A function is called even if
for all . A function is called odd if
for all . Let denote the set of real-valued even functions on and let denote the set of real-valued odd functions on . Show that:
Proof. To show that is a direct sum, it is sufficient to prove that . Let . Hence, for all , which only happens for . Hence, is a direct sum.
Let . The equation means that can be written as the sum of an odd function and an even function . Or in other words, for all :
By the definition of the odd and even functions, (2) can be written as:
The system of linear equations (1) and (3) has a unique solution and . Hence, can always be uniquely written as the sum of an odd function and an even function.