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I've just started to study general relativity and I'm a bit confused about dual basis vectors.

If we have a vector space [itex]\textbf{V}[/itex] and a basis [itex]\{\textbf{e}_i\}[/itex], I can define a dual basis [itex]\{\omega^i\}[/itex] in [itex]\textbf{V}^*[/itex] such that: [tex] \omega^i(\textbf{e}_j) = \delta^i_j [/tex]But in some pdf and documents I found this relationship: [tex]\omega^i\cdot\textbf{e}_j = \delta^i_j[/tex] So I don't understand why these two relationships are equals.

In fact I know that there's an isomophism [itex]\Phi: \textbf{V}\rightarrow\textbf{V}^*[/itex] induced by the inner product in such a way that: [tex]\tilde v(\textbf{u}) = \textbf{v}\cdot\textbf{u}\qquad\forall\textbf{u}\in\textbf{V}[/tex]Where [itex]\tilde v[/itex] is the covector associated to the vector [itex]\textbf{v}[/itex] by the isomorphism [itex]\Phi[/itex].

So I expect that the basis associated to the dual basis is exactly the reciprocal basis: [tex]\omega^i(\textbf{e}_j) = \textbf{e}^i\cdot\textbf{e}_j =\delta^i_j[/tex].So the dual basis [itex]\{\omega^i\}[/itex] seems to be equal to the reciprocal basis [itex]\{\textbf{e}^i\}[/itex].

I think I'm doing a very bad mistake.

Can anyone help me, please? Thank you!

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# I Understanding dual basis

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