\[ \newcommand{R}{\mathbb{R}} \newcommand{defarrow}{\quad \stackrel{\text{def}}{\Longleftrightarrow} \quad} \]

# Linear subspaces

Let \(X\) be a vector space. A subset \(S \subset X\) is a **subspace** of \(X\) if it is closed under linear operations, i.e.
\[
S \subset X \text{ subspace of } X \defarrow \lambda x + \mu y \in S \quad \text{ whenever } \quad x,y \in S \:\text{ and }\: \mu,\lambda \in \mathbb R \quad (\text{or } \mathbb C).
\]
In particular, \(\mathbf 0 \in S\), and \(S\) is itself a vector space (the axioms for a vector space follow from those of \(X\)).

**Ex.**

- In any vector space, \(\{\mathbf 0\}\) (the set consisting only of the zero element) is a subspace, since \[\lambda \mathbf 0 + \mu \mathbf 0 = \mathbf 0 \in \{\mathbf 0\} \quad\text{ for all scalars } \lambda, \mu.\]

- Consider \[ \R = \{(x, 0,0)\colon x \in \R\}\] as a subset of \[\R^3 = \{(x,y,z)\colon x,y,z \in \R\}. \] Then \(\R\) is a subspace of \(\R^3\), since it is non-empty and \[\lambda (x_1,0,0) + \mu (x_2,0,0) = (\lambda x_1 + \mu x_2,0,0) \in \R \subset \R^3. \]

- Similarly, the set of real-valued continuous functions on \(\R\) which vanish on some set \(S \subset \R\) is a subspace of \(C(\R,\R)\): \[ \{f\in C(\R,\R) \colon f \equiv 0 \text{ on } S \} \quad\text{ is a subspace of }\quad C(\R,\R),\] since \[ \mu f(x) + \lambda g(x) = 0 \quad\text{ if } \quad f(x) = 0\: \text{ and }\: g(x) = 0. \]

- The vector space of polynomials of degree at most \(n\), \(P_n(\R)\), endowed with the usual addition and scalar multiplication, is a subspace of the set of polynomials of degree at most \(n+1\). Indeed, \[ P_0(\R) \subset P_1(\R) \subset \ldots \subset P_n(\R) \subset P_{n+1}(\R) \subset \ldots \subset P(\R) := \bigcup_{n=0}^\infty P_n(\R)\] are all subspaces of each other and, ultimately, of
**the vector space of all polynomials**, \(P(\R)\). The isomorphisms \(P_n(\R) \cong \R^{n+1}\) induces a natural representation of \(P(\R)\): \[ P(\R) = \{ (a_0,a_1, \ldots, a_n, 0, 0, \ldots) \colon a_0, \ldots, a_n \in \R \text{ and } n \in \mathbb N\}. \]