Talk:Multilinear form

Disagree with merging with homogeneous polynomial

Multilinear forms are defined on products of vector spaces, while homogeneous polynomials are in general defined on products of fields. Also multilinear forms are linear in each of its arguments, while homogeneous polynomials not necessarily. As such, they are two different animals. Oleg Alexandrov 01:28, 25 September 2005 (UTC)

Every field is a vector space over some field.--84.161.160.48 (talk) 16:45, 8 December 2012 (UTC)

So what are you arguing? The intersection between the two seems to be the linear homogeneous polynomials of degree 1. This is only a not-so-interesting subset of each. I see no case to merge. —Quondum 11:54, 17 November 2016 (UTC)

Should we include "Linear forms" under the title "Examples" ?

As mentioned above. Dominic3203 (talk) 10:34, 9 February 2019 (UTC)

I think so. Linear forms are certainly an example and a starting point, even if it's one that doesn't have the features of the general case. There should at least be a link, if not a short section. Alsosaid1987 (talk) 19:48, 9 February 2019 (UTC)

Section on differential forms

This article has a very long discussion of differential forms under the header Examples. However, differential forms are not an example of a multilinear form. Mathwriter2718 (talk) 13:23, 11 July 2024 (UTC)

If you're right, the following doesn't matter, but the part of that section titled "Definition of differential k-forms and construction of 1-forms" 'defines' tangent spaces using all these subscript p's, without defining what the subscript p's mean. DubleH (talk) 04:21, 18 September 2024 (UTC)

Proceeding purely formally, all the relevant behaviors of things in \mathbb{R}^n_p have been described in that beginning section. Things that live in that space are, as the name suggests, tangent vectors, but the tangent space of \mathbb{R}^n is, not surprisingly, just a copy of \mathbb{R}^n. To address the original objection, that whole section on differential forms is now on its own, avoiding any suggestion that differential forms are a type of multilinear form. The presentation here is for subsets of \mathbb{R}^n, which can be handled with just the machinery of multilinear forms, while the main article is on manifolds in general, requiring considerably more background. Alsosaid1987 (talk) 06:06, 6 July 2026 (UTC)