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.
3. All $f$ with $f(0) = f(1)$.
To verify closure under addition, suppose we have two functions $f$ and $g$ in the set. Then, since $(f+g)(x) = f(x) + g(x),$ we can see that $(f + g)(0) = (f + g)(1)$ thus $f + g$ is also in the set. For closure under multiplication by real numbers, we can see that for any real $a$ that $af(0) = af(1),$ which also shows the existence of negatives if $a = -1.$ If we let $O = 0$ for all $x$, then we can see that for all $f$ in the set that $f + O = f.$ Then, for the remaining axioms, we can see that they are satisfied since all $f$ in the set are real-valued and hence behave like the set of real numbers under addition and multiplication.
The set of all $f$ where $f(0) = f(1)$ is a linear space. $\,\blacksquare$
