Definition:Pointwise Sum

Definition

Let $C$ be the set of all continuous real functions on the set of real numbers $\R$.

Let $f, g \in C$.

Let us define $f + g$ as:

$\forall x \in R: \left({f + g}\right) \left({x}\right) = f \left({x}\right) + g \left({x}\right)$

Then $f + g$ is called the pointwise sum of $f$ and $g$.

Using the language of abstract algebra, we can define this as:

$\left({C, +}\right)$ is the algebraic structure on $C$ induced by $+$.

The structure $\left({C, +}\right)$ forms a group.

Sources

• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$: Exercise $\text{U}$