Exercise 1.2.12

Question

A real-valued function $f$ deﬁned on the real line is called an even function if $f (-t) = f (t)$ for each real number $t$. Prove that the set of even functions deﬁned on the real line with the operations of addition and scalar multiplication deﬁned in Example 3 is a vector space.

Jephian C.-H. Lin · Accepted Answer

\begin{proof} We have $$f(-t)+g(-t)=f(t)+g(t)$$ and $$cf(-t)=cf(t)$$ if $f$ and $g$ are both even function. Furthermore, $f=0$ is the zero vector. And the field here should be the real numbers.
\end{proof}

Exercise 1.2.12

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Exercise 1.2.12

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