Mathematical Immaturity

1.13 Exercises

• Prove that the following identities are valid in every Euclidean space.
  $\text{(a)}\quad \|x + y\|^2 = \|x\|^2 + \|y\|^2 + (x, y) + (y, x).$
  $\text{(b)}\quad \|x + y\|^2 - \|x - y\|^2 = 2(x, y) + 2(y, x).$
  $\text{(c)}\quad \|x + y\|^2 + \|x - y\|^2 = 2 \|x\|^2 + 2 \|y\|^2.$
• Suppose $x$ and $y$ are elements of some complex linear space $V,$ then:

  $\text{(a)} \quad$ \begin{align*} \|x + y\|^2 &= (x + y, x + y) \\ &= (x + y, x) + (x + y, y) \\ &= \overline{(x, x + y)} + \overline{(y, x + y)} \\ &= \overline{(x, x)} + \overline{(x, y)} + \overline{(y, x)} + \overline{(y, y)} \\ &= (x, x) + (y, x) + (x, y) + (y, y) \\ &= \|x\|^2 + \|y\|^2 + (x, y) + (y, x) \quad \blacksquare \end{align*}

  $\text{(b)} \quad$ \begin{align*} \\ \|x - y\|^2 &= (x - y, x - y) \\ &= (x - y, x) - (x - y, y) \\ &= \overline{(x, x - y)} - \overline{(y, x - y)} \\ &= \overline{(x, x)} - \overline{(x, y)} - \overline{(y, x)} - \overline{(y, y)} \\ &= (x, x) - (y, x) - (x, y) + (y, y) \\ &= \|x\|^2 + \|y\|^2 - (x, y) - (y, x) \end{align*} Subtracting this from $\|x + y\|^2,$ we get: \begin{align*} \|x + y\|^2 - \|x - y\|^2 &= 2(x, y) + 2(y, x) \quad \blacksquare \end{align*}

  $\text{(c)} \quad$ Using the results of parts (a) and (b), we have: \begin{align*} \|x + y\|^2 + \|x - y \|^2 &= \|x\|^2 + \|y\|^2 + (x, y) + (y, x) \\ &+ \|x\|^2 + \|y\|^2 - (x, y) - (y, x) \\ &= 2\|x\|^2 + 2\|y\|^2 \quad \blacksquare \end{align*}