Constructive Existence of Minkowski Functionals

In Bishop's constructive mathematics, the framework of this paper, there are many situations where we cannot easily prove the existence of functional whose existence is a trivial consequence of classical logic. One such functional is the Minkowski functional of a convex absorbing set. We shall prove the existence of Minkowski functionals in various spaces, and apply the theorems to establish the locatedness of the kernel of linear mappings. In Bishop's constructive mathematics, the framework of this paper, there are many situations where we cannot easily prove the existence of functionals whose existence is a trivial consequence of classical logic. One such functional is the Minkowski functional of a convex absorbing set. We say that a convex subset C of a normed linear space X is absorbing if every x in X lies in rC for some r > 0 : note, for a convex absorbing subset C, that 0 e C and that 0 < s < t implies sC ç tC. The Minkowski functional p of C is defined by p(x) = inf{r > 0: x £ rC} (x £ X). It is perhaps surprising that we cannot prove constructively that every convex absorbing subset in R has a Minkowski functional. For if we could do this, then, by considering a convex absorbing subset C = UJjlri-4 o„ , 1 + a„) where {a„} is an increasing binary sequence, we would be able to prove the statement 3zz(a„ = 1) V Vzz(a„ = 0), which is called the limited principle of omniscience (LPO): in fact, if the Minkowski functional p of C exists, then either 2 > p(2) or p(2) > 1 ; in the former case, we have 2 rx for some x e C and r with 0 < r < 2 ; so there exists N such that x e (-1 a#, 1 + un) ; if a^ = 0, then 1 < 2/r = x < 1 , a contradiction; so a# = 1 ; in the latter case, if a„ — 1 for some n, then C = (-2, 2); so 1 > p(2) > 1 , a contradiction; so a„ = 0 for all n. Constructive mathematicians do not accept LPO, and intuitionists and Russian constructive mathematicians can prove that LPO is false. For further comments on LPO, see [4, Chapter I]. Received by the editors September 25, 1990 and, in revised form, February 6, 1991. 1980 Mathematics Subject Classification (1985 Revision). Primary 46R05; Secondary 03F65.