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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $(H_i)_{i \\in I}$ be a family of [[Hilbert spaces|HM2HHE3J51]] over $\\mathbb{K}$.\nThe *Hilbert sum* $\\widehat{\\bigoplus}_{i \\in I} H_i$ is the set of families $x = (x_i)_{i \\in I}$ with $x_i \\in H_i$ for each $i$ and $\\sum_{i \\in I} \\norm{x_i}^2 \u003c \\infty$.\nIts vector operations are componentwise, and its inner product is\n$$\\ip{x}{y} = \\sum_{i \\in I} \\ip{x_i}{y_i}.$$\nWith this inner product it is a Hilbert space.\n\nA Hilbert space $H$ is the *internal Hilbert sum* of a family $(V_i)_{i \\in I}$ of closed subspaces when these two conditions hold:\n\n1. $V_i \\perp V_j$ for all $i \\neq j$.\n2. The linear span of $\\bigcup_{i \\in I} V_i$ is dense in $H$.\n\n## Remarks\n\nThe series defining $\\ip{x}{y}$ converges absolutely, because $\\abs{\\ip{x_i}{y_i}} \\le \\tfrac12 (\\norm{x_i}^2 + \\norm{y_i}^2)$.\nIf $H$ is the internal Hilbert sum of $(V_i)$, then $x \\mapsto (P_i x)_{i}$, where $P_i$ is the orthogonal projection onto $V_i$, is a unitary map from $H$ onto $\\widehat{\\bigoplus} V_i$.\n\nExample: if every $H_i$ is $\\mathbb{K}$, the Hilbert sum is $\\ell^2(I)$.\nFor each $i$, let $e_i$ be the family with $1$ at $i$ and $0$ elsewhere; the lines $\\mathbb{K} e_i$ are pairwise orthogonal and their span, the finitely supported families, is dense.\n\nNon-example: for infinite $I$ and all $H_i \\neq 0$, the [[external direct sum|0S7W6EG32V]] $\\bigoplus_{i} H_i$ of finitely supported families is not complete.\nSo it is not a Hilbert space for the same inner product.\nThe Hilbert sum is its completion.\n\nFor a closed subspace $V$ of $H$, the space $H$ is the internal Hilbert sum of $V$ and its [[orthogonal complement|0XJXTBCB2T]] $V^\\perp$.\n",
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