definition. Peano space [fauser2002treatise] [hopf-0002]

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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.