MATHEMATICS 116, FALL 2015 REAL ANALYSIS, CONVEXITY AND OPTIMIZATION Proof List for the fnall exam Last modified: December 3, 2015 The final exam will have one proof chosen at random from 1-3, one from 4-6, one from 7-8, and one from 9-10. In some cases the proof listed here is only part of a longer proof that was presented in class, or I have made simplifying assumptions such as assuming a complete normed space with a closed subspace. 1. Let M be a closed subspace of a normed vector space X, let p(x) be a continuous sublinear functional defined on all of X, and let f(m) be a linear functional defined on M that satisfies f(m) ≤ p(m) for all m 2 M. Let y be a vector in X that is not in M. Prove that, by choosing an appropriate value for g(y); it is possible to define a linear functional g on the subspace [M + y] such that g(m) = f(m) for all m 2 M and g(x) ≤ p(x) for all x 2 [M + y]. Remember to consider vectors of the form x = m + αy; α > 0 and vectors of the form x = m − βy; β > 0: 2. Let K be a convex set with nonempty interior in a vector space X. Assume that θ is in the interior of K. Suppose that V is a linear variety in X containing no interior points of K. Using the extension form of the Hahn-Banach theorem, prove that there exists an element x∗ 2 X∗ such that < v; x∗ >= 1 for all v 2 V . < k; x∗ >< 1 for all k in the interior of K. Restate this result in terms of hyperplanes, and draw a diagram to illustrate it for the special case where X = R2 and V consists of a single point on the boundary of K. You may take it as proved that the Minkowski functional of a convex set is sublinear, continuous, non-negative, and finite. 1 3. You may take the extension version of the Hahn-Banach theorem as proved. Show that any norm jjxjj is a sublinear function, then, given a bounded linear functional f defined on subspace M ⊂ X, with norm jjfjjM , prove that it can be extended to a bounded linear functional F , with the same norm, defined on all of X. 4. Start with a real Banach space X and a closed subspace M. Choose x 2 X: Define d = min jjx − mjj m2M and let m0 be an element for which this minimum distance is achieved. Explain how to use the Hahn-Banach theorem to construct ∗ ? x0 2 M for which ∗ ∗ d = max < x; x >=< x; x0 > jjx∗||≤1;x∗2M ? ∗ and prove that x0 is aligned with x − m0. 5. Start with a real Banach space X and a closed subspace M. Choose x∗ 2 X∗: Define d = max < x; x∗ > jjx||≤1;x2M and let x0 2 M be an element for which this maximum value is achieved. Explain how to use the Hahn-Banach theorem to construct ∗ ? m0 2 M for which ∗ ∗ ∗ ∗ d = min jjx − m jj = jjx − m0jj m∗2M ? ∗ ∗ and prove that x − m0 is aligned with x0. 2 6. Start with the theorem, valid for a two-dimensional subspace M 2 X gen- erated by y1 and y2, d = min jjx∗ − m∗jj = max < x; x∗ > m∗2M ? jjx||≤1;x2M where x∗ is any vector in X∗ that satisfies the constraints ∗ < y1; x >= c1 ∗ < y2; x >= c2: Let D denote the linear variety (in X∗) that satisfies these constraints. Prove that the minimum norm for a vector in D is ∗ d = min jjx jj = max (a1c1 + a2c2) ∗ x 2D jja1y1+a2y2≤1jj ∗ and that an optimal x is aligned with x = a1y1 + a2y2: 7. Let Z t2 J = f(x(t); x_(t); t)dt t1 Over the set of vectors x(t) that satisfy x(t1) = c1; x(t2) = c2, show that the one that minimizes the functional J must satisfy the differential equation d f (x(t); x_(t); t) − f (x(t); x_(t); t) = 0 x dt x_ . Make any assumptions of continuity that you find convenient. 8. Suppose that f is a convex functional defined on convex subset C of a normed space X. Define the infimum µ = inf f(x): x2C Prove that (a) The subset Ω where f(x) = µ is convex. (b) If x0 is a local minimum of f, so that it is a minimum in some neigh- borhood N of x0, then x0 is also a global minimum of f on C. (c) The set above the graph of f, [f; C] = f(r; x) 2 R × X : X 2 C; r ≥ f(x)g, is convex. 3 9. Given convex set C 2 X and convex functional f(x), define the conjugate set C∗ = fx∗ 2 X∗ : sup[< x; x∗ > −f(x)] < 1g x2C and the convex functional (with domain C∗) f ∗(x∗) = sup[< x; x∗ > −f(x)] x2C . Prove that C∗ and f ∗ are convex. 10. Suppose that f is a convex functional with convex domain C and that g is a concave functional with convex domain D. Let µ = inf [f(x) − g(x)] C\D Prove that sup [g∗(x∗) − f ∗(x∗)] = µ. C∗\D∗ 4.