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  "body": "\n## Definition\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$.\nA *concrete operator space* is a linear subspace $E \\subseteq B(H)$ that is closed in the [[operator norm|06EKW1AKXF]].\nFor each $n \\ge 1$, let $M_n(E)$ be the space of $n \\times n$ matrices with entries in $E$.\nA matrix $[x_{ij}] \\in M_n(E)$ acts on the direct sum $H^n$ of $n$ copies of $H$, and its norm is the operator norm of this action:\n$$[x_{ij}] (h_1, \\ldots, h_n) = \\Big( \\sum_{j} x_{1j} h_j, \\ldots, \\sum_{j} x_{nj} h_j \\Big).$$\nThe operator space is $E$ together with this sequence of norms $(\\norm{\\cdot}_n)_{n \\ge 1}$.\n\n## Remarks\n\nExample: $B(H)$ itself, and every norm-closed subalgebra of it, are operator spaces.\nExample: the set of matrices $\\begin{pmatrix} a \u0026 b \\\\ 0 \u0026 a \\end{pmatrix}$ with $a, b \\in \\C$ is a closed subspace of $M_2(\\C) = B(\\C^2)$, since every subspace of a finite-dimensional space is closed.\nNon-example: if $H$ is infinite-dimensional, the finite-rank operators form a subspace of $B(H)$ that is not closed, so it is not an operator space.\nIts closure, the space of compact operators, is one.\nAbstractly, an operator space is a complex vector space $E$ with a complete norm on $E$ and norms $\\norm{\\cdot}_n$ on each $M_n(E)$ with two properties.\nFirst, $\\norm{x \\oplus y}_{m+n} = \\max(\\norm{x}_m, \\norm{y}_n)$ for $x \\in M_m(E)$ and $y \\in M_n(E)$.\nSecond, $\\norm{\\alpha x \\beta}_n \\le \\norm{\\alpha} \\norm{x}_m \\norm{\\beta}$ for $x \\in M_m(E)$ and scalar matrices $\\alpha$ of size $n \\times m$ and $\\beta$ of size $m \\times n$.\nRuan's theorem states that every such abstract operator space is completely isometric to a concrete one.\n",
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