Let $V$ a $K-$vector space and $V^\*$ it's dual. What is the motivation of using the notation $\left<\varphi,x\right>$ for $\varphi(x)$ ? Is it a consequence of the fact that \begin{align\*}
\left<\cdot ,\cdot \right>: V^\*\times V&\longrightarrow K\\ (\varphi,x)&\longmapsto \left<\varphi,x\right>=:\varphi(x)
\end{align\*}

would be a scalar product ? But it looks strange since a scalar product must take element form $V\times V$ or $V^\*\times V^\*$, but not of the form $V^\*\times V$.