Comparison of two types of order convergence with topological convergence in an ordered topological vector space
Authors:
Roger W. May and Charles W. McArthur
Journal:
Proc. Amer. Math. Soc. 63 (1977), 4955
MSC:
Primary 46A40
MathSciNet review:
0438078
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Abstract: Birkhoff and Peressini proved that if is a complete metrizable topological vector lattice, a sequence converges for the topology iff the sequence relatively uniformly star converges. The above assumption of lattice structure is unnecessary. A necessary and sufficient condition for the conclusion is that the positive cone be closed, normal, and generating. If, moreover, the space is locally convex, Namioka [11, Theorem 5.4] has shown that coincides with the order bound topology and Gordon [4, Corollary, p. 423] (assuming lattice structure and local convexity) shows that metric convergence coincides with relative uniform star convergence. Omitting the assumptions of lattice structure and local convexity of it is shown for the nonnecessarily local convex topology that and when is locally convex.
 [1]
Garrett
Birkhoff, Lattice theory, Third edition. American Mathematical
Society Colloquium Publications, Vol. XXV, American Mathematical Society,
Providence, R.I., 1967. MR 0227053
(37 #2638)
 [2]
Lyne
H. Carter, An order topology in ordered
topological vector spaces, Trans. Amer. Math.
Soc. 216 (1976),
131–144. MR 0390704
(52 #11527), http://dx.doi.org/10.1090/S00029947197603907042
 [3]
J. M. Ceĭtlin, Unconditional bases and semiorderedness, Izv. Vysš. Učebn. Zaved. Mat. 1966, no. 2(51), 98104; English transl., Amer. Math. Soc. Transl. (2) 90 (1970), 1725. MR 33 #6362; 41 #8191.
 [4]
Hugh
Gordon, Relative uniform convergence, Math. Ann.
153 (1964), 418–427. MR 0161123
(28 #4332)
 [5]
Graham
Jameson, Ordered linear spaces, Lecture Notes in Mathematics,
Vol. 141, SpringerVerlag, BerlinNew York, 1970. MR 0438077
(55 #10996)
 [6]
L.
V. Kantorovič, B.
Z. Vulih, and A.
G. Pinsker, Partially ordered groups and partially ordered linear
spaces, Amer. Math. Soc. Transl. (2) 27 (1963),
51–124. MR
0151532 (27 #1517)
 [7]
J.
L. Kelley and Isaac
Namioka, Linear topological spaces, With the collaboration of
W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G.
Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. The University
Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.,
1963. MR
0166578 (29 #3851)
 [8]
V.
L. Klee Jr., Boundedness and continuity of linear functionals,
Duke Math. J. 22 (1955), 263–269. MR 0069387
(16,1030b)
 [9]
W.
A. J. Luxemburg and A.
C. Zaanen, Riesz spaces. Vol. I, NorthHolland Publishing Co.,
AmsterdamLondon; American Elsevier Publishing Co., New York, 1971.
NorthHolland Mathematical Library. MR 0511676
(58 #23483)
 [10]
R. W. May, Comparison of order topologies with the topology of an ordered topological vector space, Doctoral Dissertation, Florida State Univ., Tallahassee, 1975.
 [11]
Isaac
Namioka, Partially ordered linear topological spaces, Mem.
Amer. Math. Soc. no. 24 (1957), 50. MR 0094681
(20 #1193)
 [12]
Anthony
L. Peressini, Ordered topological vector spaces, Harper &
Row, Publishers, New YorkLondon, 1967. MR 0227731
(37 #3315)
 [13]
B.
Z. Vulikh, Introduction to the theory of partially ordered
spaces, Translated from the Russian by Leo F. Boron, with the
editorial collaboration of Adriaan C. Zaanen and Kiyoshi Iséki,
WoltersNoordhoff Scientific Publications, Ltd., Groningen, 1967. MR 0224522
(37 #121)
 [1]
 G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R.I., 1967. MR 37 #2638. MR 0227053 (37:2638)
 [2]
 L. H. Carter, An order topology in ordered topological vector spaces, Trans. Amer. Math. Soc. 216 (1976), 131144. MR 0390704 (52:11527)
 [3]
 J. M. Ceĭtlin, Unconditional bases and semiorderedness, Izv. Vysš. Učebn. Zaved. Mat. 1966, no. 2(51), 98104; English transl., Amer. Math. Soc. Transl. (2) 90 (1970), 1725. MR 33 #6362; 41 #8191.
 [4]
 H. Gordon, Relative uniform convergence, Math. Ann. 153 (1964), 418427. MR 28 #4332. MR 0161123 (28:4332)
 [5]
 G. J. O. Jameson, Ordered linear spaces, Lecture Notes in Math., vol. 141, SpringerVerlag, Berlin and New York, 1970. MR 0438077 (55:10996)
 [6]
 L. V. Kantorovič, B. Z. Vulih and A. G. Pinsker, Partially ordered groups and partially ordered linear spaces, Uspehi Mat. Nauk 6 (1951), no. 3 (43), 3198; English transl., Amer. Math. Soc. Transl. (2) 27 (1963), 51124. MR 13, 361; 27 #1517. MR 0151532 (27:1517)
 [7]
 J. L. Kelley and I. Namioka, Linear topological spaces, Van Nostrand, Princeton, N.J., 1963. MR 29 #3851. MR 0166578 (29:3851)
 [8]
 V. L. Klee, Jr., Boundedness and continuity of linear functionals, Duke Math. J. 22 (1955), 263269. MR 16, 1030. MR 0069387 (16:1030b)
 [9]
 W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces, Vol. I, NorthHolland, Amsterdam, 1971. MR 0511676 (58:23483)
 [10]
 R. W. May, Comparison of order topologies with the topology of an ordered topological vector space, Doctoral Dissertation, Florida State Univ., Tallahassee, 1975.
 [11]
 I. Namioka, Partially ordered linear topological spaces, Mem. Amer. Math. Soc. No. 24 (1957). MR 20 #1193. MR 0094681 (20:1193)
 [12]
 A. L. Peressini, Ordered topological vector spaces, Harper and Row, New York, 1967. MR 37 #3315. MR 0227731 (37:3315)
 [13]
 B. Z. Vulih, Introduction to the theory of partially ordered spaces, Fizmatgiz, Moscow, 1961; English transl., Noordhoff, Groningen, 1967. MR 24 #A3494; 37 #121. MR 0224522 (37:121)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197704380789
PII:
S 00029939(1977)04380789
Keywords:
Relative uniform convergence,
order convergence,
normal cone,
generating cone,
relatively uniform topology,
order topology
Article copyright:
© Copyright 1977
American Mathematical Society
