A Study of Generalized Orthogonality Relations in Normed Linear Spaces

✍ Scribed by Raymond W. Freese; Charles R. Diminnie; Edward Z. Andalafte

• Publisher: John Wiley and Sons
• Year: 1985
• Tongue: English
• Weight: 443 KB
• Volume: 122
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19851220120

✦ Synopsis

The concept of orthogonality in normed linear spaces has been studied extensively by BIRKHOFF [3], JAMES IS], [7], [8], and the present authors [l], 151, among others. The most natural notion of orthogonality arises in the case where there is an inner product (-, -) compatible with the norm 11. 11 on a space X. In this case, is defined by sly if and only if (x, 3) = 0. Some of the major properties of this relation are listed as follows :

(1) d z if and only if z=O for all zES,

(2) d y implies a z l y for all 2, yEX, aER (Homogeneity),

(3) zly implies 2/12 for all x, V E X (Symmetry),

(4) z1y and d z implies d (y + z ) for all z, y, z € . X (Additivity), and

( 5 ) For every z, VEX, x + O , there exists a real number y such that zl ('yz+3).

For general normed linear spaces (X, 11. 11) , the above-mentioned authors formulated definitions of orthogonality which did not require the existence of an inner product. These are given as follows:

BIRKHOFF Orthogonality [3], [73, [B]: ( d y ) ( B ) provided