The purpose of this paper is to show that the mathematics quantum mechanics (QM) set partitions (which specify indefiniteness and definiteness) linearized vector spaces, particularly in Hilbert spaces. That is, math QM space version describe objective at level partitions. key analytical concepts are definiteness versus indefiniteness, distinctions indistinctions, distinguishability indistinguis...

In this paper, it is shown that a refinement of Hölder’s inequality can be established using the positive definiteness of the Gram matrix. As applications, some improvements on Minkowski’s inequality, Fan Ky’s inequality and Hardy’s inequality are given.