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


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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.


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