  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) $\Phi$-algebras, improving results by Boulabiar
and Toumi. As a consequence, the Minkowski inequality for weighted geometric
mean closed Archimedean (real or complex) $\Phi$-algebras is obtained.
