Every nontrivial linear functional is open

Question

Let $X$ be a normed linear space and let $f:X\to \mathbb K$ be a nontrivial linear functional. I want to prove that $f$ is open. I tried as follows:

Let $E$ be an open set in $X$ and let $y\in f(E)$. Then there is $x\in E$ such that $y=f(x)$. Since $E$ is open there exists $r>0$ such that $B_r(x)\subset E$. Let $z\in B_1(0)$ such that $f(z)=\delta>0$. If I can show that $(-\delta,\delta)\subset f(B_1(0))$, then I am through. But I could not show this. Please help me to resolve this.

This is true. I have taken this exercise from http://link.springer.com/book/10.1007%2F978-1-4419-7515-7 — Anupam, May 24 '15 at 04:58

score 4 · Accepted Answer · answered May 24 '15 at 05:12

4

Let $a\in (-\delta, \delta)$. Then it is easy to see that $z\in B_1(0) \Rightarrow \frac{a}{\delta}z\in B_1(0)$ and $f(\frac{a}{\delta}z) =a.$ Therefore, $(-\delta,\delta)\subseteq f(B_1(0))$

answered May 24 '15 at 05:12

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Every nontrivial linear functional is open

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