# How to use Hahn-Banach and Riesz theorems

I got stuck when reading a paper. Let $X$ be a compact Riemann surface, $L^{\infty}_{(0,1)}$(X) be a space of $(0,1)$-forms on $X$ with coefficients of class $L^{\infty}$, $H^{(1,0)}(X)$ be a space of holomorphic $(1,0)$ forms on $X$. Then author states that by Hahn-Banach and Riesz theorems for any $F \in (H^{(1,0)}(X))^{*}$ there exists $f \in L^{\infty}_{(0,1)}(X)$ such that $$\langle F, h \rangle = \int\limits_{X} f \wedge h$$ for any $h \in H^{(1,0)}(X)$. But I can't see how to derive that statement from Hanh-Banach theorem and classic Riesz representation theorem for Hilbert spaces. Please help me to understand how Hanh-Banach and Riesz theorems were used to prove that statement.

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Can you share some information of the paper you are reading? – Hui Yu Oct 29 '12 at 15:58
@HuiYu of course, math.jussieu.fr/~henkin/mp2-a-10.pdf page 12 last line – Nimza Oct 29 '12 at 16:02