What is Subspace?

Example 18.4

In $\mathcal{F}(\R, \R)$: the continuous functions, the differentiable ones, the polynomials of degree $\leq n$ (written $K_n[X]$ inside $K[X]$), the solutions of a homogeneous linear differential equation (Theorem 5.10 said just that). Non-examples: $\{f : f(0) = 1\}$ (no zero), degree exactly $n$ (not stable under addition).

Example 18.5 (Subspace or not: four verdicts, argued)

In the space of real sequences:

• $\{u : u \text{ bounded}\}$ is a subspace: $0$ is bounded, and if $\abs{u_n} \leq M$, $\abs{v_n} \leq M'$, then $\abs{u_n + \lambda v_n} \leq M + \abs\lambda M'$.
• $\{u : u_n \to 1\}$ is not: the zero sequence is missing (and the sum of two members tends to $2$).
• $\{u : u \text{ monotone}\}$ is not: $u_n = n$ and $v_n = -n + (-1)^n$ are monotone, their sum $(-1)^n$ is not; stability under addition is the axiom that fails, even though the set contains $0$ and all scalar multiples of its members.
• $\{u : u_{n+1} = u_n^2\}$ is not: it contains $0$ but $2u$ escapes as soon as $u$ is a nonzero member ($2u_{n+1} \neq (2u_n)^2$ in general) — squaring is the nonlinearity.

The working order is always the same: test $0$ first (cheapest), then stability — and to refute, one explicit counterexample pair beats any amount of doubt.

Example 18.12 (A supplementary pair in $K_n[X]$)

Fix $a \in K$ and set $F = \{P \in K_n[X] : P(a) = 0\}$, $G = \operatorname{Vect}(1)$ (the constants). Then $K_n[X] = F \oplus G$. Indeed $F \cap G$ consists of the constants vanishing at $a$, i.e. $\{0\}$; and every $P$ decomposes as

The decomposition is worth memorizing: subtracting the value at a point is the standard way to project onto “functions vanishing at $a$”. Note that $F$ is a large subspace and $G$ a small one; a supplementary pair need not be balanced in any sense.