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yeyejingjing

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[求助] show that the space of polynomials is not complete 已有1人参与

How to show that the space of polynomials is not complete. And the norm is defined by the maximum of the absolute value of the coefficient of polynomial.
For example, p(x)=\sum_{k=0}^n c_{k}x^{k}, ||p||=max_{k}|c_{k}|.

Help~!!!!
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junefi

铁杆木虫 (正式写手)

【答案】应助回帖

感谢参与,应助指数 +1
Consider a sequence of polynomials . We can see that is Cauchy (with respect to the above norm) but actually which is not in the space of polynomials.
理论改变世界!
2楼2015-10-28 15:11:39
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yeyejingjing

新虫 (初入文坛)

引用回帖:
2楼: Originally posted by junefi at 2015-10-28 15:11:39
Consider a sequence of polynomials \left\{ {{p_n};{p_n} = \sum\limits_{i = 0}^n {\frac{{{x^i}}}{{i!}}} } \right\}. We can see that \left\{ {{p_n}} \right\} is Cauchy (with respect to the above norm)  ...

这个不对吧,,如果是这个序列的话,,||pn||=max{1/i!}=1
3楼2015-10-28 15:18:05
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junefi

铁杆木虫 (正式写手)

【答案】应助回帖

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yeyejingjing(Edstrayer代发): 金币+10, good 2015-10-29 02:13:31
引用回帖:
3楼: Originally posted by yeyejingjing at 2015-10-28 15:18:05
这个不对吧,,如果是这个序列的话,,||pn||=max{1/i!}=1...

1, The definition of a complete metric space M: if every Cauchy sequence in M converges in M.
2, is Cauchy: for , .
3, is not in the space of polynomials.
理论改变世界!
4楼2015-10-28 15:28:20
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yeyejingjing

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引用回帖:
4楼: Originally posted by junefi at 2015-10-28 15:28:20
1, The definition of a complete metric space M: if every Cauchy sequence in M converges in M.
2, \left\{ {{p_n}} \right\} is Cauchy: for n<m, \left\| {{p_n} - {p_m}} \right\| = \left\| {\sum\lim ...

嗯嗯,,,我也看明白了,,非常感谢啊
5楼2015-10-28 15:59:04
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