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        "quote": "对于任意一个满足$u_{n} \\rightarrow u$ 和 $A u_{n} \\rightarrow v$ 的序列 $u_{n} \\in$ $D_{A}$，如果有 $u \\in D_{A}$ 且 $A u=v$，则我们称稠定算符 $\\left(A, D_{A}\\right)$ 是<u id=\"concept-13-closed\" class=\"concept-term\" data-concept-identity=\"urn:uuid:8da45318-5944-4aa7-afd5-245f6894ccb2\" data-concept-route=\"/concept-identities#concept-8da45318-5944-4aa7-afd5-245f6894ccb2\">闭的(closed)</u>。另一种表述是，一个算符是闭的，当且仅当它的<u id=\"concept-13-graph\" class=\"concept-term\" data-concept-identity=\"urn:uuid:48bd14ac-fca7-46be-b50e-f729bd475980\" data-concept-route=\"/concept-identities#concept-48bd14ac-fca7-46be-b50e-f729bd475980\">图像(graph)</u> $G_{A}=\\{(x, A x) \\mid x \\in$ $\\left.D_{A} \\right\\}$ 是积集 $\\mathcal{H} \\times \\mathcal{H}$ 的闭子集。封闭性的概念类似于连续性，不同之处在于，对于封闭性，我们必须保证极限 $A u_{n} \\rightarrow v$ 必须成立；而对于连续性，这一性质可以推导出来。每个连续算符显然都是闭的，但反之未必成立。",
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        "quote": "We say a densely defined operator $( A , D _ { A } )$ in $\\mathcal { H }$ is <u class=\"concept-term\" data-concept-identity=\"urn:uuid:8da45318-5944-4aa7-afd5-245f6894ccb2\" data-concept-route=\"/concept-identities#concept-8da45318-5944-4aa7-afd5-245f6894ccb2\" id=\"first-df2cde65096927db28d1-en\">closed</u> if for every sequence $u _ { n } \\in$ $D _ { A }$ such that $u _ { n } \\to u$ and $Au_n\\to v$ it follows that $u \\in D _ { A }$ and $A u = v$ . Another way of expressing this is to say that an operator is closed if and only if its <u class=\"concept-term\" data-concept-identity=\"urn:uuid:48bd14ac-fca7-46be-b50e-f729bd475980\" data-concept-route=\"/concept-identities#concept-48bd14ac-fca7-46be-b50e-f729bd475980\" id=\"first-df4287765e747e65cfc8-en\">graph</u> $G _ { A } = \\{ ( x , A x ) | x \\in$ $D _ { A } \\}$ is a closed subset of the product set $\\mathcal { H } \\times \\mathcal { H }$ . The notion of closedness is similar to continuity, but differs in that we must assert the limit $Au_n\\to v$ , while for continuity it is deduced. Clearly every continuous operator is closed, but the converse does not hold in general.",
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