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# inner product space

An *inner product space* (or *pre-Hilbert space*) is a vector space
(over $\mathbb{R}$ or $\mathbb{C}$)
with an inner product ${\langle\cdot,\cdot\rangle}$.

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 $\|x\|:=\sqrt{{\langle x,\,x\rangle}}$. This norm satisfies the parallelogram law.

If the metric $\|{x-y}\|$ induced by the norm is complete, then the inner product space is called a Hilbert space.

The Cauchy–Schwarz inequality

$\displaystyle|{\langle x,\,y\rangle}|\leq\|x\|\cdot\|y\|$ | (1) |

holds in any inner product space.

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

$\displaystyle\cos(x,\,y):=\frac{{\langle x,\,y\rangle}}{\|{x}\|\cdot\|{y}\|}.$ | (2) |

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

${\langle x,\,y\rangle}=0.$ |

## Mathematics Subject Classification

46C99*no label found*

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## Attached Articles

## Corrections

Phrasing, corrections by AxelBoldt ✓

property by matte ✓

definition of norm by boldra ✓

angle by pahio ✓