If $X$ is a Banach space of analytic functions on the unit disk $D$ which contains the space of analytic bounded functions on $D$, how can I prove that $X\setminus\{0\}$ is simply connected?
2026-03-26 22:15:25.1774563325
How to prove that a Banach space of analytic functions containing $H^\infty$ except the origin is simply connected?
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For any Banach space of dimension $>2$, $X \backslash \{0\}$ is simply connected.
EDIT: Hint: given a closed curve $\gamma: {\mathbb S^1} \to X \backslash \{0\}$, first deform to a polygonal curve. That polygonal curve is in a finite-dimensional subspace...