L p Spaces

5.1 In this section we prove several fundamental inequalities. For example, if p, q, and r with 1/p + 1/q = 1/r belong to [0, +∞], and if f, g are two functions on Ω such that N _p (f) and N _q (g) are finite, then N _r (fg) ≤ N _p (f)N _q (g) (Proposition 5.1.2). Theorem 1.1 is a generalization of Minkowski’s inequality: if f and g are two functions on Ω and p ≥ 1, then N _p (f + g) ≤ N _p (f) + N _p (g).5.2 We now generalize to Lp(µ) some of the convergence theorems established in Section 3.2, for example the dominated convergence theorem, and prove the Fischer-Riesz theorem (Proposition 5.2.3). Next, we analyze the duality of Lq and Lq when p and q are conjugate, that is, when 1/p + 1/q = 1. If F is a Banach space, and p and q are conjugate, then, for every g ∈ L_F′q(µ), f ↦ ∫ fgdµ is a continuous linear functional on L _F p(µ) with norm N _q (g) (Theorem 5.2.5). The converse will be dealt with in Chapter 10.5.3 The notion of convergence in measure is introduced. This section requires some knowledge of uniform spaces.5.4 This section, which may be omitted, deals with uniformly integrable sets.