Schwarz’s inequality

Since both F and belong to L2, the binary product F> will always exist according to Schwarz s inequality. In the following discussion, we will always assume that not all the binary products <5> F> are vanishing. For a specific pair, and F, having <4> j F> 0, it is convenient to introduce the new function <5 = 5> F> 1, which gives the so-called binormalization... 

An important property of all correlation functions follows from the Schwarz s inequality, Eq. (1.87)... 

The exact numerical value for thus, depends on the ratio of the integrals in eqn [15], a value that is not easily accessible. However, in mathematics we have Schwarz s inequality, which for this particular case gives the following relation ... 

Cauchy-Schwarz inequality This inequality (q.v.) is used in direct methods of phase determination. Cauchy s inequality states ... 

This set of densities has a property which will be of importance later, namely S is convex. With this we mean that if and n2 are elements of Cauchy-Schwarz inequality. Now we will derive constraints on the allowed set of external potentials. In order to do this we introduce some other function spaces. We say that a given function / belongs to the space IP if... 

This means that at constant entropy we cannot increase the temperature by forcing a change of the variable X. In other words, the temperature is a function of the entropy alone, i.e., T = T(S). The second equality in Eq. (3.21) arises from the inequality of Schwarz. Thus f = (X). 

---