By O.U. Schmidt

ISBN-10: 0716704315

ISBN-13: 9780716704317

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Geometrically, the plane {x | (x) = 1} is tangent to K in the sense that K lies on one side of the plane. If V = Rν and ∂K is a smooth manifold, such planes are tangent in the usual intuitive sense. 44 Let K be a convex, absorbing subset in V and x0 a point with gK (x0 ) = 1 where gK is the gauge of gK . Then there is a tangent to K at x0 . Proof Let W = {λx0 | λ ∈ R}. Define on W by (λx0 ) = λ. Since for λ ≥ 0, gK (λx0 ) = λ, and for λ < 0, gK (λx0 ) ≥ 0 > λ, we have ≤ gK on W. 38), there is L on V with L ≤ gK and L(x0 ) = (x0 ) = 1.

For y0 ∈ [0, 1], let f (y 0 ) (y) = log(|y − y0 |−1 ). Then QF (λ(f (y 0 ) − f (y 1 ) )) = ∞ if λ ≥ 1 and y0 = y1 . Thus, f (y 1 ) − f (y 0 ) ≥ 1 for all y0 = y1 in [0, 1] and so we have an uncountable set with pairwise distance at least 1. Thus, L(F ) is not separable. We next turn to the duality theory for Orlicz spaces. The Legendre transform discussed at the end of Chapter 1 will play a critical role. We begin by discussing Legendre transforms of even functions in one dimension. 14 Let F˜ be a convex function on [0, ∞) with F˜ (0) = 0, F˜ (x) ≥ 0, and A+ (F˜ ) = limx→∞ (D− F˜ )(x) = ∞.

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. 33) q For later purposes, notice that βp → 1 as p → 1 or ∞, β1/2 = 12 , and β is monotone decreasing on [1, 2] and increasing on [2, ∞]. The easiest way to see this is to note that θ → θ log θ is convex on (0, ∞), so θ → θ log θ + (1 − θ) log(1 − θ) is convex on (0, 1) and invariant under θ → 12 − θ.

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Abstract Theory of Groups by O.U. Schmidt

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