Mathematical Immaturity

1.13 Exercises

• Let $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ be arbitrary vectors in $V_n.$ In each case, determine whether $(x, y)$ is an inner product for $V_n$ if $(x, y)$ is defined by the formula given. In case $(x, y)$ is not an inner product, tell which axioms are not satisfied.
  $\text{(a)}\quad (x, y) = \sum_{i=1}^n x_i |y_i|.$
  $\text{(b)}\quad (x, y) = \left| \sum_{i=1}^n x_i y_i \right|.$
  $\text{(c)}\quad (x, y) = \sum_{i=1}^n x_i \sum_{j=1}^n y_j.$
  $\text{(d)}\quad (x, y) = \left( \sum_{i=1}^n x_i^2 y_i^2 \right)^{1/2}.$
  $\text{(e)}\quad (x, y) = \sum_{i=1}^n (x_i + y_i)^2 - \sum_{i=1}^n x_i^2 - \sum_{i=1}^n y_i^2.$

• Suppose we retain the first three axioms for a real inner product (symmetry, linearity, and homogeneity) but replace the fourth axiom by a new axiom $(4'):$ $(x, x) = 0$ if and only if $x = O.$ Prove that either $(x, x) \gt 0$ for all $x \neq O$ or else $(x, x) \lt 0$ for all $x \neq O.$
  [Hint: Assume $(x, x) \gt 0$ for some $x \neq O$ and $(y, y) \lt 0$ for some $y \neq O.$ In the space spanned by $\{x, y\},$ find an element $z \neq O$ with $(z, z) = 0.$]

Prove that each of the statements in Exercises 3 through 7 is valid for all elements $x$ and $y$ in a real Euclidean space.

• $(x, y) = 0$ if and only if $\|x + y\| = \|x - y\|.$
• $(x, y) = 0$ if and only if $\|x + y\|^2 = \|x\|^2 + \|y\|^2.$
• $(x, y) = 0$ if and only if $\|x + cy\| \geq \|x\|$ for all real $c.$
• $(x + y, x - y) = 0$ if and only if $\|x\| = \|y\|.$
• If $x$ and $y$ are nonzero elements making an angle $\theta$ with each other, then \begin{align*} \\ \|x - y\|^2 = \|x\|^2 + \|y\|^2 - 2 \|x\| \|y\| \cos \theta. \end{align*}

• In the real linear space $C(1, e),$ define an inner product by the equation $$(f, g) = \int_1^e (\log x) f(x) g(x) \, dx.$$
  $\text{(a)}\quad$ If $f(x) = \sqrt{x},$ compute $\|f\|.$
  $\text{(b)}\quad$ Find a linear polynomial $g(x) = a + bx$ that is orthogonal to the constant function $f(x) = 1.$

• In the real linear space $C(-1, 1),$ let $(f, g) = \int_{-1}^1 f(t) g(t) \, dt.$ Consider the three functions $u_1, u_2, u_3$ given by $$u_1(t) = 1, \quad u_2(t) = t, \quad u_3(t) = 1 + t.$$ Prove that two of them are orthogonal, two make an angle $\pi/3$ with each other, and two make an angle $\pi/6$ with each other.

• In the linear space $P_n$ of all real polynomials of degree $\leq n,$ define $$(f, g) = \sum_{k=0}^n f\left(\frac{k}{n}\right) g\left(\frac{k}{n}\right).$$
  $\text{(a)}\quad$ Prove that $(f, g)$ is an inner product for $P_n.$
  $\text{(b)}\quad$ Compute $(f, g)$ when $f(t) = t$ and $g(t) = at + b.$
  $\text{(c)}\quad$ If $f(t) = t,$ find all linear polynomials $g$ orthogonal to $f.$

• In the linear space of all real polynomials, define $(f, g) = \int_0^\infty e^{-t} f(t) g(t) \, dt.$
  $\text{(a)}\quad$ Prove that this improper integral converges absolutely for all polynomials $f$ and $g.$
  $\text{(b)}\quad$ If $x_n(t) = t^n$ for $n = 0, 1, 2, \ldots,$ prove that $(x_n, x_m) = (m + n)!.$
  $\text{(c)}\quad$ Compute $(f, g)$ when $f(t) = (t + 1)^2$ and $g(t) = t^2 + 1.$
  $\text{(d)}\quad$ Find all linear polynomials $g(t) = a + bt$ orthogonal to $f(t) = 1 + t.$

