In Exercises 1 through 28, determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. The functions in Exercises 1 through 17 are real-valued. In Exercises 3, 4, and 5, each function has domain containing 0 and 1. In Exercises 7 through 12, each domain contains all real numbers.
9. All odd functions.
Recall the definition of even and odd functions from Section 1.26 Exercise 24:
Let $f$ be a continuous function whose domain contains $-x$ whenever it contains $x.$ We say $f$ is an $even$ function if $f(-x) = f(x)$ and an $odd$ function if $f(-x) = -f(x)$ for all $x$ in the domain of $f.$
To verify closure under addition, let $f$ and $g$ be odd functions defined for all real $x,$ and let addition of functions be defined for any $x$ as $(f + g)(x) = f(x) + g(x).$ Then, we have
\begin{align*}
(f + g)(-x) &= f(-x) + g(-x)
\\
\\
&= -f(x) - g(x)
\\
\\
&= -[f(x) + g(x)]
\\
\\
&= -(f + g)(x)
\end{align*}which shows that the set of odd functions satisfies closure under addition.
Now, to verify closure under multiplication by real numbers, let $a$ be some real constant. Since $f$ is real-valued for all $x$ we have $af(-x) = a[-f(x)] = -af(x)$ which shows that odd functions satisfy closure under multiplication by real numbers, and by extension satisfies the existence of negatives when $a = -1.$
The function $f(x) = 0$ is odd function since $f(-x) = -f(x) = 0$ for all $x,$ thus satisfying the existence of a zero element. Then, since all functions in the set are real-valued, the remaining axioms are easily verified.
As such, we can see that the set of odd functions is a linear space. $\,\blacksquare$
