Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Using Linear Functionals to Study Nonlinear Functionals and Their Support Sets Jessica E. Stovall University of North Alabama May 4, 2018 University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets 1 Preliminaries 2 Associating a Linear Operator with T 3 Theorems Regarding Support Sets University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Background Topology Measure Theory Banach Lattices Operator Theory University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Topology Any union of open sets is open Any finite intersection of open sets is open Any intersection of closed sets is closed Any finite union of closed sets is closed University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Clopen Sets Definition A set is clopen if it is both open and closed. Any finite union of clopen sets is clopen Any finite intersection of clopen sets is clopen University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Extremally Disconnected Definition A topological space X is extremally disconnected if and only if the closure of every open set in X is open. For any space X , the following are equivalent: 1 X is extremally disconnected 2 Every two disjoint open sets in X have disjoint closures University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Separation Axioms The following are called the separation axioms. T1: Given two distinct points x and y, there is an open set that contains y but not x. T2: Given two distinct points x and y, there are disjoint open sets O1and O2 such that x 2 O1 and y 2 O2. T3: In addition to T1, given a closed set F and a point x not in F, there are disjoint open sets O1 and O2 such that x 2 O1 and F ⊂ O2. T4: In addition to T1, given two disjoint closed sets F1 and F2, there are disjoint open sets O1 and O2 such that F1 ⊂ O1 and F2 ⊂ O2. University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Spaces Satisfying the Separation Axioms A topological space satisfying T1 is called a Tychonoff space. A topological space which satisfies T2 is called a Hausdorff space. A topological space which satisfies T3 is called a regular space A topological space which satisfies T4 is called a normal space. For topological spaces, T4 ) T3 ) T2 ) T1. University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Measure Theory Definition An outer measure µ∗ is a nonnegative extended real-valued set function defined on all subsets of a space X and having the following properties: 1 µ∗(;) = 0 2 A ⊂ B ) µ∗(A) ≤ µ∗(B) 3 S1 ∗ P1 ∗ E ⊂ i=1 Ei ) µ (E) ≤ i=1 µ (Ei ) The second property above is called monotonicity and the third property is called countable subadditivity. University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Measure Theory Theorem (Monotone Convergence Theorem) Let ffng be a sequence of nonnegative measurable functions which converge almost everywhere to a function f and suppose that R R fn ≤ f for all n. Then, f = lim fn. University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Measure Theory Theorem (Riesz Representation Theorem) Let X be a compact Hausdorff space and C(X ) the space of continuous real-valued functions on X . Then to each bounded linear functional F on C(X ) there corresponds a unique finite signed Baire measure µ on X such that F (f ) = R fdµ for each f in C(X ). University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Banach Lattices Definition A real vector space E is an ordered vector space if for x; y, and z in E and for all real α > 0, the following properties hold: 1 If x ≤ y, then x + z ≤ y + z 2 If x ≤ y, then αx ≤ αy University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Banach Lattices Definition An ordered set (M; ≤) is called a lattice if any two elements in M have a least upper bound and a greatest lower bound. Definition A Riesz space or vector lattice is an ordered vector space with the property that for each pair of elements in E, both the infimum and the supremum are also in E. University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Banach Lattices Definition A vector lattice E with norm k · k with the property that jxj ≤ jyj implies kxk ≤ kyk for x and y in E and jxj = x _ (−x) is called a normed lattice. Definition If E is a normed lattice which is complete with respect to the norm, that is every Cauchy sequence converges, then E is called a Banach lattice. University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Examples of Banach Lattices Some examples of Banach lattices include: 1 The Lp-spaces, Lp(X ; B; µ), where (X ; B; µ) is a measure space and where the ordering is defined by f ≤ g if and only if f (x) ≤ g(x) almost everywhere. 2 The collection of continuous real-valued functions on a compact Hausdorff space X , denoted by C(X ), with the supremum norm kf k1 = supfjf (x)j : x 2 X g and the same ordering as in example 1. University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Properties of Banach Lattices Definition The positive cone of E, denoted E +, is all of the x in E such that x ≥ 0. Definition A subspace A of E is called order bounded if and only if A is contained in some order interval [x; y] = fz 2 E : x ≤ z ≤ yg. University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Dedekind Complete Definition E is called Dedekind complete if every non-empty order bounded set has a supremum and an infimum in E. Some examples of Dedekind complete Banach lattices include: 1 The Lp-spaces 2 C(X ) where X is extremally disconnected University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Ideals A subset A of E is called solid if jxj ≤ jyj for some y in A implies that x is in A. Additionally, every solid subspace I of E is called an ideal in E. The ideal generated by a non-empty subset A of E is the smallest, with respect to inclusion, ideal containing A. It coincides with the intersection of all ideals that contain A and is denoted by I (A). Furthermore, the ideal generated by a single element x is called the principal ideal generated by x and is denoted by I (x). Additionally, I (x) = fy 2 E : there exists λ ≥ 0 S such that jyj ≤ λjxjg = n[−nx; nx]. University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Example of a Principal Ideal Consider the Banach lattice C(X ) where X is compact and Hausdorff and the function e : X ! R defined by e(x) = 1 for all x in X . Then the principal ideal generated by e is given by I (e) = ff 2 C(X ) : there exists λ > 0 with jf j ≤ λeg. That is, I (e) = ff 2 C(X ) : there exists λ > 0 with jf (x)j ≤ λ for all x 2 X g. Since we are dealing with continuous, real-valued functions on a compact space, we have for every f in C(X ), there exists a λ > 0 such that kf k1 ≤ λ. Thus, I (e) = C(X ). University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Order Units Definition If I (e) = E, then e 2 E + is called an order unit or strong unit. That is, e is an order unit if for each x in X there exists some λ ≥ 0 such that jxj ≤ λe. University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Example of an Order Unit Let C([0; 1]) represent the continuous functions on the interval [0; 1]. Then, C([0; 1]) has multiple order units. Since any function that is continuous on [0; 1] must be bounded above by some multiple of 1 and bounded below by some multiple of 1, it follows that e = 1 is an order unit for C([0; 1]). In fact, any function bounded away from zero would serve as an order unit for C([0; 1]). University of North Alabama Jessica E. Stovall Preliminaries Associating a Linear Operator with T Theorems Regarding Support Sets Example of Space with No Order Units The space C(R) of continuous functions on the real line has no order units. Suppose the u is an order unit for C(R).