 Pointwise addition

For functions $f,g:X\to V$ taking values in the same vector space, pointwise addition is defined by $(f+g)(x)=f(x)+g(x)$ for every $x\in X$. For $f(x)=x$ and $g(x)=x^2$, the resulting function is $x\mapsto x+x^2$; at $x=2$ its value is six.

The zero function and pointwise scalar multiplication make the collection of all such functions a vector space. The vector space identities follow by evaluating at an arbitrary input and using the corresponding identities in $V$. This construction also preserves continuity and linearity when the original functions have those properties, and is the addition used in the superposition principle for a linear differential equation.