Vol. 58, No. 2, 1975

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Orthogonally additive and orthogonally increasing functions on vector spaces

Stanley P. Gudder and D. Strawther

Vol. 58 (1975), No. 2, 427–436

A real-valued function f : X R on an inner product space X is orthogonally additive if f(x + y) = f(x) + f(y) whenever x y. We extend this concept to more general spaces called orthogonality vector spaces. If X is an orthogonality vector space and If there exists an orthogonally additive function on X which satisfies certain natural conditions then there is an inner product on X which is equivalent to the original orthogonality and f(x) = ±∥x2 for all x X. We next consider a normed space X with James’ orthogonality. A function f : X R is orthogonally increasing if f(x + y) f(x) whenever x y. Orthogonally increasing functions on normed spaces are characterized.

Mathematical Subject Classification
Primary: 46B05
Received: 28 March 1974
Revised: 26 December 1974
Published: 1 June 1975
Stanley P. Gudder
D. Strawther