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Math Help - dimension

  1. #1
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    dimension

    I have a homework problem that I can't figure out...

    Is the space of all real valued continuous functions defined on the interval [0,1] finite dimensional?

    please help me! thanks.
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  2. #2
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    Quote Originally Posted by JackieJo View Post
    I have a homework problem that I can't figure out...

    Is the space of all real valued continuous functions defined on the interval [0,1] finite dimensional?

    please help me! thanks.
    No. Because the space of all real valued continous functions on [0,1] contains the space of all real polynomials over [0,1]. Can you see why it makes it infinite dimensional?
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  3. #3
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    because it has infinite linearly dependent elements? is that right?
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  4. #4
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    Quote Originally Posted by JackieJo View Post
    because it has infinite linearly dependent elements? is that right?
    You need to be more formal. If \mathcal{C}[0,1] (continous functions) were to be a finite dimensionsal vector space then every subspace would be finite dimensional also, with a smaller degree (this is a theorem). But then \mathcal{P}[0,1] (polynomials functions) would be finite dimensional. But that is impossible. Why?
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  5. #5
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    because P is a subspace of C and P is infinite so C has to also be infinite dimensional? Thanks by the way
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  6. #6
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    Quote Originally Posted by JackieJo View Post
    because P is a subspace of C and P is infinite so C has to also be infinite dimensional? Thanks by the way
    Again, why is P[0,1] infinite? You need to be more formal. Here is how you show it cannot have a finite basis. Suppose that P[0,1] has a finite basis \{ p_1(x),...,p_n(x)\} where p_i(x) are polynomials. Then it would mean any polynomial in P[0,1] can be expressed in the form a_1p_1(x)+...+a_np_n(x). Let f(x) be a polynomial with a larger degree then all of the p_i(x) polynomials. Then f(x)=a_1p_1(x)+...+a_np_n(x) is impossible because polynomials can only be equal if their have the same coefficients and the same degree, but it is impossible since the degree of f(x) exceedes the RHS.
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  7. #7
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    Thanks!

    I think I basically understand, I just didn't know how to represent it....so that's what I needed, thanks!
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