Duality of Bochner spaces by Seppo I: Hiltunen Abstract. We construct the generalized Lebesgue { Bochner spaces mvL p ( µ ; Π ) for pos- itive measures µ and for suitable real or complex topological vector spaces Π so that for 1 < p < +1 and Banachable Π with separable topology the strong dual of the classical mv p mv p∗ 0 Bochner space L ( µ ; Π ) becomes canonically represented by L ( µ ; Πσ ) . Hence we 0 need no separability assumption of the norm topology of the strong dual Π of Π . For p = 1 and for suitably restricted positive measures µ we even get a similar result without any separability of the norm topology of the target space Π . For positive Radon mea- sures on locally compact topological spaces these results are essentially contained on pages 588 { 606 in R. E. Edwards' classical Functional Analysis. Introduction and some preliminaries Our main objective in this article is the following A1 Theorem. Let 1 ≤ p < +1 , and let q = (1 − p −1 −1 if p 6= 1, and q = +1 if p = 1. Further, let µ be a positive measure on Ω, and with K 2 ftf R ; tfC g let Π 2 BaS (K ) be such that (1) or (2) or (3) or (4) or (5) or (6) below holds when mv p (D) means that µ is almost decomposable. Also let F = L (µ ; Π ), and F1 = mv q 0 L (µ ; Πσ ) if (1) or (2) or (5) or (6) below holds, otherwise letting F1 = mv q 0 Ls (µ ; Πσ ) . For R ι = h υs F × C \ f ( X ; t): 8 x 2 X ; y 2 Y ; t = Ω y : x d µ g : Y 2 υs F1 i 0 mv q 0 then ι 2 Liis (F1 ;F ) holds. In addition F1 = L (µ ; Π ) if (1) or (5) holds. (1) p = 1 and (D) and Π is reflexive, (2) p = 1 and (D) and τrd Π is a separable topology, m 1 (3) p = 1 and (D) and τrd L (µ) is a separable topology, m (4) p = 1 and (D) and a choice function c 2 L ( L +1 (µ) ; ` +1 (Ω)) exists, (5) p 6= 1 and Π is reflexive, (6) p 6= 1 and τrd Π is a separable topology. The proof is given on pages 44 { 48 below. Here we first explain the notation arXiv:1801.09059v2 [math.FA] 1 Feb 2018 appearing above, mentioning that we generally utilize the notational convention explained in [ H05 ; pp. 4 { 8 ] , [ H07 ; pp. 4 { 9 ] and [ H08 ; p. 1 ] , and further to be \polished" in [ Hfs ]. Having K 2 ftf R ; tfC g means that K is either the topological field of real num- bers or that of the complex ones. The underlying sets of these fields are R and C , respectively. Then Π 2 BaS (K ) means that Π is a Banachable, i.e. a complete 2010 Mathematics Subject Classification. Primary 46E40 , 46A20 , 46G10 , 46E30 ; Secondary 28A20 , 28B05 , 46A16 , 28C20. Key words and phrases. Bochner space, Banach space, duality, positive measure, positive Radonian, vector measure, Lebesgue space, topological vector space, suitable space, measurability, bounded variation, Dunford { Pettis. 1 2 S. HILTUNEN normable real or complex topological vector space. Thus there is a compatible norm ν on the underlying vector space σrd Π such that (σrd Π; ν ) is a normed --i −1 + Banach space. Being compatible here means that f ν ``[ 0 ; n : n 2 Z is a --i −1 filter base for the filter No Π of zero neighbourhoods. Above ν ``[ 0 ; n is the image of the closed interval [ 0 ; n −1 under the relational inverse ν --i of ν . Here ν is a function υs Π ! R+ where υs Π is the underlying set of vectors of Π . For ξ 2 υs Π one may sometimes write k ξ k for the value ν `ξ of ν at ξ . For Π1 2 t.v.s (K ) , i.e. having Π1 a real or complex topological vector space with possibly non-Hausdorff topology τrd Π1 , the exact construction of the space mv p E = L (µ ; Π1) is given in Constructions 42 (3) on page 23 below. Here we infor- mally explain the basic ideas under the additional assumption that Π1 is suitable in the sense of Definitions 15 on page 10 below. Then it suffices to consider one fixed dominating norm ν for Π1 . We consider functions x :Ω ! υs Π such that on