Revision #1064 → #1719 · back to history
modifiedInduced (canonical) norm063046f52ec6
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| anchors | [{"section":"Statement of the inequality","snippet":"Every inner product gives rise to a Euclidean"},{"type":"math_alttext","value":"{\\displaystyle \\|\\mathbf {u} \\|:={\\sqrt {\\langle \\mathbf {u} ,\\mathbf {u} \\rangle }},}"}] | — |
| mathlib.module | Mathlib.Analysis.InnerProductSpace.Defs | Mathlib.Analysis.InnerProductSpace.Basic |
modifiedSedrakyan's inequalitycc0b8e71d715
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| anchors | [{"section":"Sedrakyan's lemma – positive real numbers","snippet":"Sedrakyan's inequality , also known as"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\left(u_{1}+u_{2}+\\cdots +u_{n}\\right)^{2}}{v_{1}+v_{2}+\\cdots +v_{n}}}\\leq {\\frac {u_{1}^{2}}{v_{1}}}+{\\frac {u_{2}^{2}}{v_{2}}}+\\cdots +{\\frac {u_{n}^{2}}{v_{n}}},}"},{"type":"math_alttext","value":"{\\displaystyle {\\dfrac {\\left(\\sum \\limits _{i=1}^{n}u_{i}\\right)^{2}}{\\sum \\limits _{i=1}^{n}v_{i}}}\\leq \\sum _{i=1}^{n}{\\frac {u_{i}^{2}}{v_{i}}}.}"}] | — |
modifiedCauchy–Schwarz in the plane R²e68fe1091fed
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| anchors | [{"section":"R 2 - The plane","snippet":"then the Cauchy–Schwarz inequality becomes"},{"type":"math_alttext","value":"{\\displaystyle \\langle \\mathbf {u} ,\\mathbf {v} \\rangle ^{2}={\\bigl (}\\|\\mathbf {u} \\|\\|\\mathbf {v} \\|\\cos \\theta {\\bigr )}^{2}\\leq \\|\\mathbf {u} \\|^{2}\\|\\mathbf {v} \\|^{2},}"}] | — |
modifiedCauchy–Schwarz in Rⁿae08abc6b425
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| anchors | [{"section":"R n : n -dimensional Euclidean space","snippet":"with the standard inner product, which is the dot product"},{"type":"math_alttext","value":"{\\displaystyle {\\biggl (}\\sum _{i=1}^{n}u_{i}v_{i}{\\biggr )}^{2}\\leq {\\biggl (}\\sum _{i=1}^{n}u_{i}^{2}{\\biggr )}{\\biggl (}\\sum _{i=1}^{n}v_{i}^{2}{\\biggr )}.}"}] | — |
modifiedCauchy–Schwarz in Cⁿf267b0891777
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| anchors | [{"section":"C n : n -dimensional complex space","snippet":"is the canonical complex inner product"},{"type":"math_alttext","value":"{\\displaystyle {\\bigl |}\\langle \\mathbf {u} ,\\mathbf {v} \\rangle {\\bigr |}^{2}={\\Biggl |}\\sum _{k=1}^{n}u_{k}{\\bar {v}}_{k}{\\Biggr |}^{2}\\leq \\langle \\mathbf {u} ,\\mathbf {u} \\rangle \\langle \\mathbf {v} ,\\mathbf {v} \\rangle ={\\biggl (}\\sum _{k=1}^{n}u_{k}{\\bar {u}}_{k}{\\biggr )}{\\biggl (}\\sum _{k=1}^{n}v_{k}{\\bar {v}}_{k}{\\biggr )}=\\sum _{j=1}^{n}|u_{j}|^{2}\\sum _{k=1}^{n}|v_{k}|^{2}.}"}] | — |
modifiedCauchy–Schwarz in L²32c7d03e253c
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| anchors | [{"section":"L 2","snippet":"For the inner product space of square-integrable complex-valued functions"},{"type":"math_alttext","value":"{\\displaystyle \\left|\\int _{\\mathbb {R} ^{n}}f(x){\\overline {g(x)}}\\,dx\\right|^{2}\\leq \\int _{\\mathbb {R} ^{n}}{\\bigl |}f(x){\\bigr |}^{2}\\,dx\\int _{\\mathbb {R} ^{n}}{\\bigl |}g(x){\\bigr |}^{2}\\,dx.}"}] | — |
modifiedTriangle inequality consequencedebef77505e9
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| anchors | [{"section":"Analysis","snippet":"the triangle inequality is a consequence of the Cauchy–Schwarz inequality"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{alignedat}{4}\\|\\mathbf {u} +\\mathbf {v} \\|^{2}&=\\langle \\mathbf {u} +\\mathbf {v} ,\\mathbf {u} +\\mathbf {v} \\rangle &&\\\\&=\\|\\mathbf {u} \\|^{2}+\\langle \\mathbf {u} ,\\mathbf {v} \\rangle +\\langle \\mathbf {v} ,\\mathbf {u} \\rangle +\\|\\mathbf {v} \\|^{2}~&&~{\\text{ where }}\\langle \\mathbf {v} ,\\mathbf {u} \\rangle ={\\overline {\\langle \\mathbf {u} ,\\mathbf {v} \\rangle }}\\\\&=\\|\\mathbf {u} \\|^{2}+2\\operatorname {Re} \\langle \\mathbf {u} ,\\mathbf {v} \\rangle +\\|\\mathbf {v} \\|^{2}&&\\\\&\\leq \\|\\mathbf {u} \\|^{2}+2|\\langle \\mathbf {u} ,\\mathbf {v} \\rangle |+\\|\\mathbf {v} \\|^{2}&&\\\\&\\leq \\|\\mathbf {u} \\|^{2}+2\\|\\mathbf {u} \\|\\|\\mathbf {v} \\|+\\|\\mathbf {v} \\|^{2}~&&~{\\text{ using CS}}\\\\&={\\bigl (}\\|\\mathbf {u} \\|+\\|\\mathbf {v} \\|{\\bigr )}^{2}.&&\\end{alignedat}}}"}] | — |
modifiedAngle between vectorsee568d7a5efe
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| anchors | [{"section":"Geometry","snippet":"allows one to extend the notion of"},{"type":"math_alttext","value":"{\\displaystyle \\cos \\theta _{\\mathbf {u} \\mathbf {v} }={\\frac {\\langle \\mathbf {u} ,\\mathbf {v} \\rangle }{\\|\\mathbf {u} \\|\\|\\mathbf {v} \\|}}.}"}] | — |
modifiedCovariance inequalityb6f49dc06d57
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| anchors | [{"section":"Probability theory","snippet":"Then the covariance inequality"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {Var} (X)\\geq {\\frac {\\operatorname {Cov} (X,Y)^{2}}{\\operatorname {Var} (Y)}}.}"}] | — |
addedDiscriminant proof of Cauchy–Schwarz in Rⁿ2ad08306b7f7
addedHandshaking lemma25c8e9fe6df8