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.
11. All increasing functions.
Recall from Section 1.20 (Volume 1) that a function $f$ is said to be $increasing$ on a set $S$ if $f(x) \leq f(y)$ for every pair of points $x$ and $y$ in $S$ with $x < y.$
The set of increasing functions is not a linear space.
Let $f$ be an increasing function. Then, for any pair $x$ and $y$ in the domain of $f,$ with $x < y,$ $f(x) \leq f(y).$ Now, let $g = (-1)f$ be the negative of $f.$ We can see that $g$ is not an increasing function since $f(x) \geq f(y)$ when $x < y.$ Thus, the set does not satisfy closure under multiplication by real numbers nor the existence of negatives. $\,\blacksquare$
