\(\def\Real{\mathbb{R}}

\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}}

\def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}}

\def\eye{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}

\def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}}

\def\ker{\mathbf{ker}}\def\im{\mathbf{im}}

\def\tr{\mathrm{tr\,}}

\def\braket#1#2{\langle #1|#2\rangle}

\)

**1.**

Given a bilinear form \(Q(\cdot,\cdot)\), or, equivalently, a mapping \(Q:U\to U^*\), one can bake easily *new* bilinear forms from linear operators \(U\to U\): just take \(Q_A(u,v):=Q(u,Av)\).

This opens …