viii 196 Appendix In this appendix, we summarize some results about the integration and differ- entiation of Banach-space valued functions of a single variable. In a rough sense, vector-valued integrals of integrable functions have similar properties, often with similar proofs, to scalar-valued L1-integrals. Nevertheless, the existence of different topologies (such as the weak and strong topologies) in the range space of integrals that take values in an infinite-dimensional Banach space introduces significant new issues that do not arise in the scalar-valued case. 6.A. Vector-valued functions Suppose that X is a real Banach space with norm k·k and dual space X′. Let 0 <T < ∞, and consider functions f : (0,T ) → X. We will generalize some of the definitions in Section 3.A for real-valued functions of a single variable to vector-valued functions. 6.A.1. Measurability. If E ⊂ (0,T ), let 1 if t ∈ E, χ (t)= E 0 if t∈ / E, denote the characteristic function of E. Definition 6.13. A simple function f : (0,T ) → X is a function of the form N (6.38) f = cj χEj j=1 X where E1,...,EN are Lebesgue measurable subsets of (0,T ) and c1,...,cN ∈ X. Definition 6.14. A function f : (0,T ) → X is strongly measurable, or mea- surable for short, if there is a sequence {fn : n ∈ N} of simple functions such that fn(t) → f(t) strongly in X (i.e. in norm) for t a.e. in (0,T ). Measurability is preserved under natural operations on functions. (1) If f : (0,T ) → X is measurable, then kfk : (0,T ) → R is measurable. (2) If f : (0,T ) → X is measurable and φ : (0,T ) → R is measurable, then φf : (0,T ) → X is measurable. (3) If {fn : (0,T ) → X} is a sequence of measurable functions and fn(t) → f(t) strongly in X for t pointwise a.e. in (0,T ), then f : (0,T ) → X is measurable. We will only use strongly measurable functions, but there are other definitions of measurability. For example, a function f : (0,T ) → X is said to be weakly measurable if the real-valued function hω,fi : (0,T ) → R is measurable for every ω ∈ X′. This amounts to a ‘coordinatewise’ definition of measurability, in which we represent a vector-valued function by its real-valued coordinate functions. For finite-dimensional, or separable, Banach spaces these definitions coincide, but for non-separable spaces a weakly measurable function need not be strongly measur- able. The relationship between weak and strong measurability is given by the following Pettis theorem (1938). 6.A. VECTOR-VALUED FUNCTIONS 197 Definition 6.15. A function f : (0,T ) → X taking values in a Banach space X is almost separably valued if there is a set E ⊂ (0,T ) of measure zero such that f ((0,T ) \ E) is separable, meaning that it contains a countable dense subset. This definition is equivalent to the condition that f ((0,T ) \ E) is included in a closed, separable subspace of X. Theorem 6.16. A function f : (0,T ) → X is strongly measurable if and only if it is weakly measurable and almost separably valued. Thus, if X is a separable Banach space, f : (0,T ) → X is strongly measurable if and only hω,fi : (0,T ) → R is measurable for every ω ∈ X′. This theorem therefore reduces the verification of strong measurability to the verification of measurability of real-valued functions. Definition 6.17. A function f : [0,T ] → X taking values in a Banach space X is weakly continuous if hω,fi : [0,T ] → R is continuous for every ω ∈ X′. The space of such weakly continuous functions is denoted by Cw([0,T ]; X). Since a continuous function is measurable, every almost separably valued, weakly continuous function is strongly measurable. Example 6.18. Suppose that H is a non-separable Hilbert space whose dimen- sion is equal to the cardinality of R. Let {et : t ∈ (0, 1)} be an orthonormal basis of H, and define a function f : (0, 1) → H by f(t)= et. Then f is weakly but not strongly measurable. If K ⊂ [0, 1] is the standard middle thirds Cantor set and {e˜t : t ∈ K} is an orthonormal basis of H, then g : (0, 1) → H defined by g(t)=0 if t∈ / K and g(t)=˜et if t ∈ K is almost separably valued since |K| = 0; thus, g is strongly measurable and equivalent to the zero-function. ∞ Example 6.19. Define f : (0, 1) → L (0, 1) by f(t) = χ(0,t). Then f is not almost separably valued, since kf(t) − f(s)kL∞ = 1 for t 6= s, so f is not strongly 2 measurable. On the other hand, if we define g : (0, 1) → L (0, 1) by g(t) = χ(0,t), then g is strongly measurable. To see this, note that L2(0, 1) is separable and for every w ∈ L2(0, 1), which is isomorphic to L2(0, 1)′, we have 1 t (w,g(t))L2 = w(x)χ(0,t)(x) dx = w(x) dx. Z0 Z0 Thus, (w,g)L2 : (0, 1) → R is absolutely continuous and therefore measurable. 