[分析] 泛函分析Banach Space

看板Math作者 (林柏佐)時間14年前 (2011/05/05 22:40), 編輯推噓3(3推 0噓 13→)
留言16則, 3人參與, 最新討論串1/1
Peter Lax- Function Analysis Chap. 15 Exercise 12. (接在closed graph theorem之後) Show that for every infinite-dimensional Banach space there are linear subspaces of finite codimension that are not closed. (Hint: Use Zorn's Lemma) 請板上強者解答 -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 182.235.167.87
推
05/06 14:24, , 1F
fact: for each non-zero linear functional f,
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05/06 14:25, , 2F
Ker f is a codimension 1 subspace.
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Lemma: if f is not bdd., then either is Ker f.
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05/06 14:26, , 4F
(in fact, Ker(f) is dense in this space)
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05/06 14:26, , 5F
"neither"
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05/06 14:26, , 6F
therefore, it suffices to construct an unbdd f.
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05/06 14:27, , 7F
this is always possible, since
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05/06 14:27, , 8F
fact: a linear functional f can be constructed
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05/06 14:28, , 9F
by specifying its values on a given basis.
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05/06 14:28, , 10F
(NB: to prove the existence of linear space basis
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05/06 14:29, , 11F
we need transfinite induction like Zorn's lemma
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05/07 10:23, , 12F
Thanks a lot, and I have some idea.
05/07 10:23, 12F
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05/07 18:43, , 13F
but how could i prove the space is not closed?
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05/07 18:51, , 14F
sorry, i can prove it, thank you
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05/07 22:19, , 15F
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05/07 22:19, 15F
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05/08 12:32, , 16F
you can try to prove that Ker f is actually dense
05/08 12:32, 16F
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