The generalized sine function and geometrical properties of normed spaces

• Autorzy: Szostok T.

Identyfikatory

DOI

http://dx.doi.org/10.7494/OpMath.2015.35.L117

• Języki publikacji: EN

Abstrakty

EN

Let [formula] be a nornied space. We deal here with a function s : X x X —> R given by the formula [formula] (for x = 0 we must define it separately). Then we take two unit vectors x and y such that y is orthogonal to x in the Birkhoff-James sense. Using these vectors we construct new functions Φx,y which are defined on R. If X is an inner product space, then Φx, y = sin and, therefore, one may call this function a generalization of the sine function. We show that the properties of this function are connected with geometrical properties of the normed space X.

Słowa kluczowe

EN

geometry of normed spaces; smoothness; strict convexity; Birkhoff-James orthogonality; conditional functional equations

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2015
• Tom: Vol. 35, no. 1
• Strony: 117--126
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Szostok T.

• Silesian University Institute of Mathematics Bankowa 14, 40-007 Katowice, Poland

Bibliografia

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• [2] M.M. Day, Linear Normed Spaces, Springer Verlag, New York, 1973.
• [3] J. Ratz, On orthogonally additive mappings, Aequationes Math. 28 (1985), 35-49.
• [4] Gy. Szabó, A conditional Gauchy equation on normed spaces, Publ. Math. Debrecen 42/3-4 (1993), 265-271.
• [5] Gy. Szabó, Isosceles orthogonally additive mappings and inner product spaces, Publ. Math. Debrecen 46 (1995), 373-384.
• [6] T. Szostok, Modified version of Jensen equation and orthogonal additivity, Publ. Math. Debrecen, 58 (2001), 491-504.
• [7] T. Szostok, On some conditional functional equations, Ann. Math. Sil. 16 (2002), 65-77.
• [8] T. Szostok, On a generalization of the sine function, Glas. Mat. Ser. Ill 38(58) (2003), 29-44.