Abstract
Complete normed linear spaces are introduced, an example being the space \(C(K)\) of continuous functions defined on a compact metric space \(K\). For \(K\) the boundary of a bounded open set in \({R^N}\), this is the space to which the boundary functions for the Dirichlet problem belong. The space of polynomials of degree less than or equal to \(m\), and the subspace of polynomials homogeneous of degree \(k\), are finite dimensional spaces discussed here and which are used in the approximate solution of the Dirichlet problem. The Hahn-Banach Theorem is proved, and is a key ingredient in proving that the complex variable boundary element method will provide approximate solutions to the Dirichlet problem.
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Hromadka, T., Whitley, R. (2014). Banach Spaces. In: Foundations of the Complex Variable Boundary Element Method. SpringerBriefs in Applied Sciences and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-05954-9_3
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DOI: https://doi.org/10.1007/978-3-319-05954-9_3
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Online ISBN: 978-3-319-05954-9
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