Barrelledness in Topological and Ordered Vector Spaces by T. Husain, S.M. Khaleelulla PDF

By T. Husain, S.M. Khaleelulla

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Then qve E because the functions fi(xlf • • •, xn) are real-valued (i = 1, • • •, n). Now £ has no limit points in s because if there were such a limit point q0l then M(q0) = p by the continuity of 9ft. Thus there would be a ^>-point in 5 which would not be isolated. But J(p) # 0 by hypothesis. Hence there exists e > 0 such that the set n (aL + iblf • • •, an + ibn) / (a l5 • • • , a B ) 6 « , £ |6,| < e i= l contains the ^-points on s but no other ^-points and also such tha t the set £<2) _ j?

Let V, = [(z1,---,zq)eR<>lf(zv ••-,*„) = 0] q where R is a complex Euclidean q-space. Then if plt p2e Rq — Vf, there is a continuous path in Rq — Vf which has pl and p2 as its endpoints. PROOF. Define the polynomial in a complex variable A: (X) = f[(l-X)Pl + Xp2]. Since (f>(0) = f(pi) ^ 0 and cj>(l) = f(p2) ^ 0, polynomial is not identically zero and hence has only a finite number of zeros. Since these do not include 0 or 1, there is in the A-plane a continuous path Xt (0 g t g 1) such that A0 = 0, X1 = 1 and such that {Xt) # 0 for 0 < I < 1.

The mapping ^ would be the desired mapping except that there may be more than one g-point in an n-simplex. Now we show that there is a finite succession of simplicial subdivisions of A such that each n-simplex of the resulting complex contains at most one g-point. If xn contains more than one g-point, let o be the center of gravity of xn. 1), then each g-point in xn is in the interior of one of the simplexes of the subdivision. Let 8 be less than the minimum of Si, 82, - - -, 8m. Continue to make subdivisions as described above until the diameters of the resulting simplexes are all less than 8.

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