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Dual vectors are the members of a dual space.
In the context of tensors , we use raised indices to refer to members of the dual basis vs the underlying basis:
The dual basis vectors are defined to "pick the corresponding coordinate" out of elements of V. E.g.:
By expanding into the basis, we can put this more succinctly with the Kronecker delta as:
Note that in Einstein notation, the components of a dual vector have lower indices. This works well with the upper case indices of the dual vectors, allowing us to write a dual vector as:
In the context of quantum mechanics, the bra notation is also used for dual vectors.

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