?x∈X, if ?r>0\exists r>0?r>0, B(r,x)∩A≠?B(r,x) \cap A \ne \phiB(r,x)∩A?=?, B(r,x)∩AC≠?B(r,x)\cap A^C\ne \phiB(r,x)∩AC?=?
Interior int(A)int(A)int(A), collection of all interior points
Boundary ?A\partial A?A, collection of all boundary points
ClosureAˉ\bar{A}Aˉ, the smallest closed set containing AAA, Aˉ=intA??A\bar A = int A \sqcup \partial AAˉ=intA??A
dense in X if Eˉ=X\bar E = XEˉ=X
nowhere dense intEˉ=?int \bar E = \phiintEˉ=?
Separable has countable dense subset
Proposition 0.22 Equivalent:
Proposition 0.23 f:X1→X2f:X_1 \to X_2f:X1?→X2? conti iff f?1(U)f^{-1}(U)f?1(U) open in X1X_1X1? for all open UUU in X2X_2X2?.
Cauchy ρ(xn,xm)→0\rho(x_n,x_m) \to 0ρ(xn?,xm?)→0 as n,m→∞n,m \to \inftyn,m→∞.
Complete all Cauchy sequences are convergence
Proposition 0.24 A closed subset of a complete metric space is complete. A complete subset of an arbitrary metric space is closed.
Totally Bounded ?zj∈E\exists z_j \in E?zj?∈E, ∪j=1∞B(?,zj)?E\cup_{j=1}^{\infty}B(\epsilon,z_j) \supset E∪j=1∞?B(?,zj?)?E.
Compact = Complete+Totally Bounded, totally bounded = bounded in real space.
Theorem 0.25 Equivalent
總結
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