Basic Notions

Abstract

We recall here some basic facts about the Hilbert spaces that are used in the sequel. They can be found in textbooks on functional analysis (see, e.g., Balakrishnan [9]). A pair (H, (· , ·)) is called a real inner product space if H is a real linear space and the function (· , ·) : H × H ↦ ℜ, referred to further as the inner product, is linear in each argument when the other is fixed, symmetric under rearrangement of arguments, and positive if both arguments are identical and nonzero. These properties imply the Schwarzinequality

$$\forall g,h \in H (g,h) \leqslant {[(g,g)(h,h)]^{1/2}}$$

(2.1)

.