Abstract
Firstly, in the general normed linear space, the concepts of generalized isosceles orthogonal group, generalized Birkhoff orthogonal group, generalized Roberts orthogonal group, strong Birkhoff orthogonal group and generalized orthogonal basis are introduced. Secondly, the conclusion that any two nonzero generalized orthogonal groups must be linearly independent group is proven. And the existence of nonzero generalized orthogonal group and its linear correlation are discussed preliminarily, as well as some related properties of nonempty generalized orthogonal group in specific normed linear space namely the lp space.
Highlights
In the general normed linear space, the concepts of generalized isosceles orthogonal group, generalized Birkhoff orthogonal group, generalized Roberts orthogonal group, strong Birkhoff orthogonal group and generalized orthogonal basis are introduced
As scholars have deepened their understanding of functional analysis, especially the understanding of Banach geometric theory, generalized orthogonal theory in the normed linear space was established and corresponding studies were carried out
Is a vector group of X, if there exist αi ⊥B α j or α j ⊥B αi for any αi,α j ∈ A, the vector group A is a generalized Birkhoff orthogonal group of X
Summary
In the general normed linear space, the concepts of generalized isosceles orthogonal group, generalized Birkhoff orthogonal group, generalized Roberts orthogonal group, strong Birkhoff orthogonal group and generalized orthogonal basis are introduced. An example showing that there exist four nonzero Birkhoff orthogonal groups in two-dimensional space was given. Definition 1.2 ([3]) Let X be a normed linear space, x, y ∈ X , if they satisfy x+ y = x−y . The related properties of orthogonal groups were studied in specific normed linear space.
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