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2 Suppose 1 蠄 p, q 蠄 蜴, are conjugate exponents. (i) If g Lq, then (ii) Suppose g is integrable on all sets of finite measure, and Then g Lq, and gLq = M. For the proof of the lemma, we recall the signum of a real number defined by Proof. We start with (i). , and hence gLq 蠀 0. By HɆlder舗s inequality, we have that To prove the reverse inequality we consider several cases. 舦 First, if q = 1 and p = 蜴, we may take f(x) = sign g(x). Then, we have f L蜴 = 1, and clearly, fg = gL1. ) Then, if we take denotes the characteristic function of the set E, we see that f L1 = 1, and also This completes the proof of part (i).

Figure 2. The gauge function of the interval (a, b) in We note, for example, that if V is normed and K is the unit ball {||॥|| ` 1}, then p(॥) = ||॥||. In general, the non-negative function p completely characterizes K in that Moreover p has an important sub-linear property: In fact, if ॥ K then ॥/(1 蜢 ) K for some b 0, since K is open, which gives that p(॥) ` 1. Conversely if p(॥) ` 1, then ॥ = (1 蜢 )॥艂, for some 0 ` ` 1, and ॥艂 K. Then since ॥ = (1 蜢 )॥艂 + ƃ 0 this shows ॥ K, because 0 K and K is convex.

To begin with, we assume that X is a compact metric space. Our main result then states that if C (X)*, then there exists a finite signed Borel measure ॖ (this measure is sometimes referred to as a Radon measure) so that Before proceeding with the argument leading to this result, we collect some basic facts and definitions. Let X be a metric space with metric d, and assume that X is compact; that is, every covering of X by open sets contains a finite sub-covering. The vector space C(X) of real-valued continuous functions on X equipped with the sup-norm is a Banach space over .

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