I am very sorry if I am bothering you guys a lot but I am trying to understand some of this math and its very hard to do it alone

So I have this problem that says: Find which of the following sets are linear subspaces of the corresponding linear spaces:

I am only going to show the first one because maybe if I understand it I can do the rest(I have the answers but I have no idea how it comes out). This is the problem:

$\displaystyle X_{0}=\left \{ (x1,...\: xn)/ x1+...+xn=0 \right.\left. \right \} \subset \mathbb{R}^{n}, (\mathbb{R}^{n}, \mathbb{R}, \cdot )$

Now I understand to be a linear subspace the set must contain the null vector, be closed under multiplication and addition. My problems begin from the step that I have no idea what $\displaystyle (\mathbb{R}^{n} , \mathbb{R}, \cdot )$ is.

Also are these the vectors within $\displaystyle (x1,...\: xn)$ the set X and if so what does this do: $\displaystyle / x1+...+xn=0$