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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?

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  • $\begingroup$ $\mathbf R$ 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. $\endgroup$
    – Bernard
    Sep 24 '17 at 9:59
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    $\begingroup$ $\Bbb R\times \{0\}$ 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. $\endgroup$
    – s.harp
    Sep 24 '17 at 10:48

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