In general, multilinear mappings from a vector space to scalars are called “forms.” Here we define some forms with regard to the real vector space \({\mathbb{R}^{n}}\); more general definitions may exist in more general settings.

A form is completely symmetric if it is invariant under exchange of any two vector arguments; it is completely anti-symmetric (AKA alternating) if it changes sign under such an exchange. Some classes of forms include:

• Bilinear form: a bilinear map from two vectors to \({\mathbb{R}}\)
• Multilinear form: generalizes the above to take any number of vectors
• Quadratic form: equivalent to a symmetric bilinear form: a quadratic form is a homogeneous polynomial of degree two, i.e. every term has the same number of variables, with no power greater than 2; by considering the variables components of a vector, the polarization identity gives a 1-1 correspondence with symmetric bilinear forms.
• Algebraic form: generalization of the above to arbitrary degree; equivalent to a completely symmetric multilinear form
• 2-form: an anti-symmetric bilinear form
• Exterior form: a completely anti-symmetric multilinear form; exterior forms will be revisited from a different perspective in a subsequent section.

A real inner product is thus a positive definite quadratic form, and is the generalization to arbitrary dimension of the geometrically defined dot product in \({\mathbb{R}^{3}}\). Below we summarize the relationships between various forms.