New Version of the Daniell-Stone-Riesz Representation Theorem

New version of the Daniell-Stone-Riesz representation theorem

The traditional representation theorems after Daniell-Stone and Riesz were in a kind of separate existence until Pollard-Topsøe 1975 and Topsøe 1976 were the first to put them under common roofs. In the same spirit the present article wants to obtain a unified representation theorem in the context of the author’s work in measure and integration. It is an inner theorem like the previous ones. Th...

On the Inner Daniell-Stone and Riesz Representation Theorems

The paper deals with the context of the inner DaniellStone and Riesz representation theorems, which arose within the new development in measure and integration in the book 1997 and subsequent work of the author. The theorems extend the traditional ones, in case of the Riesz theorem to arbitrary Hausdorff topological spaces. The extension enforces that the assertions attain different forms. The ...

The Riesz representation theorem

A Banach-stone Theorem for Riesz Isomorphisms of Banach Lattices

Let X and Y be compact Hausdorff spaces, and E, F be Banach lattices. Let C(X,E) denote the Banach lattice of all continuous E-valued functions on X equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism Φ : C(X,E) → C(Y, F ) such that Φf is non-vanishing on Y if and only if f is non-vanishing on X, then X is homeomorphic to Y , and E is Riesz i...

A General Stone Representation Theorem

This note contains a Stone-style representation theorem for compact Haus-dorff spaces. The note is very much inspired by some existing representation theorems and is expository in nature. The first representation theorem is by Jung and Sünderhauf in [JS] and there is also a version of it for compact Hausdorff spaces noted by D. Moshier [Mo], the ideas of which were described to us by A. Jung. T...