Utensil's Zettelkasten-style forest of evergreen notes on math and tech.
utensil.tngl.sh/forest/
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trees/hopf-0002.tree
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trees/hopf-0002.tree
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\p{
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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
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##{
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\begin{gathered}
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{[, \ldots, .]: V \times \ldots \times V \rightarrow \mathbb{k}} \\
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{\left[x_1, \ldots, x_n\right]=\operatorname{sign}(p)\left[x_{p(1)}, \ldots, x_{p(n)}\right]} \\
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{\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]}
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\end{gathered}
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}
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#{n}-factors
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##{
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\begin{aligned}
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{\left[x_1, \ldots, x_n\right] } & =\operatorname{sign}(p)\left[x_{p(1)}, \ldots, x_{p(n)}\right] \\
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[, \ldots, .] & : V \times \ldots \times V \rightarrow \mathbb{k} \quad (n\text{-factors}) \\
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{\left[x_1, \ldots, x_n\right]} & =\operatorname{sign}(p)\left[x_{p(1)}, \ldots, x_{p(n)}\right] \\
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{\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]
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\end{aligned}
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}}