One number system may be included in other. For example, the integers **Z** is contained in the real numbers **R**. Moreover, the addition and multiplication are compatible: The equalities 1 + 2 = 3, 2×3 = 6 hold either as integer operations or real number operations. Therefore we call **Z** a sub-number system of the number system **R**. The same remark applies to other inclusions such as **Z** ⊂ **Q** or **R** ⊂ **C**. The similar situation may happen for one vector space contained in another in a compatible way.

A subset ** H** of a vector space

,`u`∈`v`⇒`H`+`u`∈`v`.`H`∈`u`,`H``c`∈**R**⇒`c`∈**u**.`H`

By taking `c` = 0 in the second property, we see that a subspace must contain the zero vector **0**. Moreover, by combining the two properties, a subspace is also closed under linear combinations.

Example In any vector space ** V**, the zero vector form the zero subspace {

Example `C`(**R**) is a subspace of `F`(**R**) because the addition and scalar multiplication of continuous functions are still continuous. More generally, `C ^{k}`(

The subset ** H** = {

The subset ** K** = {

On the other hand, the subset ** H**' = {

Example `P`_{3} is a subspace of `P`_{5}. On the other hand, ** H** = {polynomials of degree ≥ 3 and ≤ 5} is not a subspace of

Example The solutions `H`_{1} = {(`x`, `y`, `z`): `x` + `y` + `z` = 0} of a homogeneous equation is a subspace of **R**^{3}: (`x`, `y`, `z`), (`x'`, `y'`, `z'`) ∈ `H`_{1} ⇒ `x` + `y` + `z` = `x'` + `y'` + `z'` = 0 ⇒ (`x` + `x'`) + (`y` + `y'`) + (`z` + `z'`) = 0 and `cx` + `cy` + `cz` = 0.

The solutions `H`_{2} = {(`x`, `y`, `z`): `x` + `y` + `z` = 1} of a non-homogeneous equation is not a subspace of **R**^{3} because it does not contain the zero vector (0, 0, 0).

The first quadrant `H`_{3} = {(`x`, `y`): `x` ≥ 0, `y` ≥ 0} of **R**^{2} is not a subspace because ** u** = (1, 1) ∈

The following shows that a (infinite) straight line passing through the origin is a subspace, while the one not passing through the origin is not a subspace.

Example **R**^{2} is *not* a subspace of **R**^{3}. In fact, **R**^{2} is defined as pairs of numbers, while **R**^{3} is triples of numbers, so that **R**^{2} is not contained in **R**^{3}.

On the other hand, **R**^{2} can be *embedded* into **R**^{3} as (but not is) subspaces by maps such as (`x`, `y`) ⇒ (`x`, `y`, 0). This is perhaps where our intuition that **R**^{2} is somehow contained in **R**^{3} comes from. However, the intuition is ambiguous because of the existence of other embeddings, such as (`x`, `y`) ⇒ (0, `x`, `y`) or (`x`, `y`) ⇒ (`x`, `x`, `y`). Therefore, to be on the safe side, it is better not to confuse **R**^{2} with (2-dimensional) subspaces of **R**^{3}.

Example The set `Sym`(`n`) of symmetric matrices is a subspace of the vector space `M`(`n`, `n`) of `n` by `n` matrices.

In our examples of subspaces, we saw some appeared also as examples of vector spaces. Indeed we have

The proof is a matter of checking the eight properties. In fact, most of the properties are inherited
from the bigger space. For example, since ** u** +