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 alternatively stated as:

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

• $$\langle x,x\rangle \ge 0$$ but specifically:
  • $$\langle x,x\rangle=0\iff x=0$$

Properties

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

[<collapsible-expand>]

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

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$$\langle x,\lambda y+\mu z\rangle$$

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

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

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

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

As required.

From this we may conclude the following:

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

This leads to the most general form:

[<collapsible-expand>]

• $\langle au+bv,cx+dy\rangle=a\overline{c}\langle u,x\rangle+a\overline{d}\langle u,y\rangle+b\overline{c}\langle v,x\rangle+b\overline{d}\langle v,y\rangle$ - which isn't worth remembering!

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Proof:

$\langle au+bv,cx+dy\rangle$

$=a\langle u,cx+dy\rangle+b\langle v,cx+dy\rangle$

$=a\overline{\langle cx+dy,u\rangle}+b\overline{\langle cx+dy,v\rangle}$

$=a(\overline{c\langle x,u\rangle} + \overline{d\langle y,u\rangle})+b(\overline{c\langle x,v\rangle}+\overline{d\langle y,v\rangle})$

$=a\overline{c}\langle u,x\rangle+a\overline{d}\langle u,y\rangle+b\overline{c}\langle v,x\rangle+b\overline{d}\langle v,y\rangle$

As required

Notation

Typically, $\langle\cdot,\cdot\rangle$ is the notation for inner products, however I have seen some authors use $\langle a,b\rangle$ to denote the ordered pair containing $a$ and $b$. Also, notably use $(\cdot,\cdot)$ for an inner product (and $\langle\cdot,\cdot\rangle$ for an ordered pair!)

Examples

• Vector dot product

See also

• Hilbert space