Tensor

A multilinear form with a domain that looks like:

V^{m} \times V *^{n} \to R

(60)

where

V *

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:

T_{i_{1} \dots i_{m}}^{j_{1} \dots j_{n}} = T (e_{i_{1}}, \dots, e_{i_{m}}, e^{j_{1}}, \dots, e^{j_{m}})

(61)

where we remember that the raised indices refer dual vector.

Some examples:

levi-Civita symbol as a tensor
a linear map is a (1,1) tensor

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Levi-Civita symbol as a tensor

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Linear algebra
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A linear map is a (1,1) tensor
Dual space
Dual vector
Levi-Civita symbol as a tensor