In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part. Show that the function $$b : \left\lvert \, \begin{array} { r l l } \mathbf { R } ^ { n } \times \mathbf { R } ^ { n } & \rightarrow & \mathbf { R } \\ ( x , y ) & \mapsto & \int _ { 0 } ^ { + \infty } \left\langle e ^ { t a } ( x ) \mid e ^ { t a } ( y ) \right\rangle d t \end{array} \right.$$ is well-defined and that it defines an inner product on $\mathbf { R } ^ { n }$.
In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part.
Show that the function
$$b : \left\lvert \, \begin{array} { r l l } \mathbf { R } ^ { n } \times \mathbf { R } ^ { n } & \rightarrow & \mathbf { R } \\ ( x , y ) & \mapsto & \int _ { 0 } ^ { + \infty } \left\langle e ^ { t a } ( x ) \mid e ^ { t a } ( y ) \right\rangle d t \end{array} \right.$$
is well-defined and that it defines an inner product on $\mathbf { R } ^ { n }$.