Vol. 58, No. 2, 1975

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 329: 1
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
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