→ keroro321 :H1=H2=|R, xy=1,x>0,y>0 04/26 10:41
→ keroro321 :喔抱歉 看錯 是subspace 04/26 10:48
推 zombiea :P is continuous 04/26 18:43
推 jacky7987 :這樣應是拉回來closed 04/26 19:00
→ Lpspace :我知道拉回來closed,我現在就是要證明送過去到底 04/26 21:10
→ Lpspace :會不會是closed 04/26 21:11
推 herstein :直覺看起來應該是會對... 04/26 21:38
推 herstein :但直覺不一定是對的XD 04/27 02:14
→ Lpspace :樓上說出我的苦惱了XD 04/27 09:40
推 ppia :我覺得不對耶 Let e_n be a complete basis of a 05/02 15:30
→ ppia :separable Hilbert space H. Let H_1 = <{e_{2n-1}}> 05/02 15:30
→ ppia :H_2 = <{e_{2n}}>. H = H_1(+)H_2 05/02 15:31
→ ppia :Let f_n = (n e_{2n-1}+e_{2n})/(n^2+1)^{1/2} 05/02 15:32
→ ppia :Then {f_n} is orthonormal. 05/02 15:32
→ ppia :Let V be the closure of <{f_n}>. Then {f_n} is a 05/02 15:33
→ ppia :complete basis for V. 05/02 15:33
→ ppia :Set y=Sum e_{2n}/n, from n=1 to infty 05/02 15:35
→ ppia :Clearly, P(V) is dense in H_2. 05/02 15:36
→ ppia :But I claim that y does not sit in P(V). 05/02 15:37
→ ppia :This is so becuase each vector in V can be 05/02 15:38
→ ppia :expanded in terms of f_n. 05/02 15:38
→ Lpspace :y is not in P(V), then not closed?????? 05/02 19:28
推 ppia :Because P(V) is dense in H_2. 05/02 23:46
→ Lpspace :but y is not convergent (in norm) 05/03 11:10
→ Lpspace :sorry, u are right 05/03 11:10