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  "body": "\n## Definition\n\nLet $A$ be a C*-algebra.\nLet $E$ be a complex vector space that is a right $A$-module, with $\\lambda (x a) = (\\lambda x) a = x (\\lambda a)$ for $\\lambda \\in \\C$, $x \\in E$ and $a \\in A$.\nAn *$A$-valued inner product* on $E$ is a map $\\ip{\\cdot}{\\cdot} \\colon E \\times E \\to A$ such that for all $x, y, z \\in E$, $a \\in A$ and $\\lambda \\in \\C$:\n\n1. $\\ip{x}{y + \\lambda z} = \\ip{x}{y} + \\lambda \\ip{x}{z}$.\n2. $\\ip{x}{y a} = \\ip{x}{y} a$.\n3. $\\ip{y}{x} = \\ip{x}{y}^*$.\n4. $\\ip{x}{x}$ is a [[positive element|NA8DF5X3M4]] of $A$, and $\\ip{x}{x} = 0$ only for $x = 0$.\n\nThen $\\norm{x} = \\norm{\\ip{x}{x}}^{1/2}$ is a norm on $E$.\nA *Hilbert C*-module* over $A$ is such a module $E$ that is complete for this norm.\n\n## Remarks\n\nThe inner product is linear in the second argument and conjugate-linear in the first, as is usual in this theory.\nA Hilbert $\\C$-module is the same as a complex [[Hilbert space|HM2HHE3J51]] with this convention.\n\nExample: $E = A$ with $\\ip{a}{b} = a^* b$.\nClauses 1 to 3 follow from the algebra rules and $(a^* b)^* = b^* a$; the element $a^* a$ is positive, and $a^* a = 0$ forces $a = 0$.\nThe norm is $\\norm{a^* a}^{1/2} = \\norm{a}$ by the C* identity, so $E$ is complete.\n\nAnother example is $A^n$ with $\\ip{x}{y} = \\sum_{i=1}^n x_i^* y_i$.\n\nNon-example: $E = \\C$ over $A = \\C$ with $\\ip{x}{y} = -\\overline{x} y$.\nClauses 1 to 3 hold, but $\\ip{1}{1} = -1$ is not positive, so clause 4 fails.\n",
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