for almost every θ. The function belongs to the Lp space for the unit circle, and one has that Denoting the unit circle by T, and by Hp(T) the vector subspace of Lp(T) consisting of all limit functions , when f varies in Hp, one then has that for p ≥ 1, where the ĝ(n) are the Fourier coefficients of a function g integrable on the unit circle, The space Hp(T) is a closed subspace of Lp(T). Since Lp(T) is a Banach space (for 1 ≤ p ≤ ∞), so is Hp(T). The above can be turned around. Given a function , with p ≥ 1, one can regain a (harmonic) function f on the unit disk by means of the Poisson kernel Pr: and f belongs to Hp exactly when is in Hp(T). Supposing that is in Hp(T), i.e. that has Fourier coefficients (an)n∈Z with an = 0 for every n < 0,then the element f of the Hardy space Hp associated to is the holomorphic function In applications, those functions with vanishing negative Fourier coefficients are commonly interpreted as the causal solutions. Thus, the space H2 is seen to sit naturally inside L2 space, and is represented by infinite sequences indexed by N; whereas L2 consists of bi-infinite sequences indexed by Z. Connection to real Hardy spaces on the circle When 1 ≤ p < ∞, the real Hardy spaces Hp discussed further down in this article are easy to describe in the present context. A real function f on the unit circle belongs to the real Hardy space Hp(T) if it is the real part of a function in Hp(T), and a complex function f belongs to the real Hardy space iff Re(f) and Im(f) belong to the space (see the section on real Hardy spaces below). Thus for 1 ≤ p < ∞, the real Hardy space contains the Hardy space, but is much bigger, since no relationship is imposed between the real and imaginary part of the function. For 0 < p < 1, such tools as Fourier coefficients, Poisson integral, conjugate function, are no longer valid. For example, consider the function Then F is in Hp for every 0 < p < 1, and the radial limit exists for a.e. θ and is in Hp(T), but Re(f) is 0 almost everywhere, so it is no longer possible to recover F from Re(f). As a