What is $\nabla^{j}f(x)$ for $f:\mathbb{R}^{n}\rightarrow{\mathbb{C}}$ in this note which is just after Exercise 1? It is mentioned there that is $d^{j}$-dimensional vector but I am not able to get the intuitive idea behind this explanation. Could someone help me to understand this concept by explaining explicitly?
I also found that it is $d^{j}$-dimensional vector whose components are all derivative operators of total order $j$. But what does it means?
$\nabla f$ is the gradient, $\nabla^2 f$ is the Hessian matrix $f_{ij}$, in general $\nabla ^j f$ is the "tensor" with entries $f_{i_1\cdots i_j}$, where I used lower indices to mean partial derivatives. So in that passage it really mean
$$||f||_{C^k} = \sum_{j=1}^k \sum_{i_1, \cdots i_j}\sup_{x\in \mathbb R^d} |f_{i_1\cdots i_j}(x)|$$