Mathematics and Statistics 11(4): 703-709, 2023 http://www.hrpub.org DOI: 10.13189/ms.2023.110412 Self-Adjoint Operators in Bilinear Spaces Sabarinsyah 1,* , Hanni Garminia 2 , Pudji Astuti 2 , Zelvin Mutiara Leastari 1 1 Department of Mathematics, Institut Teknologi Batam, Indonesia 2 Algebra Research Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia Received February 22, 2023; Revised May 24, 2023; Accepted June 11, 2023 Cite This Paper in the Following Citation Styles (a): [1] Sabarinsyah, Hanni Garminia, Pudji Astuti, Zelvin Mutiara Leastari , "Self-Adjoint Operators in Bilinear Spaces," Mathematics and Statistics, Vol. 11, No. 4, pp. 703 - 709, 2023. DOI: 10.13189/ms.2023.110412. (b): Sabarinsyah, Hanni Garminia, Pudji Astuti, Zelvin Mutiara Leastari (2023). Self-Adjoint Operators in Bilinear Spaces. Mathematics and Statistics, 11(4), 703 - 709. DOI: 10.13189/ms.2023.110412. Copyright©2023 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract In this research, it was agreed that a bilinear form is an extension of the inner product since a symmetry bilinear form will be equivalent to the inner product over a field of real numbers. Concepts in bilinear space, such as the concept of orthogonality of two vectors, the concept of orthogonal subspace of a subspace, the concept of adjoint operators of a linear operator and the concept of closed subspace are defined according to those prevailing in the inner product space fact assumed to be extensions of the concepts applicable in the inner product space. In the context of a cap subspace, we can identify the necessary and sufficient conditions for any linear operator in a continuous Hilbert space. These results open up opportunities to introduce the concept of pseudo-continuous linear mapping in bilinear spaces. We have obtained the result that pseudo-continuous linear mapping spaces in bilinear spaces have a relationship with linear mapping spaces that have adjoint mapping. We have also obtained the result that the structure of linear operators limited to Hilbert spaces can be extended to pseudo-continuous operator structures in bilinearal spaces. In this study, we have generalized the properties of self-adjoint operators in product spaces in infinite dimensions to bilinear, including closed properties of addition operations, and scalar multiplication, commutative properties, properties of inverse operators, properties of zero operators, properties of polynomial operators over real fields, and orthogonal properties of eigenspaces of different eigenvalues. Keywords Self-Adjoint Operator, Non-Degenerated Bilinear Forms, Pseudo-Continuity 1. Introduction The bilinear forms are very closely related to the inner products. In any vector space over a field where a bilinear form is defined, it is called a bilinear space. It generally holds that a symmetry bilinear form will be equivalent to the inner product over a field of real numbers. According to this nature, in this research, it was agreed that a bilinear form is an extension of the inner product. Since in the inner product space, the concept of orthogonality of two vectors, the concept of orthogonal subspace of a subspace, the concept of adjoint operator of a linear operator and the concept of closed subspace have been known, then in bilinear space the concept is defined in the same way and can be viewed as an extension of the concept that applies in the inner product space. 2. Materials and Methods We were able to identify the necessary and sufficient condition for linear operators in Hilbert spaces being continuous in terms of closed subspaces [1]. This fact gave us the opportunity to introduce the pseudo-continuous notion of linear mappings on bilinear spaces. We obtained that the class of pseudo-continuous linear mappings on bilinear spaces is nonetheless the class of linear mappings that have adjoint mappings [2]. As a result, a class of pseudo-continuous linear operators in a bilinear space forms a subalgebra. It is interesting to see how far the structure of the subalgebra of bounded linear operators on a Hilbert space can be extended to the subalgebra of pseudo-continuous operators in a bilinear space.