Lp(Ω) Spaces

In this chapter, we deal with the spaces of Lp(Ω ). Here p may be a positive finite number or equal to ∞ and Ω is a measurable set. We start by introducing Lp(Ω ) space with its norm and provide some simple examples for different choices of Ω. Next part is devoted to several important inequalities which are stated in order as the Young, Holder and Minkowski inequalities with their corresponding reverse inequalities, the interpolation, Gronwall, Komornik and Nakao inequalities. Moreover, two types of the Green’s identities are presented. After investigating particular cases of finite p, we then consider the infinite case p = ∞, namely L∞ space. The Riesz-Fischer theorem is then analyzed with its proof in detail. In the remaining part, embedding property of Lp(Ω) space, L1loc(Ω) space, the space of continuous functions, C0∞ (Ω) space, Holder space and Cm([0; T];X) space with further theorems and properties are also given out in this chapter. Most of the theorems and propositions in that chapter are expressed with proofs. The statements without proof are addressed to the references given at the end of the book. Particularly [16–20] may be worthwhile.

Keywords: Lp(Ω) spaces, Young inequality, Holder inequality, Minkowski inequality, Gronwall inequality, Komornik inequality, Nakao inequality, Green’s identities, Riesz-Fischer theorem, L1 loc(Ω ) space, C∞ 0 (Ω ) space, Holder space.