# Some examples of subspaces – 1

As we knew, a subset ${W}$ of a vector space ${V}$ is called a subspace if it is nonempty and closed under addition and scalar multiplication.

• The vector spaces ${\mathbb{R}, \mathbb{R}^2, \mathbb{R}^3}$ are the usual Euclidean spaces of analytic geometry. There are three types of subspaces of ${\mathbb{R}^2}$: ${\{\theta\},}$ a line through the origin, and ${\mathbb{R}^2}$ itself. There are four types of subspaces of ${\mathbb{R}^3}$: ${\{\theta\}}$, a line through the origin, and ${\mathbb{R}^3}$ itself.
• Let ${P[x]}$ be the set of all polynomials in the single variable ${x}$, with coefficients from ${\mathbb{R}}$ and ${P_n[x]}$ be the subset of ${P[x]}$ consisting of all polynomials of degree ${n}$ or less. We have a result: ${P_n[x]}$ is a subspace of ${P[x]}$.
• When ${n > 0}$, the set of all polynomials of degree exactly ${n}$ is not a subspace of ${P[x]}$.
• Let ${\mathcal{F}(X, F)}$ be the set of all functions ${f: X \rightarrow F}$ (${F}$ is field ${\mathbb{R}}$ or ${\mathbb{C}}$). ${\mathcal{F}(X, F)}$ is a vector space. Why?
• ${\mathcal{F}(X, F)}$ have many subspaces, for example, we consider the case of ${X \subset \mathbb{R}^n}$ and ${F = \mathbb{R}}$. The set ${C(X)}$ of all continuous functions ${f: X \rightarrow \mathbb{R}}$ is a subspace of ${\mathcal{F}(X, \mathbb{R})}$. The set ${\mathcal{D}(X)}$ of all differentiable functions ${f: X \rightarrow \mathbb{R}}$ is a subspace ${C(X)}$ and also of ${\mathcal{F}(X, \mathbb{R})}$. Prove that?

References

D. Lay, Linear algebra and its applications, Addison-Wesley, 2012.

Nguyen Huu Viet Hung, Linear algebra (Vietnamese), VNU Publisher .