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'inner product space'
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| Title of object: |
inner product space |
| Canonical Name: |
InnerProductSpace |
| Type: |
Definition |
| Created on: |
2002-01-24 16:58:56-05 |
| Modified on: |
2002-07-16 01:38:49.071649-04 |
| Classification: |
msc:15A63 |
| Defines: |
induced norm |
Revision comment (for changes between this and next version):
| Remove "induced" stuff. Going to talk to Evandar about this. |
Preamble:
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Content:
A vector space over $\mathbb{R}$ or $\mathbb{C}$ taken with a specific inner product ($\left< x, y \right>$) forms an inner product space.
For example, the familiar dot product forms an inner product space over $\mathbb{R}^n$.
The expression $\sqrt{\left<x,x\right>}$ is written $\Vert x \Vert$ and is called the (induced) norm. This makes the inner product space also a normed vector space. That is, the inner product space also has the following properties:
\begin{enumerate}
\item $\Vert x\Vert =|c|\cdot \Vert x\Vert$ , $c \in K$.
\item $\Vert x\Vert =0$ if and only if $x=0$, $\Vert x\Vert \ge 0$.
\item $\Vert x+y\Vert \le \Vert x\Vert + \Vert y\Vert $, the triangle inequality.
\end{enumerate}
In addition, the Cauchy-Schwarz inequality
$$ |\left<x,y\right>|\le \Vert x\Vert \;\Vert y\Vert $$
holds and follows from the definition of a normed vector space. |
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