ker(\phi)$ ...

Asked Dec 30 '22 at 16:25

Active Dec 30 '22 at 16:37

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The following question is from my assignment in Functional Analysis and I am not able to make any significant progress on it.

Question: Let $X$ be a normed vector space and let $\phi\in X^{*}\setminus\{0\}$ . Show that for every $a\in X \setminus \ker(\phi) $ , we have : $||\phi||=\frac{| \phi(a)|} { \operatorname{dist}(a, \ker(\phi))}$

Here $X^{*}=${continuous linear forms $X\to \mathbb{C}$ }$=L(X,\mathbb{K})$.

I am very sorry but I don't have any idea on how to prove the norm of $\phi$ equal to the RHS. I e-mailed my professor a week ago and he is not replying (due to holidays). I read my notes of relevant lecture again but I still can't figure this out.

Please guide me on how this result should be proved. I really need help.

edited Dec 30 '22 at 16:37

Mark

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asked Dec 30 '22 at 16:25

Remember that set difference is denoted by $\setminus$, not by $/$. – Mark Dec 30 '22 at 16:38
2

https://math.stackexchange.com/questions/1162140/formula-for-the-distance-to-the-kernel-of-a-functional – Evangelopoulos Foivos Dec 30 '22 at 16:39
This (mentionned above by Evangelopoulos Phoevos) answers your question: Formula for the distance to the kernel of a functional. – Anne Bauval Dec 30 '22 at 17:49
@AnneBauval Thank you from the bottom of my heart for help. I will have a look at it soon. – Dec 30 '22 at 19:34

If $X$ is a normed vector space and let $\phi \in X^{*} $/{0} . Then for every $a\in X/ker(\phi)$ ...

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