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.
17. All solutions of a linear second-order homogeneous differential equation $y'' + P(x)y' + Q(x)y = 0$, where $P$ and $Q$ are given functions, continuous everywhere.
Let $y_1$ and $y_2$ be solutions to the second-order homogeneous differential equation $y'' + P(x)y' + Q(x)y = 0.$ Then by definition,
\begin{align*}
y_1'' + P(x)y_1' + Q(x)y_1 &= 0
\\
\\
y_2'' + P(x)y_2' + Q(x)y_2 &= 0
\end{align*}
Adding both sides of the equations gives us
\begin{align*}
(y_1 + y_2)'' + P(x)(y_1 + y_2)' + Q(x)(y_1 + y_2) &= 0
\end{align*}which means that $(y_1 + y_2)$ is also in the set of solutions to the second-order homogeneous equation $y'' + P(x)y' + Q(x)y = 0$, thus verifying closure under addition.
Closure under multiplication by real numbers is easily verified by noting that if $y$ is a solution to the equation
\begin{align*}
y'' + P(x)y' + Q(x)y &= 0
\end{align*}then $ay$ is also a solution since for any real scalar $a,$ we have
\begin{align*}
ay'' + P(x)ay' + Q(x)ay &= a\cdot 0
\\
&= 0
\end{align*}which also verifies the existence of negatives. We can trivially confirm the existence of the zero element by setting $y = 0,$ and the remaining axioms can be verified by noting that $y$ is real-valued over its domain. As such, the set of solutions to the second-order homogeneous differential equation
$$
y'' + P(x)y' + Q(x)y = 0
$$is a real linear space. $\,\blacksquare$
