real-analysis-class38 实变函数论应用数学讲义.pdf

實變函數論─應用數學系 吳培元老師  Class 38 Sec.3.4. Complete metric spaces Advanced Calculus: Cantor intersection theorem  F compact,  R n , F  F  n n  n n Note1. False if Fn not comapct 1 Ex.1. F (0, ) R F n   n  n n 1 Ex.2. F [0, ] R F 0 n   n  n n Note2. (X , ) metric space    Fn compact Fn  n Pf: Let x F n n Then x F compact   1 n F1 sequentially compact xnk  xnk x in F1   , xnk xnk 1,... x | F closed k nk    x F k nk    x F n n Thm. (X ,  ) complete metric space       F X closed, F , d (F ) 0 F x n n n  n  n Note: If only assume F bdd but d (F )  0 , then F may be  (Ex. 3.4.3 3.4.4) n n n n Pf: Existence: Let x F n n (xn , xm ) d (Fn ) 0 if m n large x Cauchy sequence   n x X  x x n F n x F n ( x , x ,... x) n  n n1  x  F |