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I am trying to show that there exist non zero bounded functional on $L^{\infty}(\mathbb{R})$ that vanishes on $C(\mathbb{R})$ (which is closed subspace under the $L^{\infty}$ norm) using Hahn Banach with the gauge $p(x):=dist(x,C(\mathbb{R}))$.

If I define a functional $f(\alpha x_0)=\alpha$, where $x_0\in L^{\infty}$ ($dist(x_0,C(\mathbb{R}))=1$) and is not continuous, I know that it can be extended to the whole $L^{\infty}(\mathbb{R})$ so that the extension, $F$ satisfies $F(x)\leq dist(x,C(\mathbb{R})) $.

In particular for every $x\in C(\mathbb{R})$, $F(x)\leq 0$.

How can I show that $F(x)=0$?

thank you

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If $x\in C(\mathbb{R} )$ then $-x\in C(\mathbb{R} )$ hence $F(-x ) \leq 0 $ but this implies that $F(x) \geq 0.$

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