Vector lattices and $f$-algebras: the classical inequalities

We prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean (real or complex) vector lattice, from which the Cauchy-Schwarz inequality follows. A reformulation of this result for sesquilinear maps with a geometric mean closed semiprime Archimedean (real or complex) $f$-algebra as codomain is also given. In addition, a sufficient and necessary condition for equality is presented. We also prove the H"older inequality for weighted geometric mean closed Archimedean (real or complex) $\mathrm{\Phi }$-algebras, improving results by Boulabiar and Toumi. As a consequence, the Minkowski inequality for weighted geometric mean closed Archimedean (real or complex) $\mathrm{\Phi }$-algebras is obtained.