Akademik Veri Yönetim Sistemi
Revista de la Union Matematica Argentina, cilt.64, sa.2, ss.271-280, 2022 (SCI-Expanded)
We show that there exist inequivalent representations of the dual space of (Formula presented)[0, 1] and of Lp[(Formula presented)] for p (Formula presented) [1, ∞). We also show how these inequivalent representations reveal new and important results for both the operator and the geometric structure of these spaces. For example, if A is a proper closed subspace of (Formula presented)[0, 1], there always exists a closed subspace A┴ (with the same definition as for L2 [0, 1]) such that A∩A┴ = {0} and A(Formula presented)A┴ = (Formula presented)[0, 1]. Thus, the geometry of (Formula presented)[0, 1] can be viewed from a completely new perspective. At the operator level, we prove that every bounded linear operator A on (Formula presented)[0, 1] has a uniquely defined adjoint A * defined on (Formula presented)[0, 1], and both can be extended to bounded linear operators on L2 [0, 1]. This leads to a polar decomposition and a spectral theorem for operators on the space. The same results also apply to Lp [(Formula presented)]. Another unexpected result is a proof of the Baire one approximation property (every closed densely defined linear operator on (Formula presented)[0, 1] is the limit of a sequence of bounded linear operators). A fundamental implication of this paper is that the use of inequivalent representations of the dual space is a powerful new tool for functional analysis.