What is a form?

Question

I have read about differential forms, bilinear forms, quadratic forms and some other r-linear forms but I still have this shred of doubt in my mind on what exactly is a form. I have an assumption that it is to a ring what a vector is to a field. I apologise if it is a repost or something though.

Answer 1

I initially upvoted Michael Hoppe's answer because it is a true statement, but I don't think it quite addresses the OP's question, since bilinear and $k$-linear forms more generally are neither linear nor have domain $V$, and quadratic forms aren't even linear in any sense.

I think perhaps the best answer so far was given by Dietrich Burde in the comments. A form is just another word for function. However that's perhaps not so illuminating.

All the examples you've listed have some things in common.

1. All of them involve a vector space $V$ over a field $K$
2. All of them are functions from a product of the vector space with itself to the ground field. I.e. they are all of the form $V\times V\times \cdots \times V=V^n\to K$.

There aren't really any more similarities than that though, and these similarities might not hold for all things called forms. The key thing that determines what the word "form" means are the adjectives in front of it.

You probably know what the specific things you mentioned are, but just to highlight their similarities and differences, and why I say the adjectives in front of the word "form" are more important than the word "form" itself, their definitions are given below.

For example, a $k$-linear or multilinear form on $V$ is a function $F(v_1,\ldots,v_k)$ from $V^k$ to $K$ such that whenever the other arguments are held constant, the map $v_i\mapsto F(v_1,\ldots,v_i,\ldots,v_k)$ is a linear map.

A bilinear form is just a $k$-linear form where $k=2$.

A quadratic form is a function $q$ from $V$ to $K$ such that $f(tv)=t^2v$ for all $t\in K$, and $v\in V$ and such that $B(v,w):=q(v+w)-q(v)-q(w)$ is a bilinear form on $V$.

Finally a differential $k$-form (well sort of) on a vector space $V$ is a $k$-linear form on $V$ that is alternating, which means that $F(v_1,\ldots,v,\ldots,v,\ldots,v_n)=0$ whenever two of the arguments of the $k$-form are equal. Usually though a differential form is an object on a manifold that smoothly assigns what I've just called a differential form to the tangent space at every point of some open subset of the manifold.

Edit: I have had a couple further thoughts that I would like to add. user1551's comment reminded me of my first surprising encounter with the word "form," which I had forgotten until now. When I read Fulton's Algebraic Curves, I was surprised to encounter his usage of form to mean a homogeneous polynomial in some number of variables. Indeed, checking out the wiki page on homogeneous polynomials, at the end of the first paragraph we find the following two sentences:

An algebraic form, or simply form, is a function defined by a homogeneous polynomial. [...] A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis.

This last sentence in particular unifies all of the above examples. All of the examples of forms that you've listed satisfy the definition given in the last sentence of that paragraph.

Nonetheless, I would like to reiterate my emphasis on the idea that the word form itself is much less important than the adjectives that precede it, since one of the related questions in the sidebar, points out that modular forms aren't forms according to this definition, being only defined on the upper half of the complex plane.

Answer 2

If $V$ is a vector space over field $F$, any $F$-valued linear map with domain $V$ is called a form.