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Topological Vector Space : Closed Kernel

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.

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.

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