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.
4. All $f$ with $2f(0) = f(1)$.
Let $f$ and $g$ be members of this set, then $(f + g)(x) = f(x) + g(x)$ which gives us
\begin{align*}
(f + g)(1) &= f(1) + g(1)
\\
\\
&= 2f(0) + 2g(0)
\\
\\
&= 2\left[f(0) + g(0)\right]
\\
\\
&= 2(f + g)(0)
\end{align*}Which verifies closure under addition. Then, let $g = af,$ where $a$ is a real scalar and $f$ is an element in the set. We then have $g(1) = af(1) = 2af(0) = 2g(0)$ which verifies closure under multiplication by real numbers (and by extension, the existence of negatives). The existence of the zero element is given by the constant function $O = 0$ for all $x,$ and the remaining axioms are satisfied by all elements being real-valued.
The set of all $f$ with $2f(0) = f(1)$ is a linear space. $\,\blacksquare$
