8 2. PRELIMINARIES Furthermore, it follows from [51] (or Corollary 9.2) also that if P and K Cr0, 1s, then x˚p1ru,1sq 0 for all but countably many u thus x˚pXq 0 a.s. which shows that X is weakly a.s. separably valued. (Cf. Theorem 9.24.) Cf. also [62, Example 3-2-2] which studies essentially the same example but as an element of L8r0, 1s. Remark 2.6. If X is Borel measurable, then }X} is measurable, since x Þ Ñ }x} is continuous. However, if X only is weakly measurable, then }X} is not always measurable without additional hypotheses. (For this reason, we will sometimes use the upper integral ¯}X}.) E If X is weakly measurable and a.s. separably valued, then }X} is measurable, e.g. by Theorem 2.3. (In particular, there is no problem when B is separable.) Furthermore, if B has the property that there exists a countable norm-determining set of linear functionals, then every weakly measurable X in B has }X} measurable Dr0, 1s is an example of such a space. Remark 2.7. Several other forms of measurability may be considered, for ex- ample using the Baire σ-field (generated by the continuous functions B Ñ R) [62] or the σ-field generated by the closed (or open) balls in B [20], [21], [6]. Note further that, in general, Bw is not the same as the σ-field generated by the weak topology. (In fact, Bw equals the Baire σ-field for the weak topology [23], [62].) When B is separable, all these coincide with the Borel σ-field. See also [23], [24], [42] and [62], where further possibilities are discussed. 2.3. Tensor products of Banach spaces We give a summary of the definitions and some properties of the two main tensor products of Banach spaces. We refer to e.g. Blei [8] or Ryan [57] for further details. We consider the general case of the tensor product of k different spaces. The tensor products we consider (both algebraic and completed) are associative in a natural way for example, B1 b B2 b B3 pB1 b B2q b B3 B1 b pB2 b B3q, and the general case may be reduced to tensor products of two spaces. (Many authors, including [57], thus consider only this case.) 2.3.1. Algebraic tensor products. The algebraic tensor product of a finite sequence of vector spaces B1,...,Bk (over an arbitrary field) can be defined in an abstract way as a vector space B1 b ¨ ¨ ¨ b Bk with a k-linear map B1 ˆ ¨ ¨ ¨ ˆ Bk Ñ B1 ¨ ¨b Bk, written px1, . . . xkq Ñ x1 ¨ ¨b xk, such that if α : B1 ˆ¨ ¨ ¨ˆ Bk Ñ A is any k-linear map, then there is a unique linear map ˜ α : B1 b ¨ ¨ ¨ b Bk Ñ A such that (2.2) αpx1,...,xkq ˜px1 α b ¨ ¨ ¨ b xkq. (All such spaces are naturally isomorphic, so the tensor product is uniquely defined, up to trivial isomorphisms.) Several concrete constructions can also be given. One useful construction is to let Bi 7 be the algebraic dual of Bi and define B1 b ¨ ¨ ¨ b Bk as a subspace of the linear space of all k-linear forms on B1 7 ˆ ¨ ¨ ¨ ˆ Bk 7 more precisely we define x1 b ¨ ¨ ¨ b xk as the k-linear form on B1 7 ˆ ¨ ¨ ¨ ˆ Bk 7 defined by (2.3) x1 b ¨ ¨ ¨ b xkpx1 ˚, . . . , xk ˚q x1 ˚px1q ¨ ¨ ¨ xk ˚pxk q, and then define B1 b ¨ ¨ ¨ b Bk as the linear span of all x1 b ¨ ¨ ¨ b xk.
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