The authors consider some applications of the Bishop--De Leeuw theorem about representing measures to the situation of certain algebras of analytic functions on unit balls of Banach spaces. In particular, the Hardy spaces \(H^2(\mu)\) are studied. Let \(B\) be the unit ball of a complex Banach space and let \(A_a(B)\) be the Banach algebra of holomorphic functions on \(B\) generated by the continuous linear forms. Applying the Bishop--De Leeuw theorem to a homomorphism \(\varphi\) on \(A_a(B),\) one gets a representing measure \(\mu\) on the maximal ideal space \({\mathcal M}(A_a(B)).\) One next constructs \(H^2_a(\mu)\) as being the completion of \(A_a(B)\) relative to \(\| f\| _\mu = \int_{{\mathcal{M}}(A_a(B))} \sqrt{| {f}| ^2 \,d\mu }.\) The authors study \(H^2_a(\mu)\) for certain measures \(\mu\) for which \(\| \cdot \| \) is a norm. In addition, conditions are studied under which \(H^2_a(\mu)\) is a reproducing kernel Hilbert space and when that reproducing kernel is determined by an analytic function. Examples of representing measures and corresponding Hardy spaces \(H^2\) for \(c_0\) and \(\ell_p, \;1 < p < \infty,\) are given. Useful references include [\textit{B.\,Cole} and \textit{T.\,W.\thinspace Gamelin}, Proc.\ Lond.\ Math.\ Soc.\ 53, 112--142 (1986; Zbl 0624.46032); \textit{D.\,Pinasco} and \textit{I.\,Zaldendo}, J.~Math.\ Anal.\ Appl.\ 308, No.\,1, 159--174 (2005; Zbl 1086.46033); the authors, Ann.\ Pol.\ Math.\ 81, No.\,2, 111--122 (2003; Zbl 1036.46030)].