On analytic continuation in Hardy spaces (Q731991)

Let \(\mathbb{D}\) be the open unit disk in \(\mathbb{C}\). The author proves that there is a dense subspace \(G\) of \(H^{\infty}(\mathbb{D})\) which is barrelled and such that every nonzero element \(f\) of \(G\) does not extend holomorphically outside \(\mathbb{D}\). Moreover, there is a dense subspace \(E\) of \(H^{p}(\mathbb{D})\), \(1 \leq p < \infty\), which is semi-Baire and such that every nonzero element \(f\) of \(E\) cannot be extended holomorphically outside \(\mathbb{D}\). To prove these results, the author studies certain properties of locally convex barrelled and semi-Baire spaces.