A real inner product space is a real Hilbert space if, with its induced norm, it is a real Banach space.
In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let be a real inner product space, regarded as a real normed space with its induced norm (The Norm of a Real Inner Product Space (iii)).
Real Hilbert space. is a real Hilbert space if this real normed space is a real Banach space.
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