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Generalized Vector Spaces. I.

Published online by Cambridge University Press:  20 November 2018

Karl Menger
Karl Menger*
Affiliation:
Illinois Institute of Technology, Chicago
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During the last fifty years, the concept of the Euclidean space (an ndimensional coordinate space with a Pythagorean distance) has undergone various profound generalizations.

Hilbert introduced the infinitely-dimensional Euclidean space whose points are infinite sequences of coordinates having from the origin, and thus from each other, finite Pythagorean distances.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 1 , 01 February 1949 , pp. 94 - 104
Copyright
Copyright © Canadian Mathematical Society 1949

References

[1] Banach, S., Fund. Math., vol. 3 (1922), 133-181;Google Scholar
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[2] Cf., in particular, “Die metrische Methode in der Variationsrechnung,” Ergebn. mathem. Kolloquiums, vol. 8 (1937), 1-32; two notes in the Proc. Nat. Acad. Sci., vol. 23 (1937), 246 and vol. 25 (1939), 474; and the lecture “Analysis and Metric Geometry“ in The Rice Institute Pamphlet, vol. 27 (1940), 1-40.Google Scholar
[3] Ergebn. mathem. Kolloquiums,vol. 8 (1937), 32.Google Scholar
[4]Cf., in particular, Pauc's résumé in the pamphlet “Les méthodes directes en Calcul des Variations,” (Paris, Herman, 1941).Google Scholar
[5] Rend. delta Ace. Naz. Line, vol. 26 (1937).Google Scholar
[6] Ergebn. math. Kolloquiums, vol. 8 (1937), 25 and Alt, loc. cit.3 Illinois Institute Google Scholar