multilinear form

for $\text{[math]}$ a positive integer, an $\text{[math]}$ -linear form on $\text{[math]}$ is a function $\text{[math]}$ that is linear in each slot when the other slots are held fixed. this means that for each $\text{[math]}$ and all $\text{[math]}$ , the function $\text{[math]}$ is a linear map from $\text{[math]}$ to $\text{[math]}$ .
the set of $\text{[math]}$ -linear forms on $\text{[math]}$ is denoted by $\text{[math]}$ .
a function $\text{[math]}$ is called a multilinear form on $\text{[math]}$ if it is an $\text{[math]}$ -linear form on $\text{[math]}$ for some positive $\text{[math]}$ .

[cite:;taken from @algebra_axler_2024 chapter 9 multilinear algebra and determinants]