By Barry Simon
Convexity is critical in theoretical features of arithmetic and in addition for economists and physicists. during this monograph the writer presents a accomplished perception into convex units and capabilities together with the infinite-dimensional case and emphasizing the analytic perspective. bankruptcy one introduces the reader to the elemental definitions and concepts that play principal roles in the course of the publication. the remainder of the booklet is split into 4 elements: convexity and topology on infinite-dimensional areas; Loewner's theorem; severe issues of convex units and comparable matters, together with the Krein-Milman theorem and Choquet concept; and a dialogue of convexity and inequalities. The connections among disparate themes are truly defined, giving the reader an intensive knowing of the way convexity turns out to be useful as an analytic instrument. a last bankruptcy overviews the subject's heritage and explores additional the various issues pointed out past. this is often a good source for someone attracted to this critical subject.
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We now turn to the main result on consequences of the Δ2 condition. 9 Let (M, μ) be a measure space with a nonatomic component and F a weak Young function. Then the following are equivalent: (i) F obeys the Δ2 condition. 7, is equivalent to) norm convergence. , L∞ is dense in L(F ) ). (iv) YF = L(F ) (v) YF is a vector space. 21) We will show (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (v) ⇒ (vi) ⇒ (i). 22) (we use μ(M ) = 1 here). 22), QF (2k g) ≤ 1 so g F ≤ 2−k . This shows if QF (gn ) → 0, then gn F → 0. 40 Convexity (ii) ⇒ (iii) Suppose (ii) holds and let f ∈ L(F ) (M, dμ).
Conversely, if limx↓0 F (x)/x = y0 > 0, then F (x) ≥ xy0 so G(y0 ) = 0. 32) is part of the deﬁnition of Young functions – it is needed for the conjugate function to be a weak Young function. 17 Let F be a Young function and G its convex conjugate. Then G is a Young function. 45, G is a convex function which is steep and nonnegative. 16, G(x) = 0 if and only if x = 0. Since F is even, G is even. 16 implies that limx↓0 G(x)/x = 0. 18 If F (x) = xp , 1 < p < ∞, the conjugate function is not y q (where p−1 + q −1 = 1) but G(y) = p−q /p q −1 y q by a simple calculation.
Iii) ⇒ (iv) (iv) ⇒ (i) is trivial. Pick x0 so F (x0 ) ≥ ε. For x ≥ x0 , F (kx) ≤ B F (x) + ε ≤ (B + 1)F (x) Now pick so k ≥ 2. Then for x ≥ x0 , F (2x) ≤ F (k x) ≤ (B + 1)F (k (i) ⇒ (v) −1 x) ≤ · · · ≤ (B + 1) F (x) Since (D− F )(x) is monotone, if (i) holds and x ≥ x0 , 2x x(D− F )(x) ≤ x D− F (y) dy ≤ F (2x) ≤ C F (x) Orlicz spaces 37 so sup x≥x 0 x(D− F )(x) ≤C F (x) Since sup 1≤x≤x 0 x(D− F )(x) <∞ F (x) (v) holds. (v) ⇒ (iv) Suppose (v) holds. Let the sup be A and let x ≥ 1 and k > 1. 18) Pick k so A(log k) ≤ 12 .