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My understanding of the Hahn-Banach theorem is as follows: given a linear subspace $U$ of $V$, a linear functional $f$ defined on $U$, and a sub-linear functional $p$ which dominates $f$, there exists a linear functional $F$ on $V$ with $F(x) \leq p(x)$ for all $x \in V$ and $F(x) = f(x)$ for $x\in U$.

In Cybenko 1989, the proof of Theorem 1, the first step uses Hahn Banach. Specifically, $R$ is a proper subspace of $C(I)$ where $I$ is the unit cube, the author states that by the Hahn Banach theorem, there exists a linear functional $L$ on $C(I)$, with the property that $L\neq 0$ but $L(R) = 0$.

How does this follow from the Hahn Banach theorem? In particular, why does the Hahn Banach theorem state that there exists an $L$ is not just zero everywhere? Also, where is the dominating sublinear functional coming into play?

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    $\begingroup$ This is a consequence of the Hahn Banach theorem. It says that if $F$ is a closed proper subspace of $V$ and $x\in V\setminus F$, then there exists a nonzero $f\in V^*$ which is identically equal to zero on $F$ and s.t. $f(x)=\text{dist} (x, F)=\delta>0$. You can find its proof in every functional analysis textbook. Basically, the idea is to set $Z=\text{span}\{F, x\}$ and then functional $f:Z\rightarrow \mathbb{R}$ for which $f(z)=f(y+\lambda x)= \lambda \delta$ is majorized by the norm. $\endgroup$ – tree detective Aug 21 '17 at 19:48
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    $\begingroup$ Thank you @treedetective...I should have figured that out. Would you like to make your comment an answer so I can accept it and close this question? $\endgroup$ – The_Anomaly Aug 22 '17 at 19:36
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This is a consequence of the Hahn Banach theorem. It says that if $F$ is a closed proper subspace of $V$ and $x\in V\setminus F$, then there exists a nonzero $f\in V^*$ which is identically equal to zero on $F$ and such that $f(x)=\text{dist} (x, F)=\delta>0$. You can find its proof in every functional analysis textbook. Basically, the idea is to set $Z=\text{span}\{F, x\}$ and consider the functional $f:Z\rightarrow \mathbb{R}$ for which $f(z)=f(y+\lambda x)= \lambda \delta$, for $z\in Z$. This functional is majorized by the norm on $Z$. Then you apply the usual form of the Hahn-Banach theorem.

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