SOME STRUCTURES OF BILINEAR SPACES AS GENERALIZATIONS OF INNER PRODUCT SPACES

• Main Author: SABARINSYAH
• Format: Dissertations
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/24160
• Institution: Institut Teknologi Bandung

In finite dimensional inner product spaces, a number of facts hold, for examples any subspace is closed, the associated space can be expressed as a direct sum of a subspace with its orthogonal subspace, any vector holds best approximation on any subspace, Riesz Representation Theorem holds for any linear functional and any linear mapping is continuous. In contrast with the above facts, in the infinite dimensional inner product spaces, particular in Hilbert spaces, the following facts hold, the best approximation holds only on a closed subspace, conditions that meet the Riesz representation theorem can be attributed to bounded linear functionals, and <br /> <br /> conditions that meet the continuity properties of linear mappings can be attributed to bounded linear mappings. <br /> <br /> <br /> In this dissertation it is presented first, an application of properties of orthogonal projection or the best approximation to a closed subspace of a Hilbert space, that is to construct a Hilbert basis of a closed subspace by utilizing a given Hilbert basis of the Hilbert space. Second, conditions that meet the Riesz representation theorem can be conected to the closedness property of the kernel of the associated linear functional. Third, conditions that meet the continuity properties of a linear mapping can be attributed to the existence of the adjoint mapping of the associated linear mapping. Fourth, conditions that meet the existence of the adjoint of a linear mapping can be associated to closed properties of subspaces. <br /> <br /> In this dissertation it is also presented extensions of the above facts on Hilbert space to bilinear spaces. Among them, first, conditions that meet the Riesz representation theorem can be attributed to the closedness property of the kernel of the associated linear functional. Second, conditions that meet the continuity properties of a linear mapping can be attributed to the existence of the adjoint mapping of the associated linear mapping. Since spaces of truncated Laurent series are bilinear spaces, it will be presented as well as what conditions for a linear functional on a truncated Laurent series space in order to satisfy the Riesz Representation Theorem.