semi-inner product
SemiInnerProduct
2008-02-08 22:54:40
2008-02-12 23:10:20
Definition
inner product
semi-inner product space
Cauchy-Schwartz inequality for semi-inner products
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\subsubsection{Definition}
Let $V$ be a vector space over a field $\mathbb{K}$, where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$.
A {\bf semi-inner product} on $V$ is a function $\;\langle \cdot , \cdot \rangle : V \times V \longrightarrow \mathbb{K}\;$ that \PMlinkescapetext{satisfies} the following conditions:
\begin{enumerate}
\item $\langle \lambda_1 v_1 + \lambda_2 v_2 , w \rangle = \lambda_1 \langle v_1 , w \rangle + \lambda_2 \langle v_2 , w \rangle\;$ for every $v_1, v_2, w \in V$ and $\lambda_1, \lambda_2 \in \mathbb{K}$.
\item $\langle v ,w \rangle = \overline{\langle w ,v \rangle}\;$ for every $v, w \in V$, where the \PMlinkescapetext{line} above means complex conjugation.
\item $\langle v ,v \rangle \geq 0$ (\PMlinkescapetext{positive} semi definite).
\end{enumerate}
Hence, a semi-inner product on a vector space is just like an inner product, but for which $\langle v ,v \rangle$ can be zero (\PMlinkescapetext{even} if $v \neq 0$).
A \emph{semi-inner product space} is just a vector space endowed with a semi-inner product.
\subsubsection{Topology}
Every semi-inner product space $V$ can be given a topology associated with the semi-inner product. In fact, a semi-norm $\| \cdot \|$ can be defined in $V$ by
\begin{displaymath}
\|v\| := \sqrt{\langle v ,v \rangle}
\end{displaymath}
\subsubsection{Cauchy-Schwarz inequality}
The Cauchy-Schwarz inequality is valid for semi-inner product spaces:
\begin{displaymath}
|\langle v , w \rangle| \leq \sqrt{\langle v,v \rangle}\sqrt{\langle w, w \rangle}
\end{displaymath}
\subsubsection{Properties}
Let $V$ be a semi-inner product space and $W:=\{v \in V : \langle v , v \rangle = 0\}$. It is not difficult to see, using the Cauchy-Schwarz inequality, that $W$ is a vector subspace.
The semi-inner product in $V$ induces a well defined semi-inner product in the \PMlinkname{quotient}{QuotientModule} $V/W$ which is, in fact, an inner product. Thus, the \PMlinkescapetext{quotient} $V/W$ is an inner product space.