35.4 The Hardy Space
Definition 35.4.1 (Hardy Space).label Let $D = B_{\complex}(0, 1)$, then the Hardy space $H^{\infty}(D)$ is the algebra of all bounded holomorphic functions on $D$, equipped with the uniform norm.
Proposition 35.4.2.label Let $f \in H^{\infty}(D)$, then $\sigma(f) = \ol{f(D)}$.
Proposition 35.4.3.label Let $\fU \subset 2^{B_\complex(0, 1)}$ be an ultrafilter and
then:
- (1)
If $\fU \to x_{0} \in \ol{D}$, then $f \in A(D)$, $\phi_{\fU}(f) = f(z_{0})$.
- (2)
$\phi_{\fU}$ is a multiplicative linear functional on $H^{\infty}(D)$.
Proof. Let $f \in H^{\infty}(D)$, then by Proposition 5.2.4, $f(\fU)$ is an ultrafilter base. Since $f$ is bounded, $f(\fU)$ converges to exactly one element of $\complex$. Hence the limit is well-defined.
(2): By Definition 33.7.4.$\square$
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