Appendix A The Bochner Integral Let X; kk be a Banach space, B.X/ the Borel -field of X and .; F;/a measure space with finite measure . A.1 Definition of the Bochner Integral Step 1: As a first step we want to define the integral for simple functions which are defined as follows. Set n ˇ n o ˇ X E ˇ 1 ; ; F;16 6 ; N WD f W ! X f D xk Ak xk 2 X Ak 2 k n n 2 kD1 and define a semi-norm kkE on the vector space E by Z kf kE WD kf k d; f 2 E: To get that E; kkE is a normed vector space we consider equivalence classes with respect to PkkE . For simplicity we will not change the notation. E n 1 For f 2 , f D kD1 xk Ak , Ak’s pairwiseP disjoint (such a representation is called n 1 ./ ;:::; normal and always exists, because f D kD1 xk Ak ,wheref Dfx1 xkg, xi 6D xj,andAk WD ff D xkg) and we now define the Bochner integral to be Z Xn f d WD xk.Ak/: kD1 © Springer International Publishing Switzerland 2015 209 W. Liu, M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, DOI 10.1007/978-3-319-22354-4 210 Appendix A (Exercise: This definition is independent of representations, and hence linear.) In this way we get a mapping E; ; int W kkE ! ZX kk f 7! f d R R which is linear and uniformly continuous since f d 6 kf k d for all f 2 E. Therefore we can extend the mapping int to the abstract completion of E with respect to kkE which we denote by E. Step 2: We give an explicit representation of E. Definition A.1.1 A function f W ! X is called strongly measurable if it is F=B.X/-measurable and f ./ X is separable. Definition A.1.2 Let 1 6 p < 1. Then we define Lp.; F;I X/ WD Lp.I X/ ˇ ˇ ˇ WD f W ! X ˇ f is strongly measurable with Z respect to F; and kf kp d<1 and the semi-norm ÂZ à 1 p p p kf kLp WD kf k d ; f 2 L .; F;I X/: p The space of all equivalence classes in L .; F;I X/ with respect to kkLp is denoted by Lp.; F;I X/ WD Lp.I X/. Claim L1.; F;I X/ D E. 1 Step 2.a: L .; F;I X/; kkL1 is complete. The proof is just a modification of the proof of the Fischer–Riesz theorem by the following proposition. Proposition A.1.3 Let .; F/ be a measurable space and let X be a Banach space. Then: (i) the set of F=B.X/-measurable functions from to X is closed under the formation of pointwise limits, and (ii) the set of strongly measurable functions from to X is closed under the formation of pointwise limits. A.2 Properties of the Bochner Integral 211 Proof Simple exercise or see [18, Proposition E.1, p. 350]. ut 1 Step 2.b E is a dense subset of L .; F;I X/ with respect to kkL1 . This will follow from the following lemma. Lemma A.1.4 Let E be a metric space with metric d and let f W ! E be strongly measurable. Then there exists a sequence fn,n2 N, of simple E-valued functions (i.e. fn is F=B.E/-measurable and takes only a finite number of values) such that for arbitrary ! 2 the sequence d fn.!/; f .!/ ,n2 N, is monotonely decreasing to zero. Proof ([26, Lemma 1.1, p. 16]) Let fek j k 2 Ng be a countable dense subset of f ./.Form 2 N and ! 2 define ˚ ˇ « ˇ dm.!/ WD min d f .!/; ek k 6 m D dist.f .!/; fek; k 6 mg/ ; ˚ ˇ « ˇ km.!/ WD min k 6 m dm.!/ D d f .!/; ek ; .!/ : fm WD ekm.!/ Obviously fm, m 2 N, are simple functions since they are F=B.E/-measurable (exercise) and fm./ fe1; e2;:::;emg: Moreover, by the density of fek j k 2 Ng, the sequence dm.!/ , m 2 N,is monotonically decreasing to zero for arbitrary ! 2 .Sinced fm.!/; f .!/ D dm.!/ the assertion follows. ut Let now f 2 L1.; F;I X/. By Lemma A.1.4 above we get the existence of a sequence of simple functions fn, n 2 N, such that fn.!