Exercise 1.3.22

Question

Let $F_{1}$ and $F_{2}$ be fields. A function $g \in \mathcal{F}\left(F_{1}, F_{2}\right)$ is called an even function if $g(-t)=g(t)$ for each $t \in F_{1}$ and is called an odd function if $g(-t)=-g(t)$ for each $t \in F_{1}$. Prove that the set of all even functions in $\mathcal{F}\left(F_{1}, F_{2}\right)$ and the set of all odd functions in $\mathcal{F}\left(F_{1}, F_{2}\right)$ are subspaces of $\mathcal{F}\left(F_{1}, F_{2}\right)$.

Jephian C.-H. Lin · Accepted Answer

\begin{proof}
The fact that it's closed has been proved in the previous exercise. And a zero function is either an even function or an odd function.
\end{proof}

Exercise 1.3.22

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Exercise 1.3.22

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