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.
13. All $f$ integrable on [0, 1] with $\int_0^1 f(x) dx = 0$.
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*}
Applying what we know from Theorem 1.16 (Volume 1, Section 1.24), we know that for any real constants $c_1$ and $c_2$ and any functions $f$ and $g$ in the set, we have
\begin{align*}
\int_0^1[c_1f(x) + c_2g(x)]\,dx &= c_1\int_0^1f(x)\,dx + c_2\int_0^1g(x)\,dx
\\
\\
&= 0
\end{align*}
which verifies that the set satisfies closure under addition and closure under multiplication by real numbers. Moreover, setting $c_1 = c_2 = -1$ verifies the existence of negatives in the set. The function $f(x) = 0$ is also a member of the set, verifying the existence of the zero element. The remaining axioms can be verified by noting that $f(x)$ is real-valued for $x$ in $[0, 1].$
As such, we can see that the set of all $f,$ integrable on $[0, 1],$ with $\int_0^1f(x)\,dx = 0$ is a linear space. $\,\blacksquare$
