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In Wald's "General Relativity", in his section on abstract tensor notation, he lets [itex]g_{ab}[/itex] denote the metric tensor. When applied to a vector [itex]v^a[/itex], we get a dual vector, because [itex]g_{ab}(v^a, \cdot)[/itex] is just a dual vector. Okay cool. But then he says that this dual vector is actually [itex]g_{ab}v^b[/itex], which is a contraction. But don't we have [itex]g_{ab}v^b = \sum\limits_{i=1}^n g(\cdot, v^i) = \sum\limits_{i=1}^n g_{ab}(v^i, \cdot)[/itex], which in general is not going to be equal to [itex]g_{ab}(v^a, \cdot)[/itex]? Where am I messing up here?

EDIT: [itex] \{v^i\}[/itex] is a basis for the tangent space.

EDIT: [itex] \{v^i\}[/itex] is a basis for the tangent space.

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