Inner Product

Suppose $V$ be a vector space over $F$.

Inner product is an operation $\langle x,y\rangle$ satisfying:

• $\langle x,y\rangle: V\times V \rightarrow F$ is a function
• $\langle x + y,z\rangle = \langle x,z\rangle + \langle x,z\rangle$
• $\langle x,y\rangle = \overline{\langle y,x\rangle}$ (complex conjucate)
• $\langle ax,y\rangle = \overline{a}\langle x,y \rangle$
• $\langle x,y\rangle \ge 0$ and $\langle x,x\rangle = 0 \iff x = \underline{0}$

Properties

• $\langle x,y+z\rangle = \langle x,y\rangle + \langle x,z\rangle$
  Invert it. Use the plus expansion on first operand. Invert it back.
• $\langle x,ay\rangle = a\langle x,y \rangle$
  Invert it. Extract $a$. Invert it back.

Inner Product Space

A vector space equipped with an inner product.