# How to understand the similarities and differences between $\textbf{subset}$ and $\textbf{subspace}$? Examples?

Let me first state my opinions.

Suppose $X$ is a linear space closed with addition and scalar multiplication.

1. A subspace must be a subset, while a subset may not be a subspace.
2. If the subset is closed with the addition and scalar multiplication, then it is a subspace.
3. The subspace must contain $\{0\}$.

Example: Let $X=\mathbb{R}^2$, then $\mathbb{R}$ is a subspace and a subset, $y=x$ is a subspace and a subset, $[0,1]\times[0,1]$ is a subset but not a subspace. Any more examples?

• $\mathbf R$ is not a subspace of $\mathbf R^2$. Also condition 3 is superfluous, as it is enough to ask in condition 1 that a subspace be a non-empty subset. Sep 24 '17 at 9:59
• $\Bbb R\times \{0\}$ is a subspace of $\Bbb R^2 = \Bbb R\times \Bbb R$. This space can be seen as an embedding of $\Bbb R$, but it is not $\Bbb R$. I wanted to elaborate on what Bernard wrote since this is often not clear to beginners. Sep 24 '17 at 10:48