[分析] 實變問題

[BA0954016 (YES)]

想請問這題應該如何著手證明??
Let T be a linear functional on a normed linear space X. Show that T is
bounded if and only if the continuity property:if {fn}→f in X, then
 {T(fn)}→T(f) holds.

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◆ From: 124.12.34.190
推 jacky7987   :bounded就continuous 那件事跟continuous等價
→ jacky7987   :這樣想可以嗎@@
→ THEJOY      :題目就是在問這件事啊大哥
→ THEJOY      :http://en.wikipedia.org/wiki/Bounded_operator
推 jacky7987   :我以為那個可以直接用阿QQ
推 herstein    :題目就是要你證bounded跟continuous等價
→ c76068      :用反證法 假設T連續 unbd,則 存在{fn},|Tfn| > n|fn|
→ c76068      :令gn=fn/n|fn|, 則gn 收斂到0 但|Tgn|>1