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Math Help - Closed set in Banach?

  1. #1
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    Closed set in Banach?

    Is set
    L=\{f\in X | \int_{-\infty}^{+ \infty} f(x)dx =0\} .
    closed in Banach space X?

    a) X = L^1 (-\infty, +\infty), b) X = L^2 (-\infty, +\infty)
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  2. #2
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    a) Is true because f\mapsto \int_{\mathbb{R}}f is a continuous linear functional on L_1(\mathbb{R}). If we denote by T this continuous mapping the space L is
    T^{-1}(0).

    b) Is false. To prove it just observe that there are dense subspaces in L_2(\mathbb{R}) formed by integrable functions and with integral 0. Look at the Haar system Haar wavelet - Wikipedia, the free encyclopedia.

    If the set of functions with integral 0 were closed, this would imply that the closure of the span of the Haar system, that is the whole L_2(\mathbb{R}), would be included in L, that certainly is not true.
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