MATHEMATICS 116, FALL 2015 REAL ANALYSIS, CONVEXITY and OPTIMIZATION Proof List for the Fnall Exam Last Modified

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 ∈ 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 ∈ M and g(x) ≤ p(x) for all x ∈ [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∗ ∈ X∗ such that < v, x∗ >= 1 for all v ∈ 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 ||x|| is a sublinear function, then, given a bounded linear functional f deﬁned on subspace M ⊂ X, with norm ||f||M , 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 ∈ X. Deﬁne

d = min ||x − m|| m∈M

and let m0 be an element for which this minimum distance is achieved. Explain how to use the Hahn-Banach theorem to construct ∗ ⊥ x0 ∈ M for which

∗ ∗ d = max < x, x >=< x, x0 > ||x∗||≤1,x∗∈M ⊥

∗ and prove that x0 is aligned with x − m0. 5. Start with a real Banach space X and a closed subspace M. Choose x∗ ∈ X∗. Deﬁne

d = max < x, x∗ > ||x||≤1,x∈M

and let x0 ∈ M be an element for which this maximum value is achieved. Explain how to use the Hahn-Banach theorem to construct ∗ ⊥ m0 ∈ M for which

∗ ∗ ∗ ∗ d = min ||x − m || = ||x − m0|| m∗∈M ⊥

∗ ∗ and prove that x − m0 is aligned with x0.

2 6. Start with the theorem, valid for a two-dimensional subspace M ∈ X gen- erated by y1 and y2,

d = min ||x∗ − m∗|| = max < x, x∗ > m∗∈M ⊥ ||x||≤1,x∈M 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 ||x || = max (a1c1 + a2c2) ∗ x ∈D ||a1y1+a2y2≤1||

∗ 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). x∈C 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] = {(r, x) ∈ R × X : X ∈ C; r ≥ f(x)}, is convex.

3 9. Given convex set C ∈ X and convex functional f(x), deﬁne the conjugate set

C∗ = {x∗ ∈ X∗ : sup[< x, x∗ > −f(x)] < ∞} x∈C and the convex functional (with domain C∗)

f ∗(x∗) = sup[< x, x∗ > −f(x)] x∈C . 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∗