# inner product space

An *inner product space ^{}* (or

*pre-Hilbert space*) is a vector space

^{}(over $\mathbb{R}$ or $\u2102$) with an inner product

^{}$\u27e8\cdot ,\cdot \u27e9$.

For example, ${\mathbb{R}}^{n}$ with the familiar dot product^{}
forms an inner product space.

Every inner product space is also a normed vector space^{},
with the norm defined by $\parallel x\parallel :=\sqrt{\u27e8x,x\u27e9}$.
This norm satisfies the parallelogram law^{}.

If the metric $\parallel x-y\parallel $
induced by the norm is complete (http://planetmath.org/Complete),
then the inner product space is called a Hilbert space^{}.

The Cauchy–Schwarz inequality

$|\u27e8x,y\u27e9|\le \parallel x\parallel \cdot \parallel y\parallel $ | (1) |

holds in any inner product space.

According to (1), one can define the angle between two non-zero vectors $x$ and $y$:

$\mathrm{cos}(x,y):={\displaystyle \frac{\u27e8x,y\u27e9}{\parallel x\parallel \cdot \parallel y\parallel}}.$ | (2) |

This provides that the scalars are the real numbers. In any case, the perpendiculatity of the vectors may be defined with the condition

$$\u27e8x,y\u27e9=0.$$ |

Title | inner product space |

Canonical name | InnerProductSpace |

Date of creation | 2013-03-22 12:14:05 |

Last modified on | 2013-03-22 12:14:05 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 23 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 46C99 |

Synonym | pre-Hilbert space |

Related topic | InnerProduct |

Related topic | OrthonormalBasis |

Related topic | HilbertSpace |

Related topic | EuclideanVectorSpace2 |

Related topic | AngleBetweenTwoLines |

Related topic | FluxOfVectorField |

Related topic | CauchySchwarzInequality |

Defines | angle between two vectors |

Defines | perpendicularity^{} |