A complete inner product space. Since every inner product defines a norm, a Hilbert space is necessarily a Banach space.
Up to equivalence of norms, there's only one distinct n-dimensional Hilbert space for each n, namely R^n.
The space of all continuous real-valued functions on the closed interval from 0 to 1, with the inner product given by the integral of pointwise absolute-value products, is an infinite-dimensional Hilbert space.
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