Finite-dimensional lattice-subspaces of $C(\Omega )$ and curves of $\mathbb {R}^n$
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- by Ioannis A. Polyrakis PDF
- Trans. Amer. Math. Soc. 348 (1996), 2793-2810 Request permission
Abstract:
Let $x_1,\dotsc ,x_n$ be linearly independent positive functions in $C(\Omega )$, let $X$ be the vector subspace generated by the $x_i$ and let $\beta$ denote the curve of $\mathbb R^n$ determined by the function $\beta (t)=\frac {1}{z(t)} (x_1(t),x_2(t),\dotsc ,x_n(t))$, where $z(t)=x_1(t)+x_2(t)+\dotsb +x_n(t)$. We establish that $X$ is a vector lattice under the induced ordering from $C(\Omega )$ if and only if there exists a convex polygon of $\mathbb R^n$ with $n$ vertices containing the curve $\beta$ and having its vertices in the closure of the range of $\beta$. We also present an algorithm which determines whether or not $X$ is a vector lattice and in case $X$ is a vector lattice it constructs a positive basis of $X$. The results are also shown to be valid for general normed vector lattices.References
- Y. A. Abramovich, C. D. Aliprantis and I. A. Polyrakis, Lattice-subspaces and positive projections, Proc. Roy. Irish Acad., 94A (1994), 237–253.
- C. D. Aliprantis and K. C. Border, Infinite dimensional analysis: A hitchhickers guide, Studies in Economic Theory, #4, Springer-Verlag, New York and Heidelberg, 1994.
- Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. MR 809372
- P. Henrotte, Existence and optimality of equilibria in markets with tradable derivative securities, Stanford Institute for Theoretical Economics, Technical Report, No. 48, 1992.
- —, Three essays in financial economics, Ph.D. Dissertation, Department of Economics, Standford University, 1993.
- Graham Jameson, Ordered linear spaces, Lecture Notes in Mathematics, Vol. 141, Springer-Verlag, Berlin-New York, 1970. MR 0438077, DOI 10.1007/BFb0059130
- David G. Kendall, Simplexes and vector lattices, J. London Math. Soc. 37 (1962), 365–371. MR 138983, DOI 10.1112/jlms/s1-37.1.365
- Shizuo Miyajima, Structure of Banach quasisublattices, Hokkaido Math. J. 12 (1983), no. 1, 83–91. MR 689259, DOI 10.14492/hokmj/1381757794
- Anthony L. Peressini, Ordered topological vector spaces, Harper & Row, Publishers, New York-London, 1967. MR 0227731
- Ioannis A. Polyrakis, Lattice Banach spaces, order-isomorphic to $l_{1}$, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 3, 519–522. MR 720802, DOI 10.1017/S0305004100000906
- Ioannis A. Polyrakis, Schauder bases in locally solid lattice Banach spaces, Math. Proc. Cambridge Philos. Soc. 101 (1987), no. 1, 91–105. MR 877703, DOI 10.1017/S0305004100066433
- Ioannis A. Polyrakis, Lattice-subspaces of $C[0,1]$ and positive bases, J. Math. Anal. Appl. 184 (1994), no. 1, 1–18. MR 1275938, DOI 10.1006/jmaa.1994.1178
- Ivan Singer, Bases in Banach spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 154, Springer-Verlag, New York-Berlin, 1970. MR 0298399, DOI 10.1007/978-3-642-51633-7
Additional Information
- Ioannis A. Polyrakis
- Affiliation: Department of Mathematics, National Technical University, 157 80 Athens, Greece
- Email: ypoly@math.ntua.gr
- Received by editor(s): April 24, 1995
- Additional Notes: This research was supported in part by the NATO Collaborative Research Grant #941059.
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 2793-2810
- MSC (1991): Primary 46B42, 52A21, 15A48, 53A04
- DOI: https://doi.org/10.1090/S0002-9947-96-01639-X
- MathSciNet review: 1355300