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Math Help - Question from Hilbert spaces

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    Question from Hilbert spaces

    e_m(t)=e^{imt}, f_n=e_{-n} + n e_n (m \in Z, n \in N) and A = \overline{Lin \{ e_n| n \in N\}}, <br />
B = \overline{Lin \{ f_n| n \in N\}}. (Clousures in L^2(-\pi, \pi)). Are A+B closed? What is \overline{A+B} ?
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  2. #2
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    Quote Originally Posted by veljko View Post
    e_m(t)=e^{imt}, f_n=e_{-n} + n e_n (m \in Z, n \in N) and A = \overline{Lin \{ e_n| n \in N\}}, <br />
B = \overline{Lin \{ f_n| n \in N\}}. (Clousures in L^2(-\pi, \pi)). Are A+B closed? What is \overline{A+B} ?
    If you include 0 in the natural numbers then A+B will contain all the trigonometric polynomials and therefore \overline{A+B}= L^2(-\pi, \pi). If you do not allow 0 as a natural number then \overline{A+B}= \{x\in L^2(-\pi, \pi): \int_{-\pi}^\pi\!\!\!x(t)\,dt=0\} (the orthogonal complement of the constant functions in L^2(-\pi, \pi)).

    To see that A+B is not closed, you only need to find some element x\in L^2(-\pi, \pi) such that x\notin A+B. For that, notice that every x in A+B must be of the form \sum_{m\in\mathbb{Z}}x_me_m with \sum_{n=1}^\infty n^2|x_{-n}|^2 < \infty (and of course \sum_{m\in\mathbb{Z}}|x_m|^2<\infty to ensure that x\in L^2(-\pi, \pi)). So for example x = \sum_{n=1}^\infty \frac{e_{-n}}n \notin A+B.
    Last edited by Opalg; April 13th 2010 at 12:19 AM. Reason: corrected typos
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