By Ron Blei

This ebook offers a radical and self-contained examine of interdependence and complexity in settings of sensible research, harmonic research and stochastic research. It makes a speciality of "dimension" as a uncomplicated counter of levels of freedom, resulting in distinct kinfolk among combinatorial measurements and diverse indices originating from the classical inequalities of Khintchin, Littlewood and Grothendieck. themes contain the (two-dimensional) Grothendieck inequality and its extensions to better dimensions, stochastic versions of Brownian movement, levels of randomness and Fréchet measures in stochastic research. This publication is basically aimed toward graduate scholars focusing on harmonic research, practical research or chance conception. It includes many workouts and is appropriate as a textbook. it's also of curiosity to computing device scientists, physicists, statisticians, biologists and economists.

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7). , 1 2 |an |2 2 12 an rn . 8) n 2 The Khintchin L1 –L2 Inequality If f ∈ L2 (Ω, P), then f L2 ≥ f L1 . 1) The reverse inequality is generally false; there exist f ∈ L2 (Ω, P) such that f L1 = 1 and f L2 = ∞. However, the L1 and L2 norms are equivalent on the span of the Rademacher system. The latter assertion – widely known as the ‘Khintchin inequality’ – is among the important tools in modern analysis. Theorem 1 (the Khintchin L1 –L2 inequality). There exists κK > 0 such that for every scalar sequence (an ) with ﬁnite support, 1 2 |an | an rn ≥ κK E 2 n .

4), maps the unit sphere in l2 (cf. 6)). Let θ = √e−2 into itself. 5) (cf. 3) x, y = A(x, y) − (e − 2) θx, θy . 6) (cf. 8)). Lemma 4 (Exercise 4). 7) k=0 where the convergence of the series is uniform in the unit sphere of l2 . Lemma 5 Let {xm : m ∈ S} and {yn : n ∈ T } be ﬁnite subsets of the unit sphere in l2 . Then, β(m, n) A(xm , yn ) ≤ (4e + 1) β F2 . 8) m∈S,n∈T Proof: Let {Zk : k ∈ N} be a system of (statistically) independent standard normal random variables. Then, for x and y in the unit sphere of l2 , E eiΣx(k)Zk e−iΣy(k)Zk = E ei{x(k)−y(k)}Zk k = e −{x(k)−y(k)}2 /2 = e−1 e x,y .

Use Lp –Lq duality. 5. See [LiTz, pp. 15–16]. 6. Use the ‘characteristic function’ method, the statistical independence of the Rademacher system, and the symmetry of its elements. Hints for Exercises 37 8. iv. 5) to Z × Z by writing βm (j, k) = 0 for all negative integers j and k. ) 11. See the proof of Theorem 4. p = ∞ for every III A Fourth Inequality 1 Mise en Sc` ene: Does the Khintchin L1 –L2 Inequality Imply the Grothendieck Inequality? Grothendieck’s th´eor`eme fondamental de la th´eorie metrique des produits tensoriels appeared ﬁrst in 1956, in a setting of topological tensor products [Gro2, p.