Mathematical Immaturity

1.5 Exercises

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.

• All rational functions.
• All rational functions $f/g,$ with the degree of $f \leq$ the degree of $g$ (including $f = 0$).
• All $f$ with $f(0) = f(1).$
• All $f$ with $2f(0) = f(1).$
• All $f$ with $f(1) = 1 + f(0).$
• All step functions defined on [0, 1].
• All $f$ with $f(x) \to 0$ as $x \to +\infty.$
• All even functions.
• All odd functions.
• All bounded functions.
• All increasing functions.
• All functions with period $2\pi.$
• All $f$ integrable on [0, 1] with $\int_0^1 f(x) dx = 0.$
• All $f$ integrable on [0, 1] with $\int_0^1 f(x) dx \geq 0.$
• All $f$ satisfying $f(x) = f(1 - x)$ for all $x.$
• All Taylor polynomials of degree $\leq n$ for a fixed $n$ (including the zero polynomial).
• All solutions of a linear second-order homogeneous differential equation $y'' + P(x)y' + Q(x)y = 0,$ where $P$ and $Q$ are given functions, continuous everywhere.
• All bounded real sequences.
• All convergent real sequences.
• All convergent real series.
• All absolutely convergent real series.
• All vectors $(x, y, z)$ in $V_3$ with $z = 0.$
• All vectors $(x, y, z)$ in $V_3$ with $x = 0$ or $y = 0.$
• All vectors $(x, y, z)$ in $V_3$ with $y = 5x.$
• All vectors $(x, y, z)$ in $V_3$ with $3x + 4y = 1,$ $z = 0.$
• All vectors $(x, y, z)$ in $V_3$ which are scalar multiples of $(1, 2, 3).$
• All vectors $(x, y, z)$ in $V_3$ whose components satisfy a system of three linear equations of the form: \begin{align*} a_{11}x + a_{12}y + a_{13}z = 0, \quad a_{21}x + a_{22}y + a_{23}z = 0, \quad a_{31}x + a_{32}y + a_{33}z = 0. \end{align*}
• All vectors in $V_n$ that are linear combinations of two given vectors $A$ and $B.$
• Let $V = R^+,$ the set of positive real numbers. Define the "sum" of two elements $x$ and $y$ in $V$ to be their product $x \cdot y$ (in the usual sense), and define "multiplication" of an element $x$ in $V$ by a scalar $c$ to be $x^c.$ Prove that $V$ is a real linear space with 1 as the zero element.
• (a) Prove that Axiom 10 can be deduced from the other axioms. (b) Prove that Axiom 10 cannot be deduced from the other axioms if Axiom 6 is replaced by Axiom 6': For every $x$ in $V$ there is an element $y$ in $V$ such that $x + y = O.$
• Let $S$ be the set of all ordered pairs $(x_1, x_2)$ of real numbers. In each case determine whether or not $S$ is a linear space with the operations of addition and multiplication by scalars defined as indicated. If the set is not a linear space, indicate which axioms are violated. \begin{align*} \text{(a)} \quad (x_1, x_2) + (y_1, y_2) &= (x_1 + y_1, x_2 + y_2), \quad a(x_1, x_2) = (ax_1, 0). \\ \text{(b)} \quad (x_1, x_2) + (y_1, y_2) &= (x_1 + y_1, 0), \quad a(x_1, x_2) = (ax_1, ax_2). \\ \text{(c)}\quad (x_1, x_2) + (y_1, y_2) &= (x_1, x_2 + y_2), \quad a(x_1, x_2) = (ax_1, ax_2). \\ \text{(d)}\quad (x_1, x_2) + (y_1, y_2) &= (|x_1 + x_2|, |y_1 + y_2|), \quad a(x_1, x_2) = (|ax_1|, |ax_2|). \end{align*}
• Prove parts (d) through (h) of Theorem 1.3.