Abstract
Completeness is a property of a topological vector space as a ‘uniform space’. We do not explicitly use uniform spaces but mention that the linear structure allows to define neighbourhoods of ‘uniform size’ for all 
by taking the translates x + U for 
. This allows to introduce the notion of Cauchy filters, and completeness requires Cauchy filters to be convergent.
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References
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Voigt, J. (2020). Completeness. In: A Course on Topological Vector Spaces. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-32945-7_9
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DOI: https://doi.org/10.1007/978-3-030-32945-7_9
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