# Constructive and Computational Methods for Differential and by D.L. Colton, R.P. Gilbert By D.L. Colton, R.P. Gilbert

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45 46 2 Thermostatics The linearity and continuity of the functionals thus defined are obvious, and it can be seen as follows that all functionals with these properties are 'of that form. By what was said earlier, a linear functional on f determines the restriction of an operator p to any finite-dimensional subspace. 2) it is necessary to ensure that jpi or respectively is bounded. 2 1), then their dual spaces are unaffected—the dual spaces of a space and of a dense subspace are the same. 21). The Banach space is thus not reflexive, so is strictly larger than If a Banach spaces is nonreflexiye.

Is weakly lower semicontinuous in a, so iflp) is weakly closed and, since p = 1, also weakly compact. 10; 3) the set Jt"(p) is convex. Moreover, (ii) By considering all the possibilities, one realizes that it is possible to write any unless a E 4'(p). *'(p) as (iii) Let a p, JI, 1> <1, iJ, p E p1 12, 1) (2, ii, where (Ii, 1> are two orthonormal systems. Let U 12, 1) = 1, I>, U, Ii, for I I I — I. U,Il,i) = 1> otherwise. 2 The Properties of Entropy 57 (iv) By the Krein—Milman theorem, (by (iii)), and e ii p.

Tr a(ln a — In b) i' convex. Our next task is to give the density matrices an ordering that indicates which of two p's corresponds to the more chaotic state. , with the greatest possible first eigenvalue. Because p = 1, two density matrices might not be strictly ordered by the natural ordering of Hermitian . operators. 9) A density matrix is said to be more mixed, or more chaotic, than p if p(n) for all n. In symbols, p (or p p3). 10) 1. This clearly defines a preordering of the density matrices.