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    One Dimensional Normed Linear Space : Completeness and Continuity

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    Suppose that E is a one-dimensional normed linear space.
    Prove that E is complete and that each linear functional on E is continous.

    © BrainMass Inc. brainmass.com March 4, 2021, 6:12 pm ad1c9bdddf
    https://brainmass.com/math/vector-calculus/one-dimensional-normed-linear-space-completeness-continuity-34456

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    Solution:

    a) E = normed one-dimensional vector space
    Let X = vector in E, that means X can be expressed as
    X = a*U (1)
    where U = one-dimensional basis of E, (a) belongs to the field on which E is defined as vector space (in general, real or complex number)
    We can choose the basis so that U = unitary vector, that is ||U|| = 1
    In this case, it is easy to see that (a) is just ||X||.
    Let's consider now a Cauchy sequence X(n) in E:
    ||X(m+n) - X(m)|| < eps , ...

    Solution Summary

    Completeness and Continuity are investigated for a One Dimensional Normed Linear Space.

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