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Let \(H\) denote a Hilbert space . By the Riesz representation theorem , the canonical map \( \Phi : H \to H^* \) is bijective and isomorphic .

Furthermore, for \( h_1, h_2 \in H \),

\[ \langle \Phi h_1, \Phi h_2 \rangle_{H^*} = \langle h_2, h_1 \rangle_H, \]

and for \( \varphi_1, \varphi_2 \in H^* \),

\begin{equation}\label{eq:dual} \langle \Phi^{-1} \varphi_1, \Phi^{-1} \varphi_2 \rangle_H = \langle \varphi_2, \varphi_1 \rangle_{H^*}. \end{equation}