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.
15. All $f$ satisfying $f(x) = f(1 - x)$ for all $x$.
Let $f$ and $g$ be functions in the set. Then, with function addition defined as $(f + g)(x) = f(x) + g(x),$ we have
\begin{align*}
(f + g)(1 - x) &= f(1 - x) + g(1 - x)
\\
&= f(x) + g(x)
\\
&= (f + g)(x)
\end{align*}Thus verifying closure under addition. Now, suppose $f$ is a function in the set and let $a$ be a real scalar. Then, $af(x) = af(1-x)$ which verifies closure under multiplication by real numbers, and by extension, the existence of negatives. Then, we can see that $f(x) = 0$ is a member of the set, satisfying the existence of a zero element. The remaining axioms can be confirmed by noting that $f(x)$ is real-valued for all real $x.$ Thus, the set of all $f$ satisfying $f(x) = f(1 - x)$ for all $x$ is a real linear space. $\,\blacksquare$
