locally convex real space and denseness of

Asked Nov 22 '22 at 17:52

Active Nov 22 '22 at 17:52

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This question was left as an exercise in my class of functional analysis and I am having a hard time proving this result.

Question: Let E be a locally convex real space and $\phi: E\to \mathbb{R}$ be a linear form. Show that if $\phi$ is not continuous then $ker(\phi) $ is dense in H.

Attempt: I am not able to make much progress on this question and I think I need some hints.

I have been following functional analysis by Rudin for this course.

asked Nov 22 '22 at 17:52

As far as I remember, it is a theorem in Rudin's book that a non-zerolinear functional is continuous if and only its kernel is not closed. – Jochen Nov 23 '22 at 21:22
This is one of the contrapositives of theorem 1.18 (c)=> (a) chapter 1 of Rudins analysis you are using – Julian Cederlöf Alamos Guerra Nov 29 '22 at 08:09
@JulianCederlöfAlamosGuerra Thank you very much! – Jan 04 '23 at 13:59

locally convex real space and denseness of

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