Biorthogonal Systems

Antonio J. Guirao, Vicente Montesinos and Václav Zizler
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Antonio J. Guirao: Universitat Politècnica de València, Departamento de Matemática Aplicada Instituto de Matemática Pura y Aplicada
Vicente Montesinos: Universitat Politècnica de València, Departamento de Matemática Aplicada Instituto de Matemática Pura y Aplicada
Václav Zizler: University of Alberta, Department of Mathematical and Statistical Sciences

Chapter Chapter 3 in Open Problems in the Geometry and Analysis of Banach Spaces, 2016, pp 51-57 from Springer

Abstract: Abstract In this chapter we review several problems on biorthogonal systems biorthogonal system in Banach spaces, i.e., families $$\{x_{\gamma },f_{\gamma }\}_{\gamma \in \Gamma }$$ in X × X ∗, where X is a Banach space, such that 〈x α , f β 〉 = δ α, β whenever α and β belong to $$\Gamma $$ . Here, δ α, β = 1 if α = β and 0 otherwise. Note that Schauder basis Schauder are biorthogonal systems bases, together with their functional coefficients, are examples of biorthogonal biorthogonal system and Schauder basis systems. The theory of biorthogonal systems biorthogonal system is crucial for understanding the structure of Banach spaces, in particular of nonseparable space nonseparable ones. Many problems in this area are widely open. In the nonseparable case the theory of biorthogonal systems biorthogonal system often goes as deep as to the roots of Mathematics, i.e., they use special axioms of Set Theory. In this respect we refer, for the most basic information, to, e.g., [HMVZ08, pp. 148 and 152] or [To06].

Keywords: Banach Space; Unconditional Basis; Separable Space; Biorthogonal System; Basis Basis (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/978-3-319-33572-8_3

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