• In the linear space of all real polynomials, determine whether or not $(f, g)$ is an inner product if $(f, g)$ is defined by the formula given. In case $(f, g)$ is not an inner product, indicate which axioms are violated. In (c), $f'$ and $g'$ denote derivatives.
  $\text{(a)} \quad (f, g) = f(1) g(1).$
  $\text{(b)} \quad (f, g) = \left| \int_0^1 f(t) g(t) \, dt \right|.$
  $\text{(c)} \quad (f, g) = \int_0^1 f'(t) g'(t) \, dt.$
  $\text{(d)} \quad (f, g) = \left( \int_0^1 f(t) \, dt \right) \left( \int_0^1 g(t) \, dt \right).$

• Let $V$ consist of all infinite sequences $\{x_n\}$ of real numbers for which the series $\sum x_n^2$ converges. If $x = \{x_n\}$ and $y = \{y_n\}$ are two elements of $V,$ define $$(x, y) = \sum_{n=1}^\infty x_n y_n.$$
  $\text{(a)}\quad$ Prove that this series converges absolutely.
  $\quad$[Hint: Use the Cauchy-Schwarz inequality to estimate the sum $\sum_{n=1}^M |x_n y_n|.$]
  $\text{(b)}\quad$ Prove that $V$ is a linear space with $(x, y)$ as an inner product.
  $\text{(c)}\quad$ Compute $(x, y)$ if $x_n = 1/n$ and $y_n = 1/(n + 1)$ for $n \geq 1.$
  $\text{(d)}\quad$ Compute $(x, y)$ if $x_n = 2^n$ and $y_n = 1/n!$ for $n \geq 1.$

• Let $V$ be the set of all real functions $f$ continuous on $[0, +\infty)$ and such that the integral $\int_0^\infty e^{-t} f^2(t) \, dt$ converges. Define $(f, g) = \int_0^\infty e^{-t} f(t) g(t) \, dt.$
  $\text{(a)}\quad$ Prove that the integral for $(f, g)$ converges absolutely for each pair of functions $f$ and $g$ in $V.$
  $\quad$[Hint: Use the Cauchy-Schwarz inequality to estimate the integral $\int_0^M e^{-t} |f(t) g(t)| \, dt.$]
  $\text{(b)}\quad$ Prove that $V$ is a linear space with $(f, g)$ as an inner product.
  $\text{(c)}\quad$ Compute $(f, g)$ if $f(t) = e^{-t}$ and $g(t) = t^n,$ where $n = 0, 1, 2, \ldots.$
  $\text{(d)}\quad$ Find all linear polynomials $g(t) = a + bt$ orthogonal to $f(t) = 1 + t.$

• In a complex Euclidean space, prove that the inner product has the following properties for all elements $x, y, z,$ and all complex $a$ and $b.$
  $\text{(a)}\quad (ax, by) = a\overline{b}(x, y).$
  $\text{(b)}\quad (x, ay + bz) = \overline{a}(x, y) + \overline{b}(x, z).$

• Prove that the following identities are valid in every Euclidean space.
  $\text{(a)}\quad \|x + y\|^2 = \|x\|^2 + \|y\|^2 + (x, y) + (y, x).$
  $\text{(b)}\quad \|x + y\|^2 - \|x - y\|^2 = 2(x, y) + 2(y, x).$
  $\text{(c)}\quad \|x + y\|^2 + \|x - y\|^2 = 2 \|x\|^2 + 2 \|y\|^2.$

• Prove that the space of all complex-valued functions continuous on an interval $[a, b]$ becomes a unitary space if we define an inner product by the formula $$(f, g) = \int_a^b w(t) f(t) \overline{g(t)} \, dt,$$ where $w$ is a fixed positive function, continuous on $[a, b].$