Theorem 6.1
Let \(V\) be an inner product space. Then for \(x, y, z \in V\) and \(c \in \mathbf{F}\), the following statements are true.
  1. \(\langle x, y+z \rangle = \langle x, y \rangle + \langle x, z \rangle\)
  2. \(\langle x, cy \rangle = \bar{c} \langle x , y \rangle\)
  3. \(\langle x, 0 \rangle = \langle 0,x \rangle = 0 \)
  4. \(\langle x, x \rangle = 0 \text{ if and only if } x = 0\)
  5. If \(\langle x, y \rangle = \langle x, z \rangle\) for all \(x \in V\), then \(y = z\)



Proof:

For (a)

$$ \begin{align*} \langle x, y+z \rangle &= \overline{\langle y+z, x \rangle} \\ &= \overline{\langle y, x \rangle + \langle z, x \rangle} \\ &= \overline{\langle y, x \rangle} + \overline{\langle z, x \rangle} \\ &= \langle x, y \rangle + \langle x, z \rangle \\ \end{align*} $$

For (b)

$$ \begin{align*} \langle x, cy \rangle &= \overline{\langle cy, x \rangle} \\ &= \overline{c\langle y, x \rangle} \\ &= \overline{c} \overline{\langle y, x \rangle} \\ &= \overline{c} \langle x, y \rangle \\ \end{align*} $$




References