[分析] 請教 C(X) 的一個問題

[wuxr (wuxr)]

請教各位先進一個問題

就是如果 X 是一個緊緻的拓墣空間

要證明(C(X), d) 是完備的賦距空間 d means the uniform norm of C(X)

關於證明我有一個疑問

Let <fn> be a Cauchy seq. in C(X)
For each x, def: f(x)=lim fn(x)  exists indeed.
Since <fn(x)> is also Cauchy in R

Next, we claim fn conv. to f uniformly.

Since  <fn> is Cauchy,
there exists N such that |fn(x)-fm(x)|< epsilon for all x and n, m > N

For each x,
there exists Nx>N such that |f_Nx(x)-f(x)|< epsilon

Hence |fn(x)-f(x)|<|fn(x)-f_Nx(x)|+|f_Nx(x)-f(x)|< 2 epsilon
for all n>N and x

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我的問題是, X是緊緻這件事情是為了保證所有的 fn 甚至是 f 是有界嗎?

謝謝^^

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