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"quote": "$A:\\mathcal{H}\\rightarrow\\mathcal{H}$是希尔伯特空间$\\mathcal{H}$上的有界线性算符，我们定义其伴随(adjoint)是算符$A^*:\\mathcal{H}\\rightarrow\\mathcal{H}$，而且有性质",
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"quote": "Let $A : \\mathcal H \\to \\mathcal H$ be a bounded linear operator on a Hilbert space $\\mathcal { H } .$ We define its adjoint to be the operator $A^*:\\mathcal H\\to\\mathcal H$ that has the property",
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"text": "希尔伯特空间上有界线性算符 $A$ 的伴随 $A^{*}$ 由内积关系唯一确定：\n\n$$\n\\langle u \\mid A v\\rangle=\\left\\langle A^{*} u \\mid v\\right\\rangle \\quad \\text{对全体} u, v \\in \\mathcal{H}\\tag{13.9}\n$$\n\n",
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"text": "作者构造伴随时固定 $u$，把 $v\\mapsto\\langle u\\mid Av\\rangle$ 视为连续线性泛函，再用 Riesz 表示定理得到唯一的 $A^{*}u$；随后验证所得算符线性且有界。",
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"summary": "作者先说明有界算符与范数连续的关系，再在希尔伯特空间上利用 Riesz 表示定理定义伴随，证明其存在、唯一、线性和有界，并列出伴随对加法、数乘、乘积及逆运算的性质。",
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