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    Topological Vector Space : Closed Kernel

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    Suppose that T is a topologocal vector space. Prove that a linear functional f on T is continuous if and only if ker(f) is closed.

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    https://brainmass.com/math/vector-calculus/topological-vector-space-closed-kernel-34000

    Solution Preview

    let f: X ---> R be a non-zero functional from X, a normed
    space (any topological vector space will do).

    The following are equivalent:

    1) f is continuous.
    2) the null space (or kernel) N(f) is closed.
    3) N(f) is not dense in X.
    4) f is bounded on some neighbourhood of 0.

    1) --> 2): this is easy, as N(f) = f^-1[{0}] and {0}
    is a closed set in ...

    Solution Summary

    It is proven that a linear functional f on a toplogical vector space is continuous if and only if ker(f) is closed.

    $2.49

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