The Alcantara-Bode equivalence ([2], 1993) of the Riemann Hypothesis (RH, 1859) consists in the injectivity of a certain Hilbert-Schmidt integral operator, result obtained from Beurling equivalent formulation of RH ([4], 1955).
Both outstanding equivalences reduced RH to a concrete problem solvable with techniques outside of the pure math.
The theory and the associated methods introduced for the investigation of the injectivity of the linear bounded operators on separable Hilbert spaces, have been used to prove the injectivity of the integral operator part in the equivalent formulation of the Riemann Hypothesis.
The Theorem 1 shows that a strict positive linear bounded operator on a dense set is injective. Lemma 1 simplifies considerably the transition from the theory to its associated methods.
The result obtained has been like in [1], this time without using the operator orthogonal projections on finite dimension subspaces and without to involve its adjoint.
As a consequence of the injectivity of the integral operator due to the equivalent formulation of Alcantara-Bode, RH holds i.e.: the non trivial zeros of the Riemann Zeta function are on the vertical line sigma = 1/2.
With this article, a reformulation and improvement of [1], we consider the 165 years old Riemann Hypothesis - a Clay Inst. Millennium Problems, solved.