Difference between revisions of "Inner product"

Revision as of 18:43, 10 July 2015

Definition

Given a vector space, $(V,F)$ (where $F$ is either $\mathbb{R}$ or $\mathbb{C}$), an inner product^[1][2][3] 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:

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:

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^[3] use $(\cdot,\cdot)$ for an inner product (and $\langle\cdot,\cdot\rangle$ for an ordered pair!)

Examples

• Vector dot product

See also

• Hilbert space

References

1. Jump up ↑ http://en.wikipedia.org/w/index.php?title=Inner_product_space&oldid=651022885
2. Jump up ↑ Functional Analysis I - Lecture Notes - Richard Sharp - Sep 2014
3. ↑ ^{Jump up to: 3.0 3.1} Functional Analysis - George Bachman and Lawrence Narici