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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.

Solution Summary

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

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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 , ...

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