À la recherche de la preuve perdue: a simple proof of the Ficken theorem

Dariusz Cichoń

Abstract


A new proof of the Ficken criterion is given together with a comment concerning the known proofs and related results.

Keywords


inner product; inner product norm

Mathematics Subject Classification (2010)


46C15

References


Amir, D. "Characterizations of inner product spaces." Operator Theory: Advances and Applications 20. Basel: Birkhäuser Verlag, 1986.

Besicovitch, A.S. "A problem on a circle." J. London Math. Soc. 36 (1961): 241-244.

Carathéodory, C. "Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen." Rend. Circ. Mat. Palermo 32 (1911): 193-217.

Danelich, A. "Normed spaces which satisfy Apollonius’ theorem." Mat. Zametki 20 (1976): 247-252.

Day, M.M. "Normed Linear Spaces>" Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 21. New York-Heidelberg: Springer-Verlag, 1973.

Falkner, N. "A characterization of inner product spaces." Amer. Math. Monthly 100.3 (1993): 246-249.

Ficken, F.A. "Note on the existence of scalar products in normed linear spaces." Ann. of Math. (2) 45 (1944): 362-366.

Istratescu, V.I. "Theory and applications." Mathematics and its Applications 25. Dordrecht: D. Reidel Publishing Co., 1987.

Lorch, E.R. "On certain implications which characterize Hilbert space." Ann. of Math. (2) 49 (1948): 523-532.


Full Text: PDF

Download statistics: 1950

Licencja Creative Commons
This article by Dariusz Cichoń is governed by the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported licence.


e-ISSN: 2300-133X, ISSN: 2081-545X

Since 2017 Open Access in De Gruyter and CrossCheck access cofinanced by The Ministry of Science and Higher Education - Republic of Poland - DUN 775/P-DUN/2017 see more

The Journal is indexed in:
and others see Abstracting and Indexing list

AUPC SM is on the List of the Ministry’s scored journals with 20 points for 2019

Deklaracja dostępności cyfrowej