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yeyejingjing

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[ÇóÖú] show that the space of polynomials is not complete ÒÑÓÐ1È˲ÎÓë

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

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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.
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2Â¥2015-10-28 15:11:39
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yeyejingjing

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ÒýÓûØÌû:
2Â¥: 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)  ...

Õâ¸ö²»¶Ô°É£¬£¬Èç¹ûÊÇÕâ¸öÐòÁеϰ£¬£¬||pn||=max{1/i!}=1
3Â¥2015-10-28 15:18:05
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junefi

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yeyejingjing(Edstrayer´ú·¢): ½ð±Ò+10, good 2015-10-29 02:13:31
ÒýÓûØÌû:
3Â¥: Originally posted by yeyejingjing at 2015-10-28 15:18:05
Õâ¸ö²»¶Ô°É£¬£¬Èç¹ûÊÇÕâ¸öÐòÁеϰ£¬£¬||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.
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4Â¥2015-10-28 15:28:20
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yeyejingjing

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ÒýÓûØÌû:
4Â¥: 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 ...

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5Â¥2015-10-28 15:59:04
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