**Problem**

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:

*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.*

**Proof.**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 :

(1)

(2)

By the definition of the odd and even functions, (2) can be written as:

(3)

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.