6.A.2. Integration. The definition of the Lebesgue integral as a supremum of integrals of simple functions does not extend directly to vector-valued integrals because it uses the ordering properties of R in an essential way. One can use duality to define X-valued integrals f dt in terms of the corresponding real-valued integrals hω,fi dt where ω ∈ X′, but we will not consider such weak definitions of R an integral here. Instead,R we define the integral of vector-valued functions by completing the space of simple functions with respect to the L1(0,T ; X)-norm. The resulting in- tegral is called the Bochner integral, and its properties are similar to those of the Lebesgue integral of integrable real-valued functions. For proofs of the results stated here, see e.g. [44]. 198 Definition 6.20. Let N f = cj χEj j=1 X be the simple function in (6.38). The integral of f is defined by T N f dt = cj |Ej | ∈ X 0 j=1 Z X where |Ej | denotes the Lebesgue measure of Ej . The value of the integral of a simple function is independent of how it is rep- resented in terms of characteristic functions. Definition 6.21. A strongly measurable function f : (0,T ) → X is Bochner integrable, or integrable for short, if there is a sequence of simple functions such that fn(t) → f(t) pointwise a.e. in (0,T ) and T lim kf − fnk dt =0. n→∞ Z0 The integral of f is defined by T T f dt = lim fn dt, n→∞ Z0 Z0 where the limit exists strongly in X. The value of the Bochner integral of f is independent of the sequence {fn} of approximating simple functions, and T T f dt ≤ kfk dt. 0 0 Z Z Moreover, if A : X → Y is a bounded linear operator between Banach spaces X, Y and f : (0,T ) → X is integrable, then Af : (0,T ) → Y is integrable and T T (6.39) A f dt = Af dt. Z0 ! Z0 More generally, this equality holds whenever A : D(A) ⊂ X → Y is a closed linear T operator and f : (0,T ) → D(A), in which case 0 f dt ∈ D(A). Example 6.22. If f : (0,T ) → X is integrableR and ω ∈ X′, then hω,fi : (0,T ) → R is integrable and T T ω, f dt = hω,fi dt. * Z0 + Z0 Example 6.23. If J : X ֒→ Y is a continuous embedding of a Banach space X into a Banach space Y , and f : (0,T ) → X, then T T J f dt = Jf dt. Z0 ! Z0 Thus, the X and Y valued integrals agree, and we can identify them. 6.A. VECTOR-VALUED FUNCTIONS 199 The following result, due to Bochner (1933), characterizes integrable functions as ones with integrable norm. Theorem 6.24. A function f : (0,T ) → X is Bochner integrable if and only if it is strongly measurable and T kfk dt < ∞. Z0 Thus, in order to verify that a measurable function f is Bochner integrable one only has to check that the real valued function kfk : (0,T ) → R, which is necessarily measurable, is integrable. Example 6.25. The functions f : (0, 1) → H in Example (6.18) and f : (0, 1) → L∞(0, 1) in Example (6.19) are not Bochner integrable since they are not strongly measurable. The function g : (0, 1) → H in Example (6.18) is Bochner integrable, and its integral is equal to zero. The function g : (0, 1) → L2(0, 1) in 1/2 Example (6.19) is Bochner integrable since it is measurable and kg(t)kL2 = t is integrable on (0, 1). We leave it as an exercise to compute its integral. The dominated convergence theorem holds for Bochner integrals. The proof is the same as for the scalar-valued case, and we omit it. Theorem 6.26. Suppose that fn : (0,T ) → X is Bochner integrable for each n ∈ N, fn(t) → f(t) as n → ∞ strongly in X for t a.e. in (0,T ), and there is an integrable function g : (0,T ) → R such that kfn(t)k≤ g(t) for t a.e. in (0,T ) and every n ∈ N. Then f : (0,T ) → X is Bochner integrable and T T T fn dt → f dt, kfn − fk dt → 0 as n → ∞.