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In this paper we show that that on a strongly pseudoconvex domain $D$ theprojective limit of all Poletsky--Stessin Hardy spaces $H^p_u(D)$, introducedin \cite{PS}, is isomorphic to the space $H^\infty(D)$ of bounded holomorphicfunctions on $D$ endowed with a special topology. To prove this we show thatCarath\'eodory balls lie in approach regions, establish a sharp inequality forthe Monge--Amp\'ere mass of the envelope of plurisubharmonic exhaustionfunctions and use these facts to demonstrate that the intersection of allPoletsky--Stessin Hardy spaces $H^p_u(D)$ is $H^\infty(D)$.

Projective limits of Poletsky--Stessin Hardy spaces