/ f .!/ # 0 for all ! 2 as n !1: n!1 Hence fn ! f in kkL1 by Lebesgue’s dominated convergence theorem. A.2 Properties of the Bochner Integral Proposition A.2.1 (Bochner Inequality) Let f 2 L1.; F;I X/.Then Z Z k f dk 6 kf k d: 212 Appendix A Proof We know the assertion is true for f 2 E,i.e.intW E ! X is linear, continuous with kint f k 6 kf kE for all f 2 E, so the same is true for its unique continuous extension int W E D L1.I X/ ! X,i.e.forallf 2 L1.X;/ Z Z k f dkD int f 6 kf kE D kf k d: ut Proposition A.2.2 Let f 2 L1.; F;I X/.Then Z ÂZ à L ı f d D L f d holds for all L 2 L.X; Y/, where Y is another Banach space. Proof Simple exercise or see [18, Proposition E.11, p. 356]. ut Proposition A.2.3 (Fundamental Theorem of Calculus) Let 1 < a < b < 1 and f 2 C1 Œa; bI X .Then Z ( R t 0 1Œ ; .u/f .u/ duifs6 t f .t/ f .s/ D f 0.u/ du WD R s t 0 s 1Œt;s.u/f .u/ duotherwise for all s; t 2 Œa; b where du denotes the Lebesgue measure on B.R/. Proof R . / t 0. / Œ ; 0. / 0. / Claim 1: If we set F t WD s f u du, t 2 a b , we get that F t D f t for all t 2 Œa; b. Forthatwehavetoprovethat 1 h!0 k F.t C h/ F.t/ f 0.t/k ! 0: h X To this end we fix t 2 Œa; b and take an arbitrary ">0 .Sincef 0 is continuous Œ ; ı>0 0. / 0. / <" Œ ; on a b there exists a such that f u f t X for all u 2 a b with ju tj <ı. Then we obtain that Z 1 1 tCh 0 0 0 k F.t C h/ F.t/ f .t/kX Dk f .u/ f .t/ dukX h h t Z 1 t_.tCh/ 6 0. / 0. / <" f u f t X du jhj t^.tCh/ if t C h 2 Œa; b and jhj <ı. A.2 Properties of the Bochner Integral 213 Claim 2: If FQ 2 C1 Œa; bI X is a further function with FQ 0 D F0 D f 0 then there exists a constant c 2 X such that F FQ D c. LetusfirstassumethatF.a/ D 0 D FQ .a/. Then for all L 2 X D L.X; R/ . Q / 0 0 we define gL WD L F F .ThengL D and therefore gL is constant cL.But gL.a/ D L.F.a/ FQ .a// D 0,socL D 0.SinceX separates the points of X,by the Hahn–Banach theorem (see [3, Satz 4.2, p. 114]) this implies that F FQ D 0. In the general case we apply the above to F F.a/ and FQ FQ .a/ to obtain the assertion. ut Appendix B Nuclear and Hilbert–Schmidt Operators Let U; h ; iU and H; h ; i be two separable Hilbert spaces. The space of all bounded linear operators from U to H is denoted by L.U; H/; for simplicity we write L.U/ instead of L.U; U/. If we speak of the adjoint operator of L 2 L.U; H/ we write L 2 L.H; U/.AnelementL 2 L.U/ is called symmetric if hLu;viU Dhu; LviU for all u;v 2 U. In addition, L 2 L.U/ is called nonnegative if hLu; ui > 0 for all u 2 U. Definition B.0.1 (Nuclear Operator) An element T 2 L.U; H/ is said to be a nuclear operator if there exists a sequence .aj/j2N in H and a sequence .bj/j2N in U such that X1 Tx D ajhbj; xiU for all x 2 U jD1 and X kajkkbjkU < 1: j2N The space of all nuclear operators from U to H is denoted by L1.U; H/. If U D H and T 2 L1.U; H/ is nonnegative and symmetric, then T is called trace class. Proposition B.0.2 The space L1.U; H/ endowed with the norm n ˇ 1 o X ˇ X kTkL1 .U;H/ WD inf kajkkbjkU ˇ Tx D ajhbj; xiU; x 2 U j2N jD1 is a Banach space. © Springer International Publishing Switzerland 2015 215 W. Liu, M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, DOI 10.1007/978-3-319-22354-4 216 Appendix B Proof [60, Corollary 16.25, p. 154]. ut Definition B.0.3 Let T 2 L.U/ and let ek, k 2 N, be an orthonormal basis of U. Then we define X tr T WD hTe k; ekiU k2N if the series is convergent.