※ 引述《gp3gp3 (gp3gp3)》之銘言:
: 原題目如下:
: Consider a sequence of functions f (x) (以下用fn(x)表示)
: n
: ,each fn(x) is a bounded function. If fn(x) converges pointwise
: to f(x), is it true that f(x) is a bounded function?
: If fn(x) converges uniformly to f(x),
: is it true that f(x) is a bounded function? (題目完)
: 第二個部分(converges uniformly的情形)我有證出來,
: 然後 我直覺覺得第一個converges pointwise時 答案應該是 不一定
: 可是又想不到反例@@ 請教了~ 謝謝!!
先隨便找個 |R 上的 unbounded function,譬如 f(x) = x.
定義 f_n(x) = / f(x) ,if x in {s| |f(s)| <= n}
|
\ n ,elsewhere.
顯然 |f_n(x)| <= n,且 f_n(x) ---> f(x) for any x in |R.
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※ 編輯: Serge45 來自: 140.114.232.55 (04/20 01:15)