Hardy space

Summary

Let $0 < p \le \infty$. The Hardy space $H^p$ of the unit disc is the space of holomorphic functions whose $L^p$ means over the circles $\abs{z} = r$ stay bounded as $r \to 1$. For $p \ge 1$ it is a Banach space, and $H^2$ is the Hilbert space of power series $\sum a_n z^n$ with $\sum \abs{a_n}^2 < \infty$. By Fatou's theorem, functions in $H^p$ have boundary values almost everywhere on the circle. So $H^p$ can be seen as a subspace of $L^p$ of the circle. Hardy spaces connect complex analysis with Fourier analysis and operator theory, through inner functions, Blaschke products and Toeplitz operators.