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.
2. All rational functions $f/g$, with the degree of $f \leq$ the degree of $g$ (including $f = 0$).
By definition, the set $V$ is a linear space if it satisfies the following ten axioms.
$\textit{Closure axioms}$
$\text{Axiom 1.}\quad$ $\text{Closure Under Addition. }$ For every pair of elements $x$ and $y$ in $V,$ there corresponds a unique element in $V$ called the sum of $x$ and $y,$ denoted by $x + y.$
$\text{Axiom 2.}\quad$ $\text{Closure Under Multiplication. }$ For every $x$ in $V$ and every real number $a$ there corresponds an element in $V$ called the product of $a$ and $x,$ denoted by $ax.$
$\textit{Axioms for addition}$
$\text{Axiom 3.}\quad$ $\text{Commutative Law. }$ For all $x$ and $y$ in $V,$ we have $x + y = y + x.$
$\text{Axiom 4.}\quad$ $\text{Associative Law. }$ For all $x,$ $y,$ and $z$ in $V,$ we have $(x + y) + z = x + (y + z).$
$\text{Axiom 5.}\quad$ $\text{Existence of Zero Element. }$ There is an element in $V,$ denoted by $O,$ such that
$$
x + O = x
\quad
\text{for all $x$ in $V$.}
$$
$\text{Axiom 6.}\quad$ $\text{Existence of Negatives. }$ For every $x$ in $V,$ the element $(-1)x$ has the property
$$
x + (-1)x = O
$$
$\textit{Axioms for multiplication by numbers}$
$\text{Axiom 7.}\quad$ $\text{Associative Law. }$ For every $x$ in $V$ and all real numbers $a$ and $b,$ we have
$$
a(bx) = (ab)x.
$$
$\text{Axiom 8.}\quad$ $\text{Distributive Law for Addition in $V$. }$ For all $x$ and $y$ in $V$ and all real $a,$ we have
$$
a(x + y) = ax + ay.
$$
$\text{Axiom 9.}\quad$ $\text{Distributive Law for Addition of Numbers. }$ For all $x$ in $V$ and all real $a$ and $b,$ we have
$$
(a + b)x = ax + bx.
$$
$\text{Axiom 10.}\quad$ $\text{Existence of Identity. }$ For every $x$ in $V,$ we have $1x = x.$
Since all functions in this set are real-valued, they satisfy the axioms for multiplication by numbers the same way the set of real numbers does, and with the inclusion of $f = 0,$ they satisfy the axioms for addition as well. We can easily verify that all functions in the set satisfy closure under multiplication by real numbers since multiplying $f$ by any real number does not increase its degree, hence for any real $a,$ $(af)/g$ is a memeber of the set. Now, to verify closure under addition, we wish to show that for any functions $x$ and $y$ in the set, the sum $x + y$ is yet another rational function $f/g$ with the degree of $f$ smaller than or equal to the degree of $g.$
Suppose we have two rational functions $x$ and $y$ with the degrees of their respective numerators being $a$ and $b$ and the degrees of their respective denominators being $c$ and $d.$ (In other words, $a \leq c$ and $b \leq d.$) If we take the sum $x + y$ then the degree of the numerator will be the greater of $a + d$ or $b + c,$ but both of these values are less than or equal to the degree of the denominator $c + d.$ Thus, the sum of $x + y$ is in the set of rational functions $f/g$ with the degree of $f$ $\leq$ the degree of $g.$
Hence, the set of rational functions $f/g$ with the degree of $f$ $\leq$ the degree of $g,$ including $f = 0,$ is a linear space. $\,\blacksquare$
