To add to @Peyton:
Regard $H$ as all $L$ tuples, whose $i$th component is chosen from the real vector space $H_i$. Then for $h \in H$ $$\Vert h \Vert = \Vert (h_1, \ldots, h_L)\Vert = \sqrt{\Vert h_1 \Vert^2 + \ldots \Vert h_L \Vert^2}$$ where the norm of each $h_i$ is from its respective space. Though I don't know the whole context, I think it is safe to assume that the norms on each $H_i$ is also the Euclidean norm $$\Vert h_i \Vert = \sqrt {\sum_j h_{i, j}^2 }$$ where $h_{i,j}$ is the $j$-coordinate of $h_i$. That way
$$\Vert h \Vert = \sqrt {\sum_{i,j} h_{i, j}^2 }$$
and the entire space is just the Euclidean space of dimension $\sum_i \dim H_i$. If you give the composing subspaces a fancy norm, you can get norms on the direct sum. But it would be a weird object, given your paper's title.