Inner product

Definition

Given a vector space, [ilmath](V,F)[/ilmath] (where [ilmath]F[/ilmath] is either [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath]), an inner product is a map:

• [math]\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{R}[/math] (or sometimes [math]\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{C}[/math])

Such that:

• [math]\langle x,y\rangle = \overline{\langle y, x\rangle}[/math] (where the bar denotes Complex conjugate)
  • Or just [math]\langle x,y\rangle = \langle y,x\rangle[/math] if the inner product is into [ilmath]\mathbb{R} [/ilmath]
• [math]\langle\lambda x+\mu y,z\rangle = \lambda\langle y,z\rangle + \mu\langle x,z\rangle[/math] ( linearity in first argument )

  This may be better stated as:

  • [math]\langle\lambda x,y\rangle=\lambda\langle x,y\rangle[/math] and
  • [math]\langle x+y,z\rangle = \langle x,z\rangle + \langle y,z\rangle[/math]

• [math]\langle x,x\rangle \ge 0[/math] with [math]\langle x,x\rangle=0\iff x=0[/math]

Properties

Notice that [math]\langle\cdot,\cdot\rangle[/math] is also linear in its second argument as:

• [math]\langle x,\lambda y+\mu z\rangle = \overline{\langle \lambda y+\mu z, x\rangle}[/math][math]=\overline{\lambda\langle y,x\rangle + \mu\langle z,x\rangle}[/math][math]=\bar{\lambda}\overline{\langle y,x\rangle}+\bar{\mu}\overline{\langle z,x\rangle}[/math][math]=\bar{\lambda}\langle x,y\rangle+\bar{\mu}\langle x,z\rangle[/math]

From this we may conclude the following:

• [math]\langle x,\lambda y\rangle = \bar{\lambda}\langle x,y\rangle[/math] and
• [math]\langle x,y+z\rangle = \langle x,y\rangle + \langle x,z\rangle[/math]

Examples

• Vector dot product