Let \(H\) be the hyperbolic complex plane and let \(\Xi\) be the divisors of zero region. An addition sub-semi-group \(S\) of \(H\) is called hyperbolic semi-linear space if \(o\in S\) and there exists an operation of multiplication by non-negative real numbers having the following properties: 1. \((ab)X = a(bX)\) 2. \((a+b)X = aX + bX\); 3. \(a(X+Y) = aX + aY\); 4. \(1X = X;\) \(a,b\geq 0,\) \(X,Y\in S\) Firstly, the author finds the all hyperbolic semi-linear spaces of \(H\) and establishes the relationships by conjugation between these spaces. The usual definition of inner product is adapted in a hyperbolic semi-linear space, where the nul element is replaced by \(\Xi\). Consequently, specific metric properties are obtained. The author defines the hyperbolic Cauchy point range in \(H\) and also, the notion of imaginary-distance metric space in \(H\). Finally, several properties of orthogonality are presented. The hyperbolic functional analysis generated by hyperbolic Hilbert spaces can be used to obtain an unification of relativity theory and quantum mechanics.