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There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$ $L^{1}/H^{1}_{0}$ is weakly sequentially complete, i.e., to say if we have a sequence of weak* continuous linear functionals $\lbrace\Lambda_n;n\in\mathbb{Z}_{+}\rbrace$ such that $\Lambda\varphi:=\lim\Lambda_n\varphi$ exists for each $\varphi\in H^{\infty}(\mathbb{D}).$ Then $\Lambda$ is weak* continuous. I have following questions:

  1. is the conclusion still holds true if the sequential limit exists apriori only for $\varphi\in A(\mathbb{D})?$

  2. or if there are some results about the maximal set where if the sequential limit exits the conclusion of the above theorem still holds?

Remark: $H^{\infty}(\mathbb{D})$ and $A(\mathbb{D})$ respectively represents the Banach spaces of bounded analytic functions and the holomorphic functions which are continuous upto the boundary of $\mathbb{D}.$

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I don't about the second question, but the answer to the first question is "no," since one can take a sequence $\Lambda_n$ of $L^1$ functions converging (weak*, in the dual of $C(\mathbb T)$) to the point mass at 1, say. Then take $\Lambda$ to be a Hahn-Banach extension to $H^\infty$ of the evaluation functional $\varphi\to \varphi(1)$ on $A(\mathbb D)$. We then have $\Lambda_n\to \Lambda$ pointwise on $A(\mathbb D)$ but $\Lambda$ is not weak* continuous.

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