| Version 6 |
Version 5 |
| A vector space over $\mathbb{R}$ or $\mathbb{C}$ taken with a specific inner product ($\left< x, y \right>$) forms an inner product space. |
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 $\mathbb{R}^n$. |
For example, the familiar dot product forms an inner product space over $\mathbb{R}^n$. |
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The expression $\sqrt{\left<x,x\right>}$ is written $\Vert x \Vert$ and is called the norm. This makes the inner product space also a normed vector space. That is, the inner product space also has the following properties:
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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:
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| \begin{enumerate} |
\begin{enumerate} |
| \item $\Vert x\Vert =|c|\cdot \Vert x\Vert$ , $c \in K$. |
\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 $\Vert x\Vert =0$ 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. |
\item $\Vert x+y\Vert \le \Vert x\Vert + \Vert y\Vert $, the triangle inequality. |
| \end{enumerate} |
\end{enumerate} |
| In addition, the Cauchy-Schwarz inequality |
In addition, the Cauchy-Schwarz inequality |
| $$ |\left<x,y\right>|\le \Vert x\Vert \;\Vert y\Vert $$ |
$$ |\left<x,y\right>|\le \Vert x\Vert \;\Vert y\Vert $$ |
| holds and follows from the definition of a normed vector space. |
holds and follows from the definition of a normed vector space. |
