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Homework Help: Space of continuous functions

  1. Apr 14, 2008 #1
    1. The problem statement, all variables and given/known data
    Let Ce([0,1], R) be the set of even functions in C([0,1], R), show that Ce is closed and not dense in C.


    2. Relevant equations



    3. The attempt at a solution

    I think I can solve this if I can show that even functions converge to even functions, but I can't quite figure out how to go about doing this...
     
  2. jcsd
  3. Apr 14, 2008 #2

    quasar987

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    This question does not make sense to me.

    A function is "even" is f(t)=f(-t). Here, for any t in (0,1], -t is out of [0,1] and thus f(-t) is not even defined.

    What do you mean by even?
     
  4. Apr 14, 2008 #3

    Dick

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    What does C([0,1],R) mean?
     
  5. Apr 14, 2008 #4

    quasar987

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    That would be the space of continuous functions on [0,1] no doubt.

    But do you see what it means for a fct to be even in this setting? :confused:
     
  6. Apr 14, 2008 #5

    Dick

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    I was wondering if it could mean continuous functions from [0,1]xR->R with 'even' meaning f(x,y)=f(x,-y). The failure of the question to make any obvious sense otherwise was giving me doubts.
     
  7. Apr 15, 2008 #6
    I'm sorry all. I meant that the space is [-1,1] rather than [0,1]. Sorry again and do appreciate any help.
     
  8. Apr 15, 2008 #7

    Dick

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    Then it's super easy. Take a sequence of even functions f_i converging to a function f. Take any point x, then f_i(x)->f(x) and f_i(-x)->f(-x). Need I say more?
     
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