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Math Help - Extending to closed space

  1. #1
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    Extending to closed space

    Let X be a normed vector space and M\subseteq X a closed subspace. Show that M+\mathbb{C}x is also closed.

    I was told to try using the Hahn-Banach theorem. So I know that there exists a function f\in X^{*} such that \parallel f\parallel =1 and f(x)=\delta. Where x\in X\cap M^c and \delta=\inf_{y\in M}\parallel x-y\parallel

    However, I don't see how to use this at all.
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  2. #2
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    Quote Originally Posted by putnam120 View Post
    Let X be a normed vector space and M\subseteq X a closed subspace. Show that M+\mathbb{C}x is also closed.

    I was told to try using the Hahn-Banach theorem. So I know that there exists a function f\in X^{*} such that \parallel f\parallel =1 and f(x)=\delta. Where x\in X\cap M^c and \delta=\inf_{y\in M}\parallel x-y\parallel

    However, I don't see how to use this at all.
    To show that M+\mathbb{C}x is closed, suppose that you have a convergent sequence m_k+\lambda_kx\to y. You need to show that the limit point y is also in M+\mathbb{C}x. From the continuity of your function f above, \lambda_k\delta = f(m_k+\lambda_kx)\to f(y) as k\to\infty. Deduce that m_k\to y-\delta^{-1}f(y)x. Since M is closed, it follows that z = y-\delta^{-1}f(y)x\in M, so that y = z + \delta^{-1}f(y)x\in M+\mathbb{C}x.
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  3. #3
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    Thanks. In the time between our post I had figured out that I would have to show that a sequence of that form converged to a point in M. However, I was stuck at that part. (I was thinking about using the fact that the bounded linear functionals separate points somehow)
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