Orthogonality and linear functionals in normed linear spaces

Abstract

The natural definition of orthogonality of elements of an abstract Euclidean space is that x ly if and only if the inner product (x, y) is zero. Two definitions have been given [11](2) which are equivalent to this and can be generalized to normed linear spaces, preserving the property that every twodimensional linear subset contain nonzero orthogonal elements. The definition which will be used here (Definition 1.2) has the added advantage of being closely related to the theories of linear functionals and hyperplanes. The theory and applications of this orthogonality have been organized in the following sections, which are briefly outlined: 1. Fundamental definitions. An element x is orthogonal to an element y if and only if j|x+kyjj > ||x|l for all k. This orthogonality is homogeneous, but is neither symmetric nor additive. 2. Existence of orthogonal elements. An element x of a normed linear space is orthogonal to at least one hyperplane through the origin, while for elements x and y there is at least one number a for which ax+y Ix or ||ax+yj| is minimum (Theorems 2.2-2.3). 3. Orthogonality in general normed linear spaces. The limits N?(x; y) =+lim,+.?|lnx+yil jjnxjj =liml0?o [||x+hyjj -lixil ]/hexistandsatisfyweakened linearity conditions. Also, x Iax+y if and only if N_(x; y) < -a||x|| <N+(x; y), while N+(y; x) =0 for all nonzero x and y satisfying N+(x; y) =0 if and only if orthogonality is symmetric and N_(x; y) N+(x; y) (Theorems 3.2 and 3.5). 4. Types of uniqueness of orthogonality. An element x of a normed linear space is orthogonal to only one hyperplane through the origin if and only if orthogonality is additive, or if and only if the norm is Gateaux differentiable (Theorems 4.1-4.2). The space is strictly convex if and only if there is a unique number a for which ax +y Ix (Theorem 4.3). 5. Hyperplanes and linear functionals. Conditions for orthogonality can be given in terms of hyperplanes, while there is a unique number a with xo Iaxo+y if and only if there is a unique linear functional f with I lfI = 1 and f(xo) = ljxojj, or if and only if the sphere l|x|| < lixoll has a tangent hyper-