Inner product

Definition

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

• $$\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{R}$$ (or sometimes $$\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{C}$$ )

Such that:

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

  This may be better stated as:

  • $$\langle\lambda x,y\rangle=\lambda\langle x,y\rangle$$ and
  • $$\langle x+y,z\rangle = \langle x,z\rangle + \langle y,z\rangle$$

• $$\langle x,x\rangle \ge 0$$ with $$\langle x,x\rangle=0\iff x=0$$

Properties

Notice that $$\langle\cdot,\cdot\rangle$$ is also linear in its second argument as:

• $$\langle x,\lambda y+\mu z\rangle = \overline{\langle \lambda y+\mu z, x\rangle}$$ $$=\overline{\lambda\langle y,x\rangle + \mu\langle z,x\rangle}$$ $$=\bar{\lambda}\overline{\langle y,x\rangle}+\bar{\mu}\overline{\langle z,x\rangle}$$ $$=\bar{\lambda}\langle x,y\rangle+\bar{\mu}\langle x,z\rangle$$

From this we may conclude the following:

• $$\langle x,\lambda y\rangle = \bar{\lambda}\langle x,y\rangle$$ and
• $$\langle x,y+z\rangle = \langle x,y\rangle + \langle x,z\rangle$$

Examples

• Vector dot product