This paper is a continuation of four others under the same title†. The paragraphs are numbered following on to those of the fourth paper of the series. In § XXIV we show that, if λ(A) is a linear functional, then there exists a resolution Eμ such that λ(A) = ∫μdr(AEμ), and if B = ∫μdEμ is bounded, then λ(A) = τ(AB)‡ for all A, where τ is the trace. This implies that τ(A) is a linear functional, and that the conjugate space ℒ, i.e. the space of the linear functionals, has a subset ℒ′ which is in (1, 1) correspondence with the original set of operators, and that in this correspondence the linear functional τ(A) is associated with the unit operator.