Dedekind cut

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Dedekind cut

(German ˈdedəˌkɪnt)
n
1. (Mathematics) a method of according the same status to irrational and rational numbers, devised by Julius Wilhelm Dedekind (1831-1916)
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Then P is an orthogonally-additive polynomial if and only if there exist two positive orthosymmetric polynomials [P.sub.1], [P.sub.2]: A [right arrow] [B.sup.[partial derivative]] the Dedekind completion of B) such that
Then there exists an order bounded operator T : [[PI].sub.n] (A) [right arrow] [B.sub.[partial derivative]] (the Dedekind completion of B) such that
Let K be the Dedekind completion of [I.sub.P], by Theorem 24.2 in Luxemburg and Zaann (1971:131), [I.sub.p.sup.[perpendicular to]] [direct sum] ([I.sub.p.sup.[perpendicular to]])[.sup.[perpendicular to]] = K and [I.sub.p.sup.[perpendicular to]] [intersection] ([I.sub.p.sup.[perpendicular to]])[.sup.[perpendicular to]] = {0}.