2 Duality and conventions
The article chooses to have a non canonical identification between \(V\) and its bidual : \({\lt}x,y{\gt}=-{\lt}y,x{\gt}\). Some properties only rely on this identification (see 86 for an example).
2.1 Setting up the conventions
We define a bilinear form on \(V^{**}\times V\) by \((x,y)\mapsto - y(x)\).
We check the bilinearity.
We set up also a simp lemma for the evaluation of the bilinear form.
We define a linear map from \(V\) to \(V^{**}\) by \(v\mapsto (\varphi \mapsto -\varphi (v))\).
We check the linearity of the map.
We set up also a simp lemma for the evaluation of the map.
If \(V\) is reflexive, then the linear map defined in 76 is a bijective linear map from \(V\) to \(V^{**}\).
We check it’s a bijective map by giving the explicit inverse map.
We set up also a simp lemma for the evaluation of the bijective map.
2.2 Some results about the commutator bilinear form
If \(V\) is reflexive, the map \(((x_1,y_1),(x_2,y_2))\mapsto y_1(x_2)-y_2(x_1)\) is a bilinear form on \(V\times V^*\).
We check the bilinearity.
Suppose it is degeneracy. Then there exists \(h:=(x,y)\in V\times V^*\) such that \(h\ne 0\) and \(y(x')-y'(x)=0\) for all \((x',y')\in V\times V^*\). In particular, for \(y'=0\), \(y(x')=0\) for all \(x'\), so \(y=0\). Then, for \(x'=0\), \(y'(x)=0\) for all \(y'\), so \(x=0\). Thus \(h=0\). Contradiction.