The -Angle in Normed Spaces

Abstract

Vector angles constitute a foundational element in the analysis of the geometric architecture of linear spaces. Within the framework of inner product spaces, a natural definition of angle emerges through the inner product itself, giving rise to a number of geometrically significant properties. In general normed spaces, however, the absence of an inner product renders the formulation of a suitable notion considerably more complex. Prior research introduced a novel form of orthogonality, designated ᵽ_LM-orthogonality, constructed upon the foundation of ᵽ_*-orthogonality, which itself is derived from norm derivatives. Extending this framework, the present study proposes a novel concept of angle between two nonzero vectors in real normed spaces, termed the ᵽ_LM-angle. The formulation is grounded in the functional ᵽ_LM, combined with an appropriate normalization scheme that accounts for its homogeneity characteristics. A series of fundamental properties pertaining to the ᵽ_LM-angle are subsequently derived and established. Among the results obtained, it is demonstrated that the ᵽ_LM-angle adheres to essential geometric characteristics, namely boundedness, symmetry, and invariance with respect to scalar multiplication. Furthermore, a precise correspondence between orthogonality and the right angle is established, wherein ᵽ_LM-orthogonality is shown to be equivalent to an angle of π/2. The investigation further examines the connection between the ᵽ_LM-angle and linear dependence. It is established that linearly dependent vectors necessarily yield a zero -angle. Nevertheless, counterexamples drawn from classical spaces, such as l^1, reveal that the converse does not hold in general, thereby exposing a distinctly non-Euclidean geometric behavior inherent to general normed spaces