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"quote": "\n\n> [!theorem] 定理 5.4 · 柯西–施瓦茨不等式 (Cauchy–Schwarz inequality)\n>\n> （柯西–施瓦茨 不等式）对于内积空间中的任意一对向量 $u$、$v$\n>\n> \n>\n> $$\n> | \\langle u | v \\rangle | \\leq \\| u \\| \\| v \\|.\\tag{5.13}\n> $$\n\n> [!proof]- 证明\n>\n> 由 (IP3)、(IP2) 及式 (5.10)，我们对所有 $\\lambda \\in \\mathbb { C }$ 有\n>\n> $$\n> \\begin{aligned}\n> 0 &\\leq \\langle u + \\lambda v | u + \\lambda v \\rangle \\\\[0.4em]\n> &= \\langle u | u \\rangle + \\lambda \\langle u | v \\rangle + \\overline {{\\lambda}} \\langle v | u \\rangle + \\lambda \\overline {{\\lambda}} \\langle v | v \\rangle .\n> \\end{aligned}\n> $$\n>\n> 代入特定值\n>\n> $$\n> \\lambda = - \\frac {\\langle v | u \\rangle}{\\langle v | v \\rangle}\n> $$\n>\n> 给出不等式\n>\n> $$\n> \\begin{aligned}\n> 0 &\\leq \\langle u | u \\rangle - \\frac {\\langle v | u \\rangle \\langle u | v \\rangle}{\\langle v | v \\rangle} - \\frac {\\overline {{\\langle v | u \\rangle}} \\langle v | u \\rangle}{\\langle v | v \\rangle} + \\frac {| \\langle v | u \\rangle | ^ {2}}{\\langle v | v \\rangle} \\\\[0.4em]\n> &= \\langle u | u \\rangle - \\frac {| \\langle v | u \\rangle | ^ {2}}{\\langle v | v \\rangle}.\n> \\end{aligned}\n> $$\n>\n> 因此，由(IP1)可得，\n>\n> $$\n> | \\langle u | v \\rangle | ^ {2} = | \\langle v | u \\rangle | ^ {2} \\geq \\langle u | u \\rangle \\langle v | v \\rangle\n> $$\n>\n> 而所需结果由式 (5.11) 对该不等式两边取平方根得到。\n",
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