The Related Properties of Generalized Orthogonal Group in Specific Normed Linear Spaces

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.

complete. As a result, the study achievements within these fields have played an important role in the improvement of the inner product space theory. Naturally, scholars are making efforts to extend related theories to general normed linear spaces. At the same time, the application of related orthogonality of nonorthogonal function expansion is also rising [1].
In this paper, based on the research achievements of orthogonal theory, the concepts of generalized orthogonal group and generalized orthogonal basis in the general normed linear space were introduced. The conclusion that any two nonzero generalized orthogonal groups must be linearly independent groups was proven. An example showing that there exist four nonzero Birkhoff orthogonal groups in two-dimensional space was given. Also, this example further drew our attention to the study of generalized orthogonal groups' properties.
Next, the existence of nonzero generalized orthogonal groups and its linear correlation were discussed, as well as some related questions of nonzero generalized orthogonal groups in lp space.
The research on orthogonal and isometric mapping has attracted much attention as early as the beginning of the 20th century. In 1934, B. D. Roberts proposed the concept of Roberts orthogonality [2]. The specific definition is as follows. Definition 1.1 ( [2]) Let X be a normed linear space, , x y X ∈ , if the equation Definition 1.2 ( [3]) Let X be a normed linear space, , x y X ∈ , if they satisfy x y x y + = − . Then x is isosceles orthogonal or James orthogonal to y, denoted by I x y ⊥ . According to the property of "The perpendicular segment between the point and the line is shortest", the concept of Birkhoff orthogonality in normed linear space was proposed by G. Birkhoff  x y X ∈ and α ∈  , ⋅ satisfies the following conditions: Then ⋅ is called a norm on X, and ( ) , X ⋅ is called a normed linear space.
When the norm is not emphasized, X is used to represent ( ) , X ⋅ . A real finite-dimensional normed linear space is called a Minkowski space, and a two-dimensional Minkowski space is called a Minkowski plane.
Based on the definition and related theorems of orthogonality, as scholars keep studying in the field of orthogonal elements, conclusions in specific Orlicz sequence space were correspondingly drawn [7] [8]. Along with the further understanding of orthogonal elements, the concept of orthogonal groups was proposed for the reason that orthogonal elements are not commutative. In the inner product space, the generalized orthogonal theory was relatively complete. And in algebra, studies on vector groups were already carried out. Naturally, the generalized orthogonal groups whose properties are derived from orthogonality while keep their own differences were studied in this paper. Considering of different orthogonal properties, different orthogonal groups were correspondingly proposed. And the related properties of orthogonal groups were studied in specific normed linear space.
Based on these definitions, 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 correspondingly as follows: Definition 1.7 Let X be a normed linear space, ( ) then the vector group A is a generalized Roberts orthogonal group of X. Definition 1.11 Let A be a generalized orthogonal group on normed linear space X. If A is also a set of basis on X, then A is a generalized orthogonal basis of X. Theorem 1.1 Any two elements in any nonzero generalized orthogonal group in a normed linear space are linearly independent.
Proof In the following, we distinguish four cases.
x y is linearly related, then there exist y kx = , This is a contradiction to the definition of Birkhoff orthogonal. Thus, if ( ) , x y is a nonzero Birkhoff orthogonal group, then it is linearly independent.
x y is a nonzero strong Birkhoff orthogonal group. Assuming x y is linearly related, then there exist y kx = , This is a contradiction to the definition of Birkhoff orthogonal. Thus, if ( ) , x y is a nonzero strong Birkhoff orthogonal group, then it is linearly independent.
x y is a nonzero isosceles orthogonal group. Similarly, assum- This is a contradiction to the definition of isosceles orthogonal. Thus, if ( ) , x y is a nonzero isosceles orthogonal group, then it is linearly independent. The generalized orthogonal group of two elements is linearly independent, so it is natural for us to consider whether there are linear independent orthogonal group of three or more elements in two-dimensional space.
Reasoning 1.1 Any generalized orthogonal group containing two nonzero elements in a two-dimensional space is a generalized orthogonal basis.
However, it should be noted that in a three-dimensional space, a generalized orthogonal group containing three elements is not necessarily a generalized orthogonal basis. Specific examples are given below. ) For any convex quadrilateral abcd on the Minkowski plane X, the sum of its diagonals is no less than the sum of lengths of any set of opposite sides, that is Example 1.3 shows that the number of elements in a nonzero generalized orthogonal group can be greater than the number of elements in the basis.

Generalized Orthogonal Group in l p Space
In this part, the existence of nonzero generalized orthogonal groups and its linear correlations were applied in discussing the related questions in the specific normed linear space namely the l p space. The existences of generalized orthogonal elements and generalized orthogonal groups in  It follows that J  is one option of standard dual mapping. , ⋅ ⋅ is a semi-inner product of a generation norm on X.
Assume , X u v S ∈ , and we have We have B u w ⊥ which implies that u,w is a set of orthogonal basis.

Proposition 2.1 ([2]
) If X is a Minkowski plane, then for any vector x in X, there exists a corresponding H in X (H is a hyperplane passing through the origin) so that any vector in H is Birkhoff orthogonal to x (denoted by ). However, when the dimension of the space is greater than two, the situation is completely different.

Lemma 2.1 ([2]) Let X be a normed linear space with dimensions great than or equal to three. If H exists for any vector x such that
(H is a hyperplane through the origin), then x is an inner product space.
Next, we were trying to find the generalized orthogonal element in Hence , , 0, , 0 y y y =  and 1  Since y kx b = + , we have that ( ) which is an orthogonal point pair. Hence Thus, , , . It is proved that A, B and C are generalized Roberts orthogonal groups of X, and 1 x , 2 x , 1 y , 2 y , 1 z , 2 z are mutually commutative. Hence A, B and C are called generalized commutative Roberts orthogonal groups of X. . It is proved that A is a generalized Birkhoff orthogonal group of X. However, 1 x , 2 x , 3 x are not mutually Brikhoff orthogonal, then A cannot be called as a generalized commutative Birkhoff orthogonal group of X.
However, there still remains further study and discussion on the existence of generalized orthogonal groups in other normed linear spaces.

Conclusion
Based on the concept of orthogonal group in inner product space and some related properties, the definition of generalized orthogonal group in general normed linear space is introduced in this paper. Furthermore, the existence and linear correlation of nonzero generalized orthogonal groups are discussed. Some related problems of nonzero generalized orthogonal groups in specific normed linear space namely the lp space are discussed, and corresponding conclusions are drawn. Also, generalized orthogonal basis in the three-dimensional Orlicz sequence space is discussed, and the isosceles orthogonal basis is extended especially.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.