Proposition

Let $E_i, i = 1, \cdots, k$ be Euclidean spaces of various dimensions.

1. If $f$ is multilinear and $i \ne j$, show that for $h = (h_1, \cdots, h_k)$, with $h_l \in E_l$, we have \(\begin{align*} \lim_{h \rightarrow 0} \frac{\abs{f(a_1, \cdots, h_i, \cdots, h_j, \cdots, a_k)}}{\abs{h}} = 0. \end{align*}\)
2. TODO

Solution

1

Let $g(x, y) = f(a_1, \cdots, x, \cdots, y, \cdots, a_k)$. Then we can just use the results we showed for bilinear functions because $g$ is bilinear.

2

TODO