Since we write $\text{L}(V,W)$ for the set of linear maps from $V$ to $W$, it seems to suggest itself to denote multilinear maps from $V_1\times\ldots\times V_k$ to $W$ by $\text{ML}(V_1\times\ldots\times V_k,W)$. Is this notation already used for something else or is there some other reason not to use it?
Edit: I know that it is common to use the notation $\text{L}(V_1,\ldots,V_n,W)$ for $n$-linear maps from $V_1\times\ldots\times V_n$ to $W$, but authors who use this notation (e.g. John Lee in his introduction to manifolds) constantly have to use underbraces, e.g. \begin{equation} \text{L}(\underbrace{U,\ldots,U}_{m\text{ copies}},\underbrace{V,\ldots,V}_{n\text{ copies}},W). \end{equation} I was searching for a way to avoid this, especially since this can't be used in inline equations. My idea was to write $\text{ML}(U^m\times V^n,W)$ - more generally, $\text{ML}(V_1^{n_1}\times\ldots\times V_k^{n_k},W)$.
I just realized that the notation is ambiguous: It's not clear whether the elements of $\text{ML}(V_1^{n_1}\times\ldots\times V_k^{n_k},W)$ are $k$-linear or $(n_1+\ldots+n_k)$-linear (consider for example bilinear maps from $\mathbf{R}^n\times\mathbf{R}^n$ to $\mathbf{R}$).
I guess we'll have to stick to the common notation $L(V_1,\ldots,V_n,W)$. One idea would be to define \begin{equation} L(V_1\colon n_1,\ldots,V_k\colon n_k,W):=L(\underbrace{V_1,\ldots,V_1}_{n_1\text{ copies}},\ldots,\underbrace{V_k,\ldots,V_k}_{n_k\text{ copies}},W) \end{equation} to avoid having to use underbraces all the time.