# Riesz Representation Theorem: isomorph

Riesz' Representation Theorem states that every linear functional can be represented by a vector. This shows that the Dual can be ANTILINEARLY and norm preserving identified with the Hilbert Space itself.

I'm now wondering: Is it also possible (maybe in a completely different fashion) to LINEARLY and norm preserving identify the Dual (not the "Antidual") with the Hilbert Space? In any, case is there a proof for it?

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Fix a total orthogonal system $(e_i)_i$ on your Hilbert space $H$ and consider the conjugation of coordinations: $$\varphi:\sum_i\alpha_ie_i \ \mapsto \ \sum_i\bar{\alpha_i}e_i$$ this is an antilinear isomorphism of $H$ to itself, so composing it with the Riesz presentation $H^*\to H$ gives a linear isomorphism $H^*\to H$.