Given two Hilbert spaces (mathscr{H}) and (mathscr{G}), we call a mapping (A: mathscr{H} ightarrow mathscr{G}) a

Given two Hilbert spaces \(\mathscr{H}\) and \(\mathscr{G}\), we call a mapping \(A: \mathscr{H} \rightarrow \mathscr{G}\) a Hilbert space isomorphism if it is

 

(i) a linear map; that is, \(A(a f+b g)=a A(f)+b A(g)\) for any \(f, g \in \mathscr{H}\) and a, \(b \in \mathbb{R}\).

(ii) a surjective map; and

(iii) an isometry; that is, for all \(f, g \in \mathscr{H}\), it holds that \(\langle f, gangle_{\mathscr{H}}=\langle A f, A gangle_{\mathscr{G}}\).

Let \(\mathscr{H}=\mathbb{R}^{p}\) (equipped with the usual Euclidean inner product) and construct its (continuous) dual space \(\mathscr{G}\), consisting of all continuous linear functions from \(\mathbb{R}^{p}\) to \(\mathbb{R}\), as follows: (a) For each \(\boldsymbol{\beta} \in \mathbb{R}^{p}\), define \(g_{\beta}: \mathbb{R}^{p} \rightarrow \mathbb{R}\) via \(g_{\beta}(\boldsymbol{x})=\langle\boldsymbol{\beta}, \boldsymbol{x}angle=\boldsymbol{\beta}^{\top} \boldsymbol{x}\), for all \(\boldsymbol{x} \in\) \(\mathbb{R}^{p}\). (b) Equip \(\mathscr{G}\) with the inner product \(\left\langle g_{\beta}, g_{\gamma}\rightangle_{\mathscr{G}}:=\boldsymbol{\beta}^{\top} \gamma\).

Show that \(A: \mathscr{H} \rightarrow \mathscr{G}\) defined by \(A(\boldsymbol{\beta})=g_{\beta}\) for \(\boldsymbol{\beta} \in \mathbb{R}^{p}\) is a Hilbert space isomorphism. \rightarrow \mathbb{R}\) via \(g_{\beta}(\boldsymbol{x})=\langle\boldsymbol{\beta}, \boldsymbol{x}angle=\boldsymbol{\beta}^{\top} \boldsymbol{x}\), for all \(\boldsymbol{x} \in\) \(\mathbb{R}^{p}\).

(b) Equip \(\mathscr{G}\) with the inner product \(\left\langle g_{\beta}, g_{\gamma}\rightangle_{\mathscr{G}}:=\boldsymbol{\beta}^{\top} \gamma\).
Show that \(A: \mathscr{H} \rightarrow \mathscr{G}\) defined by \(A(\boldsymbol{\beta})=g_{\beta}\) for \(\boldsymbol{\beta} \in \mathbb{R}^{p}\) is a Hilbert space isomorphism.