Prove that the vector space $F((-\infty, 0)\cup (0, \infty))$ can be written as a direct sum in two different ways using these pairs of subspaces:
(a) even functions and odd functions
(b) functions $f :(-\infty, 0)\cup (0, \infty)$ such that $f(x) = 0$ for all $x < 0$ and
functions g : $g :(-\infty, 0)\cup (0, \infty)$ such that $g(x) = 0$ for all $x > 0$.
For this question, why the direct sum of even and odd functions does not include 0 in vector? And why $f(x) = 0$, $g(x) = 0$ is nessessary in (b)?
The question can be understood in graph, but can someone give me hints about how to write a formal proof? Thx.
Best Answer
For (a):
Observe that any function $h \in F(\Bbb{R} \setminus\{0\})=V$ can be written as $$h(x)=\underbrace{\frac{h(x)+h(-x)}{2}}_{\in E}+\underbrace{\frac{h(x)-h(-x)}{2}}_{\in D},$$ where $E$ is the subspace of even functions and $D$ is the subspace of odd functions. Observe that for a direct sum we want two things: $V=E+D$ and $E \cap D=\{\mathbf{0}\}$ (only the zero function should be in both). By above we have demonstrated the first condition. The second condition is easy to verify as only the zero function is both even and odd.
For (b).
For any function $h \in F(\Bbb{R} \setminus\{0\})=V$, let us define $$f(x)=\begin{cases}h(x), & \text{ if } x >0\\ 0, & \text{ if } x < 0\end{cases} \qquad g(x)=\begin{cases}0, & \text{ if } x >0\\ h(x), & \text{ if } x < 0\end{cases}.$$ Now $h$ can be written as $$h(x)=\underbrace{f(x)}_{\in R}+\underbrace{g(x)}_{\in L},$$ where $R$ is the subspace of all functions which are zero for $x<0$ and $L$ is the subspace of all the functions which are $0$ for $x>0$.
The reason we want to have $L$ and $R$ to have functions which disappear on a certain interval is to make sure when functions are added then they won't interfere with each other (once again this is to ensure that both the properties $V=R+L$ and $R \cap L=\{\mathbf{0}\}$ associated with the direct sum can be achieved).