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  "body": "\n## Summary\n\nThe Hilbert sum of a family of Hilbert spaces consists of the families of vectors, one from each space, with finite sum of squared norms.\nIt is the right notion of direct sum for Hilbert spaces, because the algebraic direct sum of infinitely many of them is not complete.\nEvery Hilbert space with an orthonormal basis indexed by $I$ is the Hilbert sum of $I$ copies of the scalar field.\n\n## Explanations\n\n### L1 human\n\nThink of putting several Hilbert spaces side by side at right angles to each other.\nA vector of the sum has one component in each space, and its squared length is the sum of the squared lengths of the components.\nWith infinitely many spaces, one only keeps families for which this sum is finite.\nFor example, the Hilbert sum of countably many copies of $\\R$ is the space $\\ell^2$ of square-summable real sequences.\n",
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