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Math Help - linear functionals, closed sets, convergence

  1. #1
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    linear functionals, closed sets, convergence

    This is the last round of questions, I promise.

    (1) Identify the space of all continuous linear functionals on the space c_0.

    (2) X=(C([0, 2], \mathbb{R}), || . ||_2),
    Y=\{f \in C([0, 2], \mathbb{R}: f(1-x)=f(1+x), x \in [0,1]\}
    Is Y closed vector space in X? What is the complement of Y?

    (3) Prove that the series
    \sum_{k=1}^{\infty}x_k b_k converge for every vector
    x=(x_k) \in l^p, 1<p<\infty if and only if b=(b_k) \in l^q,  1/p +1/q=1.

    These are the last ones I don't know what to do with, and if anyone has the time and is willing to help, I would be really grateful.
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  2. #2
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    Quote Originally Posted by marianne View Post
    (1) Identify the space of all continuous linear functionals on the space c_0.
    The dual space of c_0 is (isometrically isomorphic to) l^1. If x=(x_n)\in c_0 and y=(y_n)\in l^1 then the value of the functional y at the point x is \sum x_ny_n. This is a very standard result which you should be able to find in any relevant textbook.

    Quote Originally Posted by marianne View Post
    (2) X=(C([0, 2], \mathbb{R}), || . ||_2),
    Y=\{f \in C([0, 2], \mathbb{R}): f(1-x)=f(1+x), x \in [0,1]\}
    Is Y closed vector space in X? What is the complement of Y?
    This is more interesting. Take the second question first. If g(x) is in the orthogonal complement of Y, then \int_0^2f(x)g(x)\,dx=0 for all f in Y. But \int_0^2f(x)g(x)\,dx = \int_0^1f(x)g(x)\,dx + \int_1^2f(x)g(x)\,dx. If you change variables in the two integrals, putting y=1-x in the first and y=1+x in the second, then you find that \int_0^1f(1-y)\bigl(g(1-y)+g(1+y)\bigr)dy=0 (remembering that f(1+y)=f(1-y)). This holds for all continuous functions on [0,1], so you conclude that g(1+y)=–g(1-y) for all y in [0,1].

    Thus the orthogonal complement of Y is Z=\{g \in C([0, 2], \mathbb{R}): g(1-x)=-g(1+x), x \in [0,1]\}. If you repeat the same argument on the subspace Z, you will find that its orthogonal complement is Y. Therefore Y is equal to its second orthogonal complement, which is closed in X. Hence Y is closed.

    Quote Originally Posted by marianne View Post
    (3) Prove that the series
    \sum_{k=1}^{\infty}x_k b_k converge for every vector
    x=(x_k) \in l^p, 1<p<\infty if and only if b=(b_k) \in l^q,  1/p +1/q=1.
    This is another standard piece of bookwork. Look it up in any good textbook. The proof uses Hölder's inequality.
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