ABSTRACT
In Modern Functional Analysis the concept of a Hilbert space plays a fundamental role. It creates a natural background for solving numerous problems in mathematics, physics, engineering, and science.
The purpose of this chapter is to present some of the fundamental inequalities that involve inner products and norms. Various inequalities related to the Schwarz, triangle, and Bessel inequalities are surveyed. More recent results due to Boas-Bellman’s and Bombieri’s generalisations of the Bessel inequality are presented. The generalisations due to Kurepa, Buzano, and Precupanu of the Schwarz inequality as well as the Dunkl-Williams inequality are also given. Last but not least, an account on recent advancement of the Gru¨ss inequality in inner product spaces is provided as well. These, as usual, have been complemented by numerous remarks and comments that engender further research and application of the results.
