欢迎访问 生活随笔！
TAG聚合

生活随笔

当前位置： 首页 > 编程资源 > 编程问答 >内容正文

编程问答

UA MATH523A 实分析1 度量空间 概念与定理总结

发布时间：2025/4/14 编程问答 42 豆豆
生活随笔 收集整理的这篇文章主要介绍了 UA MATH523A 实分析1 度量空间 概念与定理总结 小编觉得挺不错的,现在分享给大家,帮大家做个参考.

UA MATH523A 实分析1 集合论基础 概念与定理总结

    • 序关系
    • 度量空间

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

序关系

Partial order ∀x,y∈X\forall x,y \in X∀x,y∈X, RRR relation such that

  • xRxxRxxRx，∀x∈X\forall x \in X∀x∈X
  • xRy,yRx⇒x=yxRy, yRx \Rightarrow x=yxRy,yRx⇒x=y
  • xRy,yRz⇒xRzxRy, yRz \Rightarrow xRzxRy,yRz⇒xRz

    • 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尔积也是非空集合
    Zorn’s Lemma：如果偏序集的所有全序子集都有一个上界，那么这个偏序集有最大元
    Hausdorff Maximal Principle：每个偏序集都有一个最大的全序子集
    Well Ordering Principle (by Cantor 1883)：任意非空集合上都可以定义一个良序使之成为良序集

    度量空间

    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

  • ρ(x,y)=0\rho(x,y)=0ρ(x,y)=0 iff x=yx=yx=y
  • ∀x,y∈X\forall x,y \in X∀x,y∈X,ρ(x,y)=ρ(y,x)\rho(x,y)=\rho(y,x)ρ(x,y)=ρ(y,x)
  • ∀x,y,z∈X\forall x,y,z \in X∀x,y,z∈X, ρ(x,z)≤ρ(x,y)+ρ(y,z)\rho(x,z) \le \rho(x,y)+\rho(y,z)ρ(x,z)≤ρ(x,y)+ρ(y,z)

    • 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≠ϕB(r,x) \cap A \ne \phiB(r,x)∩A​=ϕ, B(r,x)∩AC≠ϕB(r,x)\cap A^C\ne \phiB(r,x)∩AC​=ϕ

    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=ϕint \bar E = \phiintEˉ=ϕ
    Separable has countable dense subset

    Proposition 0.22 Equivalent:

  • x∈Eˉx \in \bar Ex∈Eˉ
  • B(r,x)∩E≠ϕ,∀r>0B(r,x) \cap E \ne \phi, \forall r>0B(r,x)∩E​=ϕ,∀r>0
  • ∃{xn}⊂E\exists \{x_n\} \subset E∃{xn​}⊂E, xn→xx_n \to xxn​→x

    • 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(ϵ,zj)⊃E\cup_{j=1}^{\infty}B(\epsilon,z_j) \supset E∪j=1∞​B(ϵ,zj​)⊃E.

    Compact = Complete+Totally Bounded, totally bounded = bounded in real space.

    Theorem 0.25 Equivalent

  • EEE compact
  • Bolzano-Weierstrass: every sequence in E has a subsequence converges to a point in E
  • Heine-Borel: every open covering of E, exists a sub open covering of E

    • 总结

    以上是生活随笔为你收集整理的UA MATH523A 实分析1 度量空间 概念与定理总结的全部内容，希望文章能够帮你解决所遇到的问题。

    如果觉得生活随笔网站内容还不错，欢迎将生活随笔推荐给好友。