What is Sum, direct sum?

Example 18.8 (A sum of two lines)

In $\R^3$, let $F = \operatorname{Vect}\bigl((1,0,1)\bigr)$ and $G = \operatorname{Vect}\bigl((0,1,1)\bigr)$. Their sum is

the plane through the origin containing both lines. It is strictly bigger than the union $F \cup G$ (the mere cross of the two lines): the vector $(1, 1, 2) = (1,0,1) + (0,1,1)$ lies in the sum but on neither line. And $F \cap G = \{0\}$ (a common vector requires $a(1,0,1) = b(0,1,1)$, whose first two coordinates force $a = b = 0$): the sum is direct, and $F \oplus G$ is exactly that plane.

Example 18.11

In $\mathcal{F}(\R, \R)$, the even functions $\mathcal{P}$ and the odd functions $\mathcal{I}$ are supplementary: any $f$ writes

and a function both even and odd is zero. (Applied to $\exp$, this is the pair $(\cosh, \sinh)$ of Chapter 4.)

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.