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In a topological vector space, every Schauder basis is assumed countable, by definition. Supposing we drop the countability condition, we call this a [what goes here?] basis?

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Ivan Singer discusses some notions in Bases in Banach Space Vol. 2, Chapter 17: A complete set $E$ in $X$ is an $ER$-set (Enflo-Rosenthal set) if every countable subfamily of $E$ has an ordering that is a basis of $X$. This is the analogue for unconditional bases. If the index set of a set $E$ is given a well-ordering, a notion of basis is defined and simply called a "trasfinite basis". – David Mitra Jan 7 '14 at 14:27

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