inner productInnerProduct2002-01-24 02:24:232006-10-22 01:16:35Definition% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\0}{{{\bf 0}}}An \emph{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 $||\v|| := \sqrt{(\v,\v)}$.