By Marvin J. Greenberg

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**Extra resources for Algebraic topology: a first course**

**Example text**

Fn ∈ X(μ); see [30, p. s. 61) for all such n ∈ N and fj ’s (j = 1, . . , n) is called the q-convexity constant of X(μ) and is denoted by M(q) [X(μ)]. 61) yields M(q) [X(μ)] ≥ 1. 23. s. with quasi-norm · X(μ) . (i) Let 0 < p ≤ 1. s. (ii) Let 0 < p < ∞. s. X(μ) is p-convex if and only if its p-th power X(μ)[p] admits a lattice norm equivalent to · X(μ)[p] . Moreover, it is possible to select an equivalent lattice norm η[p] on X(μ)[p] satisfying η[p] (f ) ≤ f X(μ)[p] ≤ M(p) [X(μ)] p · η[p] (f ), f ∈ X(μ)[p] .

A) = 0), then χA = 0 in X(μ) ⊆ L0 (μ), and hence, λη (A) = χA , η = 0. So, λη is absolutely continuous with respect to μ; see [42, Ch. 1]. Let g ∈ L1 (μ) denote the Radon– Nikod´ ym derivative dλη /dμ, that is, λη (A) = A g dμ for A ∈ Σ. We claim that g ∈ X(μ) and Ω f g dμ = f, η for every f ∈ X(μ). In fact, ﬁx f ∈ X(μ). 6. So, sn g → f g pointwise as n → ∞, and the limit f χA , η = lim sn χA , η = lim n→∞ n→∞ sn dλη = lim n→∞ A sn g dμ A exists for all A ∈ Σ. 17 applies to conclude that f g ∈ L1 (μ) and f g dμ = f, η .

51) holds. (ii) Clearly X(μ)[p] is closed under scalar multiplication. Let f, g ∈ X(μ)[p] . 44) we have |f + g|1/p ≤ 2(1/p)−1 |f |1/p + |g|1/p and |f + g|1/p ≤ |f |1/p + |g|1/p when 0 < p < 1 and 1 ≤ p < ∞, respectively. In both cases, f + g ∈ X(μ)[p] , that is, X(μ)[p] is a linear subspace of L0 (μ). So, from the deﬁnition of X(μ)[p] , it is clear that X(μ) is an ideal of L0 (μ) with sim Σ ⊆ X(μ). 53) as follows. 52). 2. The p-th power of a quasi-Banach function space 41 Next, let 1 ≤ p < ∞. 53) holds.