Ciro Santilli @cirosantilli 35

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 $e^i$ 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 $f$ as:

In the context of quantum mechanics, the bra notation is also used for dual vectors.

A Linear map where the image is the underlying field of the vector space, e.g. ${\mathbb{R}}^n\to \mathbb{R}$.

The set of all linear forms over a vector space forms another vector space called the dual space.

A multilinear form with a domain that looks like:where $V\ast$ is the dual space.

Because a tensor is a multilinear form, it can be fully specified by how it act on all combinations of basis sets, which can be done in terms of components. We refer to each component as:where we remember that the raised indices refer dual vector.

Some examples:

Explain it properly bibliography: