Let \(V\) be a linear space of finite dimension \(n\). Let lower case \(x_i\) denote elements of \(V\), which we will call also letters. We define a bracket as an alternating multilinear scalar valued function \[ \begin {aligned} [, \ldots , .] & : V \times \ldots \times V \rightarrow \mathbb {k} \quad (n\text {-factors}) \\ {\left [x_1, \ldots , x_n\right ]} & =\operatorname {sign}(p)\left [x_{p(1)}, \ldots , x_{p(n)}\right ] \\ {\left [x_1, \ldots , \alpha x_r+\beta y_r, \ldots , x_n\right ] } & =\alpha \left [x_1, \ldots , x_r, \ldots , x_n\right ]+\beta \left [x_1, \ldots , y_r, \ldots , x_n\right ] \end {aligned} \]

The sign is due to the permutation \(p\) on the arguments of the bracket. The pair \(\mathcal {P}=(V,[., \ldots ,])\). is called a Peano space.

Of course, this structure is much weaker as e.g. a normed space or an inner product space. It does not allow to introduce the concept of length, distance or angle. Therefore it is clear that a geometry based on this structure cannot be metric. However, the bracket can be addressed as a volume form. Volume measurements are used e.g. in the analysis of chaotic systems and strange attractors.

A standard Peano space is a Peano space over the linear space \(V\) of dimension \(n\) whose bracket has the additional property that for every vector \(x \in V\) there exist vectors \(x_2, \ldots , x_n\) such that \[ \left [x, x_2, \ldots , x_n\right ] \neq 0 . \]

In such a space the length of the bracket, i.e. the number of entries, equals the dimension of the space, and conversely. We will be concerned here with standard Peano spaces only.