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Download e-book for kindle: Tensors and Their Applications by Nazrul Islam

By Nazrul Islam

ISBN-10: 8122418384

ISBN-13: 9788122418385

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Extra info for Tensors and Their Applications

Example text

14]) and thus can be thought of as Basic concepts in the geometry of Banach spaces 25 a space of integrable functions on a finite measure space (S-2, #) with u* (x) = f x d # for x in X. One can assume that u* (u) = 1 = Ilu II and Ilu* II ~< 2. Since X is order continuous, one has for each x ~> 0 in X that (nu)/x x converges to x as n --+ cx~, so that Xu is dense in X. By replacing the underlying a-algebra with the smallest cr-algebra for which all the functions in X are measurable, it can be assumed that X is dense in L1 (#).

Indeed, take {Xn }n~=l dense in the unit sphere of X and for each n pick a norm one functional x n* which achieves its norm at Xn. It is easy to check that u* " - ~ 2 -n Ix~*l is a strictly positive functional. From this comment and the representation above it follows that any lattice inequality involving finitely many vectors which is true for lattices of functions must be true in a general Banach lattice. This eliminates the tedium of verifying "obvious" inequalities (for example, V( ~ i =" 1 ~:iXi 9 ~i - - -+'1} -~- ff-~i=l ]Xi[) directly from the axioms for a Banach lattice.

A function from s --~ X of the form ~in=l xi 1ai with each Ai a measurable subset of s is called a simple function. A function f : ~2 --+ X is called strongly measurable or just measurable if it is the limit almost everywhere of a sequence of simple functions. A function f is called scalarly measurable provided x * f is measurable for each x* in X*. A function f is measurable if and only if it is scalarly measurable and the range of f is essentially separable, which means that for some set A of measure zero, f[X2 ~ A] is separable.

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