Editing Even and odd functions (section)

==Further algebraic properties==
* Any [[linear combination]] of even functions is even, and the even functions form a [[vector space]] over the [[Real number|real]]s. Similarly, any linear combination of odd functions is odd, and the odd functions also form a vector space over the reals. In fact, the vector space of ''all'' real functions is the [[Direct sum of vector spaces|direct sum]] of the [[Linear subspace|subspaces]] of even and odd functions. This is a more abstract way of expressing the property in the preceding section.
**The space of functions can be considered a [[graded algebra]] over the real numbers by this property, as well as some of those above.

*The even functions form a [[algebra over a field|commutative algebra]] over the reals. However, the odd functions do ''not'' form an algebra over the reals, as they are not [[Closure (mathematics)|closed]] under multiplication.