theorems and hyperrefs

[lectures/latex.git] / physics_compact / math_app.tex

diff --git a/physics_compact/math_app.tex b/physics_compact/math_app.tex
index 5f2b0ca..f1f0ae3 100644 (file)
--- a/physics_compact/math_app.tex
+++ b/physics_compact/math_app.tex
@@ -5,7 +5,7 @@
 \subsection{Vector space}
 \label{math_app:vector_space}
 
-\begin{definition}
+\begin{definition}[Vector space]
 A vector space $V$ over a field $(K,+,\cdot)$ is an additive abelian group $(V,+)$ and an additionally defined scalar multiplication of $\vec{v}\in V$ by $\lambda\in K$, which fullfills:
 \begin{itemize}
 \item $\forall \vec{v} \, \exists 1$ with: $\vec{v}1=\vec{v}$
@@ -36,7 +36,7 @@ The addition of two vectors is called vector addition.
 
 \subsection{Dual space}
 
-\begin{definition}
+\begin{definition}[Dual space]
 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{ .}
@@ -55,13 +55,9 @@ The map $V^{\dagger}\times V \rightarrow K: [\varphi,\vec{v}]=\varphi(\vec{v})$
 \subsection{Inner and outer product}
 \label{math_app:product}
 
-\begin{definition}
+\begin{definition}[Inner product]
 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
+$(\cdot,\cdot):V\times V \rightarrow K$, which satisfies
 \begin{itemize}
 \item $(\vec{u},\vec{v})=(\vec{v},\vec{u})^*$
       (conjugate symmetry, symmetric for $K=\mathbb{R}$)
@@ -117,7 +113,7 @@ or the conjugate transpose in matrix formalism
 In doing so, the conjugate transpose is associated with the dual vector.
 \end{remark}
 
-\begin{definition}
+\begin{definition}[Outer product]
 If $\vec{u}\in U$, $\vec{v}\in V$ are vectors within the respective vector spaces and $\vec{\varphi}\in V^{\dagger}$  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 $\vec{\varphi}$ and $\vec{u}$,
 which constitutes a map $A:V\rightarrow U$ by