every set A of finite measure, --i i.e. for A 2 µ ``R+ it holds that outside some set N of measure zero, i.e. with N 2 µ --i ``f0g we have x a pointwise limit of a sequence of simple functions, with convergence in the sense of the topology τrd Π. In the case p < +1 we then take the subset of those x such that the generally nonmeasurable function p p abs ◦ ν ◦ x :Ω 3 η 7! (ν ◦ x`η ) 2 R+ is dominated by some integrable function α :Ω ! [ 0 ; +1 ] . With the pointwise vector operations from σrd Π the set of these x becomes a vector substructure X Ω]vs of σrd Π . Then we take E = (X=vs N0 ; T ) when N0 is the set of all x 2 vs X --i R such that for all u 2 L (Π; K ) and A 2 µ ``R+ we have A u ◦ x d µ = 0 . Here L (Π; K ) is the set of all continuous linear maps Π ! K . Furthermore, we take the topology T so that a filter of zero neighbourhoods is formed by the sets R p vs (X=vs N0 ) \ f X : 9 x 2 X ; abs ◦ ν ◦ x d µ < " g + for " 2 R . Here we have the upper integral of the not-necessarily measurable p R function abs ◦ ν ◦ x that is defined as the infimum of the set of all Ω α d µ with α as above. For p = +1 we take the \obvious" modification. mv p The space E 1 = Ls (µ ; Π1) is constructed otherwise similarly except that we instead require the functions x to be such that u◦x j (A nN ) is measurable, that is, --i --i for every A 2 µ ``R+ we require existence of some N 2 µ ``f0g such that for all u 2 L (Π; K ) it holds that u ◦ x j (A n N ) is a measurable real or complex valued function on A n N. Then every vector of E is contained in some vector of E 1 , but we need not have υs E ⊆ υs E 1 . Note above that υs E = vs (X=vs N0) , and that X=vs N0 is the quotient vector space structure of X by the linear subspace N0 . Having now informally explained the general construction of our generalized 0 Bochner spaces, we note that if Π is Banachable, then Πσ is its weak dual space, and that (σrd Π; ν ) is a Banach space for any compatible norm ν for Π . At least σ mv p for finite positive measures µ then (σrd L (µ ; Π ) ; ν2 ) is a classical Bochner R p p−1 space when ν 2 is defined by X 7! Ω abs ◦ ν ◦ x d µ) for any x 2 X . The appearing ι is the function υs F1 3 Y 7! ι `Y with ι `Y given by R υs F 3 X 7! Ω y : x d µ for any x 2 X and y 2 Y . Here y : x is the function Ω 3 η 7! y`η`(x`η ) = (y`η )`(x`η ) 2 υs K 2 f R ; C g . 0 The message of Theorem A 1 is then that ι 2 Liis (F1 ;F ) holds, i.e. that ι is 0 0 a linear homeomorphism F1 ! F where F is the normable, hence Banachable 0 strong dual of F with υs (F ) = L (F; K ) . For the spaces appearing in (3) and Duality of Bochner spaces 3 m p mv p (4) note that we define L (µ) = L (µ ; tf R) . Below note that in the usual man- ner we have p ∗ = (1 − p −1 −1 for 1 < p < +1 and p ∗ = +1 for p = 1 , and also p ∗ = 1 in the case where p = +1 holds. The last part in condition (4) means that there is a continuous linear map c : m +1 +1 m +1 L (µ) ! ` (Ω) such that c`X 2 X holds for all X 2 υs L (µ) . Continuity here being equivalent to the property that for some M 2 R+ it holds that for m +1 --i --i x 2 X 2 υs L (µ) and for all A 2 µ ``R+ there is N 2 µ ``f0g such that j c`X `η j ≤ M j x`η j holds for all η 2 A n N, our condition is weaker than the requirement (b) in [ E ; Theorem 8.18.2, p. 588 ] that mL +1 (µ) can be \lifted". 1 Remarks. At first sight, it may seem that Theorem A 1 is less general than the results contained in [ E ; Theorems 8.18.2, 8.18.3, pp. 588, 590 ] when p = 1 since in Edwards' presentation there is stated no assumption on any kind of \decompos- ability". However, one should note that in [ E ] one considers only positive measures that are positive Radonian in the sense of Definitions 28 (4) on page 15 below, and that by Proposition 29 these are \automatically" almost decomposable.