{disambiguate=mathematics}

A form is a <function> from a <vector space> to elements of the <underlying field of the vector space>.

Examples:
* <linear form>
* <bilinear form>
* <multilinear form>


Form

Form (mathematics)

{wiki}

A <multilinear form> with a <domain (function)> that looks like:
$$
V^m \times {V*}^n \to \R
$$
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:
$$
T_{i_1 \ldots i_m}^{j_1 \ldots j_n} = T(e_{i_1}, \ldots, e_{i_m}, e^{j_1}, \ldots, e^{j_m})
$$
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>

Explain it properly bibliography:
* https://www.reddit.com/r/Physics/comments/7lfleo/intuitive_understanding_of_tensors/
* https://www.reddit.com/r/askscience/comments/sis3j2/what_exactly_are_tensors/
* https://math.stackexchange.com/questions/10282/an-introduction-to-tensors?noredirect=1&lq=1
* https://math.stackexchange.com/questions/2398177/question-about-the-physical-intuition-behind-tensors
* https://math.stackexchange.com/questions/657494/what-exactly-is-a-tensor
* https://physics.stackexchange.com/questions/715634/what-is-a-tensor-intuitively


Ciro Santilli @cirosantilli 34

 Incoming links: Multilinear form