Having set down elementary facts in the previous chapter, we take a detailed look at Besov spaces and Triebel–Lizorkin spaces, which are the main theme of this book. Chapter 2 is devoted to the introduction of elementary definitions together with some fundamental properties. First we define the Besov space Bspq(ℝn) with 1 ≤ p, q ≤∞ and s∈ ℝ in the spirit of Peetre, although Besov introduced Besov spaces in [171, 172]. After the Besov space Bspq(ℝn) for such a restricted case we define Aspq(ℝn), which unifies the Besov space Bspq(ℝn) with 0 < p, q ≤∞ and s∈ ℝ and the Triebel–Lizorkin space Fspq(ℝn) with 0 < p < ∞, 0 < q ≤∞ and s∈ ℝ.