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.
12. All functions with period $2\pi$.
Recall from section 2.5 (Volume 1) that a function is said to be $periodic$ with period $p \neq 0$ if its domain contains $x + p$ whenever it contains $x$ and $f(x + p) = f(x)$ for every $x$ in the domain of $f.$
Since every $f$ in the set is defined for all real $x,$ we know that $x + p$ is in the domain of $f$ whenever $x$ is. Now, to show closure under addition, let $f$ and $g$ be periodic functions with $p = 2\pi.$ Then, with $(f + g)(x) = f(x) + g(x)$ and $f(x + 2\pi) = f(x),$ we have
\begin{align*}
(f + g)(x + 2\pi) &= f(x + 2\pi) + g(x + 2\pi)
\\
\\
&= f(x) + g(x)
\\
\\
&= (f + g)(x)
\end{align*}which shows that $f + g$ is periodic with $p = 2\pi.$ Closure under multiplication by real numbers can be verified by noting that for any $f$ in the set and any real $a,$ $af(x + 2\pi) = af(x).$ This can also be extended to the case of $a = -1,$ verifying the existence of negatives. The function $f(x) = 0$ is also periodic for all $p,$ satisfying the existence of the zero element. The remaining axioms can be verified by noting that $f(x)$ is real-valued for all $x.$
Thus, we can see that the set of all functions with period $2\pi$ is a linear space. $\,\blacksquare$
