What is the meaning of "endow with the Euclidean norm" From the paper titled, Distributed Asynchronous Deterministic and Stochastic Gradient Optimization Algorithms, the author defines:

Can someone explain what the phrase endow with the Euclidean norm means?
 A: It means:

Henceforth, we will use $H$ to denote the inner product space whose vector space is what we previously called $H$ and whose inner product is the Euclidean norm. (we may also continue use $H$ to refer to the vector space as well)

A: The space $H$ itself does not carry any 'distance information' itself. Endowing a norm on $H$ simply means that you define a norm on $H$ in a canonical way. It is simply telling you which norm you are working with.
Example: The set $\mathbb R^2$ can be endowed with the Euclidean norm, meaning that distances, ..., are computed using the Euclidean norm. One might instead endow $\mathbb R^2$ with the maximum norm ($\|\mathbf x\| := \max_i |x_i|$), implying that distances are now computed differently, namely according to the maximum norm. Depending on the application, different norms lead to different spaces!
For this reason, normed vector spaces are often denoted by a pair $(H,\|\cdot\|)$, where $H$ contains the information about the set of elements of the space. Only the pairing makes up the normed space.
A: 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.
