[分析] closed subspace

看板Math作者 (Danny)時間12年前 (2012/04/26 09:32), 編輯推噓6(6推 0噓 21→)
留言27則, 7人參與, 最新討論串1/1
Let H=H1×H2 both these spaces are Hilbert spaces. P:H→H by P(x,y)=(0,y) (orthogonal projection) V is a closed subspace of H Then P(V) is also a closed subspace. Prove or disprove it. 小弟個人覺得這個論述是對的，但是不知道怎麼證明，請版上的大大幫忙！ -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 140.112.51.110
→
04/26 10:41, , 1F
H1=H2=|R, xy=1,x>0,y>0
04/26 10:41, 1F
→
04/26 10:48, , 2F
喔抱歉 看錯 是subspace
04/26 10:48, 2F
推
04/26 18:43, , 3F
P is continuous
04/26 18:43, 3F
推
04/26 19:00, , 4F
這樣應是拉回來closed
04/26 19:00, 4F
→
04/26 21:10, , 5F
我知道拉回來closed,我現在就是要證明送過去到底
04/26 21:10, 5F
→
04/26 21:11, , 6F
會不會是closed
04/26 21:11, 6F
推
04/26 21:38, , 7F
直覺看起來應該是會對...
04/26 21:38, 7F
推
04/27 02:14, , 8F
但直覺不一定是對的XD
04/27 02:14, 8F
→
04/27 09:40, , 9F
樓上說出我的苦惱了XD
04/27 09:40, 9F
推
05/02 15:30, , 10F
我覺得不對耶 Let e_n be a complete basis of a
05/02 15:30, 10F
→
05/02 15:30, , 11F
separable Hilbert space H. Let H_1 = <{e_{2n-1}}>
05/02 15:30, 11F
→
05/02 15:31, , 12F
H_2 = <{e_{2n}}>. H = H_1(+)H_2
05/02 15:31, 12F
→
05/02 15:32, , 13F
Let f_n = (n e_{2n-1}+e_{2n})/(n^2+1)^{1/2}
05/02 15:32, 13F
→
05/02 15:32, , 14F
Then {f_n} is orthonormal.
05/02 15:32, 14F
→
05/02 15:33, , 15F
Let V be the closure of <{f_n}>. Then {f_n} is a
05/02 15:33, 15F
→
05/02 15:33, , 16F
complete basis for V.
05/02 15:33, 16F
→
05/02 15:35, , 17F
Set y=Sum e_{2n}/n, from n=1 to infty
05/02 15:35, 17F
→
05/02 15:36, , 18F
Clearly, P(V) is dense in H_2.
05/02 15:36, 18F
→
05/02 15:37, , 19F
But I claim that y does not sit in P(V).
05/02 15:37, 19F
→
05/02 15:38, , 20F
This is so becuase each vector in V can be
05/02 15:38, 20F
→
05/02 15:38, , 21F
expanded in terms of f_n.
05/02 15:38, 21F
→
05/02 19:28, , 22F
y is not in P(V), then not closed??????
05/02 19:28, 22F
推
05/02 23:46, , 23F
Because P(V) is dense in H_2.
05/02 23:46, 23F
→
05/03 11:10, , 24F
but y is not convergent (in norm)
05/03 11:10, 24F
→
05/03 11:10, , 25F
sorry, u are right
05/03 11:10, 25F
→
08/13 16:49, , 26F
H1=H2=|R, x https://muxiv.com
08/13 16:49, 26F
→
09/17 14:45, , 27F
sorry, u ar https://daxiv.com
09/17 14:45, 27F
‣ 更多 Lpspace 的文章
文章代碼(AID): #1FcANA_B (Math)