author hackbard Fri, 10 Feb 2012 08:29:18 +0000 (09:29 +0100) committer hackbard Fri, 10 Feb 2012 08:29:18 +0000 (09:29 +0100)

index 79e4ec9..d097e70 100644 (file)
@@ -36,11 +36,32 @@ The addition of two vectors is called vector addition.

\subsection{Dual space}

+\begin{definition}
+The dual space $V^{\dagger}$ of vector space $V$ over field $K$ is defined as the set of all linear maps from the vector space $V$ into its field $K$
+\begin{equation}
+\varphi:V\rightarrow K \text{ .}
+\end{equation}
+These type of linear maps are termed linear functionals.
+The dual space $V^{\dagger}$ becomes a vector space over $K$ itself by the following additional definitions
+\begin{eqnarray}
+(\varphi+\psi)(\vec{v}) & = & \varphi(\vec{v})+\psi(\vec{v}) \\
+(\lambda\varphi)(\vec{v}) & = & \lambda\varphi(\vec{v})
+\end{eqnarray}
+for all $\vec{v}\in V$, $\varphi,\psi\in V^{\dagger}$ and $\lambda\in K$.
+
+The map $V^{\dagger}\times V \rightarrow K: [\varphi,\vec{v}]=\varphi(\vec{v})$ is termed dual pairing of a functional $\varphi\in V^{\dagger}$ and an elemnt $\vec{v}\in V$.
+\end{definition}
+
\subsection{Inner and outer product}
\label{math_app:product}

\begin{definition}
-The inner product on a vector space $V$ over $K$ is a map $(\cdot,\cdot):V\times V \rightarrow K$, which satisfies
+The inner product on a vector space $V$ over $K$ is a map
+\begin{equation}
+(\cdot,\cdot):V\times V \rightarrow K
+\text{ ,}
+\end{equation}
+which satisfies
\begin{itemize}
\item $(\vec{u},\vec{v})=(\vec{v},\vec{u})^*$
(conjugate symmetry, symmetric for $K=\mathbb{R}$)
@@ -51,6 +72,7 @@ The inner product on a vector space $V$ over $K$ is a map $(\cdot,\cdot):V\times (positive definite) \end{itemize} for$\vec{u},\vec{v}\in V$and$\lambda\in K$. +Taking the complex conjugate$(\cdot)^*$is the map from$K\ni z=a+bi\mapsto a-bi=z^*\in K$. \end{definition} \begin{remark} @@ -61,10 +83,22 @@ This is called a sesquilinear form. \lambda^*(\vec{v}',\vec{u})^*+\lambda^*(\vec{v}'',\vec{u})^*= \lambda^*(\vec{u},\vec{v}')+\lambda^*(\vec{u},\vec{v}'') \end{equation} + +The inner product$(\cdot,\cdot)$provides a mapping +\begin{equation} +V\rightarrow V^{\dagger}:\vec{v}\mapsto \vec{v}^{\dagger} +\end{equation} +given by +\begin{equation} +v^{\dagger}() +\end{equation} +indicating structural identity (isomorphism) of$V$and$V^{\dagger}$. + In physics and matrix algebra, the inner product is often defined with linearity in the second argument and conjugate linearity in the first argument. -This allows to express the inner product$(\vec{u},\vec{v})$as a product of vector$\vec{v}$with the dual vector or linear functional of dual space$V^{\dagger}$+This allows to express the inner product$(\vec{u},\vec{v})$as a product of vector$\vec{v}$with a dual vector or linear functional of dual space$V^{\dagger}$\begin{equation} (\vec{u},\vec{v}) \rightarrow \vec{u}^{\dagger}(\vec{u})\vec{v} +\text{ CHECK ! ! !} \end{equation} or the conjugate transpose in matrix formalism \begin{equation} @@ -74,14 +108,14 @@ In doing so, conjugacy is associated with duality. \end{remark} \begin{definition} -If$\vec{u}\in U$,$\vec{v}\in V$are vectors within the respective vector spaces and$\vec{y}^{\dagger}\in V^{\dagger}$is a linear functional of the dual space$V^{\dagger}$of$V$, -the outer product$\vec{u}\otimes\vec{v}$is defined as the tensor product of$\vec{y}^{\dagger}$and$\vec{u}$, +If$\vec{u}\in U$,$\vec{v}\in V$are vectors within the respective vector spaces and$\vec{\varphi}^{\dagger}\in V^{\dagger}$is a linear functional of the dual space$V^{\dagger}$of$V$, +the outer product$\vec{u}\otimes\vec{v}$is defined as the tensor product of$\vec{\varphi}^{\dagger}$and$\vec{u}$, which constitutes a map$A:V\rightarrow U$by \begin{equation} -\vec{v}\mapsto\vec{y}^{\dagger}(\vec{v})\vec{u} +\vec{v}\mapsto\vec{\varphi}^{\dagger}(\vec{v})\vec{u} \text{ ,} \end{equation} -where$\vec{y}^{\dagger}(\vec{v})$denotes the linear functional$\vec{y}^{\dagger}\in V^{\dagger}$on$V$when evaluated at$\vec{v}\in V$, a scalar that in turn is multiplied with$\vec{u}\in U$. +where$\vec{\varphi}^{\dagger}(\vec{v})$denotes the linear functional$\vec{\varphi}^{\dagger}\in V^{\dagger}$on$V$when evaluated at$\vec{v}\in V$, a scalar that in turn is multiplied with$\vec{u}\in U$. In matrix formalism, with respect to a given basis${\vec{e}_i}$of$\vec{u}$and${\vec{e}'_i}$of$\vec{v}$, if$\vec{u}=\sum_i^m \vec{e}_iu_i$and$\vec{v}=\sum_i^n\vec{e}'_iv_i\$,