+\begin{remark}
+Due to conjugate symmetry, linearity in the first argument results in conjugate linearity (also termed antilinearity) in the second argument.
+This is called a sesquilinear form.
+\begin{equation}
+(\vec{u},\lambda(\vec{v}'+\vec{v}''))=(\lambda(\vec{v}'+\vec{v}''),\vec{u})^*=
+\lambda^*(\vec{v}',\vec{u})^*+\lambda^*(\vec{v}'',\vec{u})^*=
+\lambda^*(\vec{u},\vec{v}')+\lambda^*(\vec{u},\vec{v}'')
+\end{equation}
+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 $V^{\dagger}$
+\begin{equation}
+(\vec{u},\vec{v}) \rightarrow \vec{u}^{\dagger}(\vec{u})\vec{v}
+\end{equation}
+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.
+\end{remark}
+