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  "body": "\n## Statement\n\n[[Cauchy–Schwarz inequality|0RFWW56Y22]]: An upper bound on the [[inner product|01EK923SKP]] between two [[vectors|0QN3D80AWP]] in an inner product space in terms of the product of the vector [[norms|0V2WP9B1MF]]. It is considered one of the most important and widely used inequalities in mathematics.\n\nThe Cauchy–Schwarz inequality states that for all vectors $\\mathbf{u}$ and $\\mathbf{v}$ of an [[inner product space|01EK923SKP]]\n\n$$\\left \\vert \\langle \\mathbf u, \\mathbf v \\rangle \\right \\vert \\leq \\langle \\mathbf u, \\mathbf u \\rangle \\cdot \\langle \\mathbf v, \\mathbf v \\rangle$$\n\nwhere $\\langle \\cdot, \\cdot \\rangle$ is the inner product. Examples of inner products include the real and complex [[dot product|0S52TNNTH9]]; see the [[examples in inner product|01EK923SKP]]. Every inner product gives rise to a Euclidean $l_2$ [[norm|0V2WP9B1MF]], called the canonical or [[induced norm|01EK923SKP]], where the norm of a vector $\\mathbf{u}$ is denoted and defined by\n\n$$\\|\\mathbf{u}\\| := \\sqrt{\\langle \\mathbf{u}, \\mathbf{u} \\rangle},$$\n\nwhere $\\langle \\mathbf{u}, \\mathbf{u} \\rangle$ is always a non-negative real number (even if the inner product is complex-valued). By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form in terms of the norm:\n\n$$\\left \\vert \\langle \\mathbf u, \\mathbf v \\rangle \\right \\vert \\leq \\langle \\mathbf u, \\mathbf u \\rangle \\cdot \\langle \\mathbf v, \\mathbf v \\rangle$$\n\nMoreover, the two sides are equal if and only if $\\mathbf{u}$ and $\\mathbf{v}$ are [[linearly dependent|0CRMGPAH83]].\n",
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