How are the spaces L2 and ℓ2 isometric?

The spaces L2 and ℓ2 are both Hilbert spaces. In fact, by choosing a Hilbert basis (i.e., a maximal orthonormal subset of L2 or any Hilbert space), one sees that all Hilbert spaces are isometric to ℓ2(E), where E is a set with an appropriate cardinality.
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