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.
14. All $f$ integrable on [0, 1] with $\int_0^1 f(x) dx \geq 0$.
Recall Theorem 1.16 from Section 1.24 (Volume 1):
$\text{Theorem 1.16.$\quad$ Linearity with Respect to the Integrand.}\quad$ If both $f$ and $g$ are integrable on $[a, b],$ so is $c_1f + c_2g$ for every pair of constants $c_1$ and $c_2.$ Furthermore, we have:
\begin{align*}
\int_a^b[c_1f(x) + c_2g(x)]\,dx &= c_1\int_a^bf(x)\,dx + c_2\int_a^bg(x)\,dx
\end{align*}
Let $f$ be a member of the set such that $\int_0^1f(x)\,dx > 0.$ Then, if we set $c_1 < 0,$ we can see that $c_1\int_0^1f(x)\,dx < 0,$ which violates closure under multiplication by real numbers. Thus, the set of all $f$ integrable on $[0, 1]$ with $\int_0^1f(x)\,dx \geq 0$ is not a linear space. $\,\blacksquare$
