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| Title of object: |
inner product |
| Canonical Name: |
InnerProduct |
| Type: |
Definition |
| Created on: |
2002-01-24 02:24:23-05 |
| Modified on: |
2002-02-25 02:27:26-05 |
| Classification: |
msc:15A63, msc:11E39 |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\renewcommand{\v}{{{\bf v}}}
\newcommand{\w}{{{\bf w}}}
\newcommand{\0}{{{\bf 0}}} |
Content:
An inner product on a vector space $V$ over a field $K$ (which must be either the field $\mathbb{R}$ of real numbers or the field $\mathbb{C}$ of complex numbers) is a function $(\ ,\ ): V \times V \longrightarrow K$ such that, for all $k_1,k_2 \in K$ and $\v_1, \v_2, \v, \w \in V$, the following properties hold:
\begin{enumerate}
\item $(k_1 \v_1 + k_2 \v_2, \w) = k_1 (\v_1, \w) + k_2 (\v_2, \w)$ (linearity\footnote{A small minority of authors impose linearity on the second coordinate instead of the first coordinate.})
\item $(\v, \w) = \overline{(\w, \v)}$, where $\overline{\ \ \ \ }$ denotes complex conjugation (conjugate symmetry)
\item $(\v, \v) \geq 0$, and $(\v, \v) = 0$ if and only if $\v = \0$ (positive definite)
\end{enumerate}
(Note: Rule 2 guarantees that $(\v,\v) \in \mathbb{R}$, so the inequality $(\v,\v) \geq 0$ in rule 3 makes sense even when $K=\mathbb{C}$.)
The standard example of an inner product is the dot product on $K^n$:
((x_1,\dots,x_n), (y_1,\dots,y_n)) := \sum_{i=1}^n x_i \overline{y_i}
Every inner product space is a normed vector space, with the norm being defined by $||\v|| := \sqrt{(\v,\v)}$. |
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