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diff --git a/physics_compact/math_app.tex b/physics_compact/math_app.tex
index d097e70a3fbe5a81754107e1022ef1ba823671b4..687f928df9145c3377402563658d5001d1090de2 100644 (file)
--- a/physics_compact/math_app.tex
+++ b/physics_compact/math_app.tex
@@ -86,13 +86,19 @@ This is called a sesquilinear form.
 
 The inner product $(\cdot,\cdot)$ provides a mapping
 \begin{equation}
-V\rightarrow V^{\dagger}:\vec{v}\mapsto \vec{v}^{\dagger}
+V\rightarrow V^{\dagger}:\vec{v}\mapsto \varphi_{\vec{v}}
+\quad
+\text{ defined by }
+\quad
+\varphi_{\vec{v}}(\vec{u})=(\vec{v},\vec{u}) \text{ .}
 \end{equation}
-given by 
+Since the inner product is linear in the first argument, the same is true for the defined mapping.
 \begin{equation}
-v^{\dagger}()
+\lambda(\vec{u}+\vec{v}) \mapsto
+\varphi_{\lambda(\vec{u}+\vec{v})}=
+\lambda\varphi_{\vec{u}}+\lambda\varphi_{\vec{v}}\\
 \end{equation}
-indicating structural identity (isomorphism) of $V$ and $V^{\dagger}$.
+The kernel is $\vec{v}=0$, structural identity (isomorphism) of $V$ and $V^{\dagger}$ is .
 
 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 a dual vector or linear functional of dual space $V^{\dagger}$
@@ -104,7 +110,7 @@ or the conjugate transpose in matrix formalism
 \begin{equation}
 (\vec{u},\vec{v}) \rightarrow \vec{u}^{\dagger}\vec{v} \text{ .}
 \end{equation}
-In doing so, conjugacy is associated with duality.
+In doing so, the conjugate transpose is associated with the dual vector.
 \end{remark}
 
 \begin{definition}