| Version 16 |
Version 15 |
| \PMlinkescapeword{norm} |
\PMlinkescapeword{norm} |
|
|
|
An \emph{inner product space} (or \emph{pre-Hilbert space}) is a vector space
|
An \emph{inner product space} is a vector space
|
| (over $\mathbb{R}$ or $\mathbb{C}$) |
(over $\mathbb{R}$ or $\mathbb{C}$) |
| with an inner product $\ip{\cdot,\cdot}$. |
with an inner product $\ip{\cdot,\cdot}$. |
|
|
| For example, $\mathbb{R}^n$ with the familiar dot product |
For example, $\mathbb{R}^n$ with the familiar dot product |
| forms an inner product space. |
forms an inner product space. |
|
|
| Every inner product space is also a normed vector space, |
Every inner product space is also a normed vector space, |
| with the norm defined by $\Vert x \Vert=\sqrt{\ip{x,x}}$. |
with the norm defined by $\Vert x \Vert=\sqrt{\ip{x,x}}$. |
| The norm in turn induces a metric $\Vert x-y\Vert$; |
The norm in turn induces a metric $\Vert x-y\Vert$; |
| if this metric is \PMlinkname{complete}{Complete} |
if this metric is \PMlinkname{complete}{Complete} |
| then the inner product space is called a Hilbert space. |
then the inner product space is called a Hilbert space. |
|
|
| The Cauchy-Schwarz inequality |
The Cauchy-Schwarz inequality |
| \[ |
\[ |
| |\ip{x,y}|\le \Vert x\Vert \cdot\Vert y\Vert |
|\ip{x,y}|\le \Vert x\Vert \cdot\Vert y\Vert |
| \] |
\] |
| holds in any inner product space. |
holds in any inner product space. |
