| Version 18 |
Version 17 |
| \PMlinkescapeword{induced} |
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| \PMlinkescapeword{norm} |
\PMlinkescapeword{norm} |
| \PMlinkescapeword{satisfies} |
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| An \emph{inner product space} (or \emph{pre-Hilbert space}) is a vector space |
An \emph{inner product space} (or \emph{pre-Hilbert 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 $\ip{\cdot,\cdot}$. |
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| 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. |
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| 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 $\Vert x \Vert=\sqrt{\ip{x,x}}$. |
| This norm satisfies the parallelogram law. |
This norm satisfies the parallelogram law. |
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| If the metric $\Vert x-y\Vert$ |
If the metric $\Vert x-y\Vert$ |
| induced by the norm is \PMlinkname{complete}{Complete}, |
induced by the norm is \PMlinkname{complete}{Complete}, |
| then the inner product space is called a Hilbert space. |
then the inner product space is called a Hilbert space. |
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| 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. |
