Abstract
We introduce the weak approach structure for an arbitrary locally convex approach space and generalize the results from [1] about the weak approach structure of a normed space. Hereto we carefully develop the notion of a closed dual unit ball in an abstract setting (as a special kind of absolutely convex subset) because it is this kind of structure on the algebraic dual that induces, in a duality-compatible way, a locally convex approach structure on the original space.
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References
R. Lowen and M. Sioen. Approximations in functional analysis. Resultate der Mathematik, 37:345–372, 2000.
R. Lowen. Approach spaces: The Missing Link in the Topology-Uniformity-Metric Triad. Oxford Mathematical Monographs. Oxford University Press, 1997.
R. Lowen and S. Verwulgen. Approach vector spaces. Houston Journal of Mathematics, To appear.
M. Sioen and S. Verwulgen. Locally convex approach spaces. Applied General Topology, 4(2):263–279, 2003.
W. Rudin. Functional Analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, 1991.
H. H. Schaefer. Topological Vector Spaces. Graduade texts in mathematics. Springer-Verlag, 1999.
A. Grothendieck. Produit tensoriels topologiques et éspaces nucléaires. Memoirs of the American Mathematical Society, 16, 1955.
B. Windeis. Uniform Approach Theory. PhD thesis, University of Antwerp, 1997.
D. Pumplün and H. Röhrl. Banach spaces and totally convex spaces i. Communications in Algebra, 12(8):953–1019, 1984.
Dieter Pumplün and H. Röhrl. Banach spaces and totally convex spaces ii. Communications in Algebra, 13(5):1047–1113, 1985.
W. Sydow. On horn-functors and tensor products of topological vector spaces. Lecture Notes in Mathematics, 962:292–301, 1982.
R. Schatten. A theory of cross spaces, volume 26 of Annals of mathematical studies. Princeton university press, 1950.
J. von Neumann. On infinite direct products. Compositio Mathematica, 6:1–77, 1938.
N. Bourbaki. Livre V, Espaces vectoriels topologique. Eléments de mathématique. Hermann, 1964.
Z. Semadeni. Banach spaces of continuous functions, volume 55 of Monografie matematyczne. Polish scientific publishers, 1971.
R. A. Ryan. Introduction to Tensor Product of Banach Spaces. Springer Monographs in Mathematics. Springer Verlag, 2002.
R. Lowen and B. Windeis. Aunif, a common supercategory of pmet and unif. International Journal of Mathematics and mathematical sciences, 21:1–18, 1998.
R. Lowen and B. Windeis. Approach groups. Rocky Mountain Journal of Mathematics, 30(3):1057–1074, 2000.
A. Wilansky. Modern methods in topological vector spaces. Mc Graw-Hill, 1978.
R. Lowen and M. Sioen. A note on separation in AP. Applied General Topology, to appear.
H. Brézis. Analyse functionelle, théorie et applications. Collection mathématiques appliquées pour la maîtrise. Masson, 1983.
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Sioen, M., Verwulgen, S. Quantified functional analysis: recapturing the dual unit ball. Results. Math. 45, 359–369 (2004). https://doi.org/10.1007/BF03323389
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DOI: https://doi.org/10.1007/BF03323389