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.
7. All $f$ with $f(x) \to 0$ as $x \to +\infty$.
Recall Theorem 3.1 which gives basic theorems on limits:
$\text{Theorem 3.1.}\quad $ Let $f$ and $g$ be functions such that
\begin{align*}
\lim_{x \to p}f(x) &= A,
\quad
\lim_{x \to p} g(x) = B.
\end{align*}Then we have
\begin{align*}
\text{(i)}
\quad
\lim_{x \to p}\,[f(x) + g(x)] &= A + B
\\
\\
\text{(ii)}
\quad
\lim_{x \to p}\,[f(x) - g(x)] &= A - B
\\
\\
\text{(iii)}
\quad
\lim_{x \to p}\,f(x) \cdot g(x) &= A \cdot B
\\
\\
\text{(iv)}
\quad
\lim_{x \to p}\,f(x) / g(x) &= A / B
\\
\\
\end{align*}
To verify the closure under addition, we wish to show that as $x \to \infty,$ $(f + g)(x) \to 0.$ But we know that addition of functions is defined as $(f + g)(x) = f(x) + g(x).$ And from Theorem 3.1, we know that if $\lim_{x \to p}f(x) = A$ and $\lim_{x \to p}g(x) = B$,
\begin{align*}
\lim_{x \to p}\,[f(x) + g(x)] &= A + B
\end{align*}
which means that $\lim_{x \to \infty}(f + g)(x) = 0,$ so the set satisfies closure under addition.
To verify closure under multiplication by real numbers, we use a special case of Theorem 3.1 (iii) where $f(x)$ is some real constant $A$ for all $x.$ Then, if $g(x)$ is a member of the set, its limit goes to zero as $x$ goes to infinity, and we get
\begin{align*}
\lim_{x \to \infty}\,f(x) \cdot g(x) &= A \cdot 0
\\
&= 0.
\end{align*}which verifies closure under multiplication by real numbers.
We can see that $f(x) = 0$ is also in this set, which satisfies the existence of a zero element. And since the set satisfies closure under multiplication by real numbers, it satisfies the existence of negatives by setting $A = -1.$ Then, the remaining axioms are easily verified since all functions in the set are real valued for all $x$.
Thus, we have verified that the set of all $f$ with $f(x) \to 0$ as $x \to \infty$ is a linear space. $\, \blacksquare$
