/ / / / / /

上一篇 下一篇 同標題 發表文章 文章列表

作者  jksmay (哈姆) 站內  97RA_I
標題  CH.2 Exercise 12
時間  2008/12/09 Tue 12:34:54

972201009
972201013

12.
                         1  d
Show that ther are f屬於L (R ) and
                              1  d                      1
a sequence {f_n} with f_n屬於L (R ) such that ∥f-f_n∥L →0,

but f_n(x)→f(x) for no x.

pf:

first,we construct a sequence {In} of intervals on R as follows:

Step1.I_1=[-1,0],I_2=[0,1]

Step2.I_3=[-2,-3/2],...,I_9=[1,3/2],I_10=[3/2,2]

.
.
.

Stepk.Divede [-k,k] into 2k^2 subintervals with length 1/k,

and enumerate these subintervals from left to right

and label them in the way of continuing the indices

of intervals constructed in previous steps.
                               ∞
Let k→∞, we obtain {In} with ∪ In=R and |In|→0 as n→∞
                               n=1
Let f_n=χ_In for all n and f≡0 on R
               1                  1
then f_n,f屬於L (R) and ∥f-f_n∥L = ∫f_n = |In| →0 as n→∞
                                     R
However,for all x屬於R,there exists {n_k} and {m_k} with n_k,m_k→∞

such that f   (x)=1 and f   (x)=0
           n_k           m_k

Hence lim  f_n(x) does not exist.
      n→∞




--
發信站 [中央數學  織夢天堂 bbs.math.ncu.edu.tw]
  •FROM [jksmay 從 pc219.math.ncu.edu.tw 發表]

上一篇 下一篇 同標題 發表文章 文章列表