Definition 40.4.1 (Hardy Space).label Let $D = B_{\complex}(0, 1)$ and $p \in [1, \infty]$. For each $f \in H(D; \complex)$, let
\[\norm{f}_{H^p(D)}= \sup_{r \in [0, 1)}\braks{\frac{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^p d\theta}^{1/p}\]
if $p < \infty$, and $\norm{f}_{H^\infty(D)}= \norm{f}_{u}$ if $p = \infty$. Then the Hardy space is the space
\[H^{p}(D) = \bracsn{f \in H(D; \complex)|\ \normn{f}_{H^p(D; \complex)} < \infty}\]
In particular, $H^{\infty}(D)$ is a Banach algebra under pointwise multiplication.
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