It is named for its discoverer, the Hungarian mathematician Gábor Szegő.
Define the Hardy space H2(∂Ω) to be the closure in L2(∂Ω) of the restrictions of elements of A(Ω) to the boundary.
The Poisson integral implies that each element ƒ of H2(∂Ω) extends to a holomorphic function Pƒ in Ω.
Furthermore, for each z ∈ Ω, the map defines a continuous linear functional on H2(∂Ω).
In fact, if φi is an orthonormal basis of H2(∂Ω) consisting entirely of the restrictions of functions in A(Ω), then a Riesz–Fischer theorem argument shows that