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For example, is $S=\{x,y\vert\ x,y\in \mathbb{R},\ x+y=0\}$ a 1-dimensional linear subspace of $\mathbb{R^2}$? And is it correct that $S=S+S$?

What are other examples of linear subspaces of $\mathbb{R^2}$?

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    $S=S+S$ is correct for every linear space, hence for every subspace (which is a linear space on its own). – 2017-02-10
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    Basically, all lines through (0,0) are 1-dimensional subspaces, and the point (0,0) is a (rather trivial) 0-dimensional subspace. Is your problem with the concept of a subspaces in general? – 2017-02-10

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For every subspace $S$ it has to be $S=S+S$, because if $s_1$ and $s_2$ are elements of $S$, then $s_1+s_2\in S$. Also, $0\in S$ for every subspace, so if $s$ is an element of $S$, we can write $s=s+0\in S+S$.

It is correct that $S=\{(x,y): x+y=0\}$ is a 1-dimensional linear subspace of $\mathbb R^2$. Other subspaces or $\mathbb R^2$ are, for example: $$V=\{(x,y): x=0\}$$ $$W=\{(x,y): y=0\}$$

and in general, every set that is defined like this:

$$S=\{(x,y): ax+by=0\}$$ where $a$ and $b$ are real numbers.