Pointwise multiplication ($(fg)(x)=f(x)g(x)$)

For functions $f,g:X\to \mathbb{R}$ or $\mathbb{C}$, their pointwise multiplication is the function $fg$ with $(fg)(x)=f(x)g(x)$. Both values are evaluated at the same input before multiplication. If $f(x)=x+1$ and $g(x)=x^2$, then $(fg)(2)=3\cdot 4=12$ and $(fg)(x)=x^3+x^2$.

Together with pointwise addition, this makes the continuous functions on a topological space into an algebra over a field. Closure properties still need checking for other function spaces: two L2 space functions need not have an L2 space product. For example, $f(x)=g(x)=x^{-1/3}$ on $(0,1)$ have integrable squares, but $(fg{)}^2=x^{-4/3}$ is not integrable. The ambient function space matters.