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"quote": "对一个希尔伯特空间 $\\mathcal{H}$ 上的线性算符$A$ 而言，若满足如果对于任何 $M>0$ ，存在一个向量 $u$ 使得 $\\|A u \\| \\geq M\\|u\\|$，则算符称为无界的(unbounded)。在整个 $\\mathcal{H}$ 上的无界算符中的有趣示例很少 ， 例如自伴算符就根本没有。因此，通常认为无界算子 $A$ 不一定定义在整个 $\\mathcal{H}$ 上，而只定义在某个向量子空间 $D_{A} \\subseteq \\mathcal{H}$ 称为 $A$ 的定义域(domain)。它的值域(range)定义为映射到的向量集合，$R_{A}=A\\left(D_{A}\\right)$。一般来说，我们会用一对 $\\left(A, D_{A}\\right)$，其中 $D_{A}$ 是 $\\mathcal{H}$的向量子空间 且$A: D_{A} \\rightarrow R_{A} \\subseteq \\mathcal{H}$ 是一个线性映射，来指代一个$\\mathcal{H}$ 中的算符。当定义域 $D_{A}$没有歧义时，我们通常会简单地用算符 $A$来指代。",
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"quote": "A linear operator $A$ on a Hilbert space is unbounded if for any $M > 0$ there exists a vector $u$ such that $\\| A u \\| \\geq M \\| u \\|$ . Very few interesting examples of unbounded operators are defined on all of $\\mathcal H$ – for self-adjoint operators, there are none at all. It is therefore usual to consider an unbounded operator $A$ as not being necessarily defined over all of $\\mathcal H$ but only on some vector subspace $D _ { { \\cal A } } \\subseteq { \\mathcal { H } }$ called the domain of $A$. Its range is defined as the set of vectors that are mapped onto, $R _ { A } = A ( D _ { A } )$ . In general we will refer to a pair $( A , D _ { A } )$ , where $D _ { A }$ is a vector subspace of $\\mathcal H$ and $A : D _ { A } \\to R _ { A } \\subseteq \\mathcal { H }$ is a linear map, as being an operator in $\\mathcal H$. Often we will simply refer to the operator $A$ when the domain $D _ { A }$ is understood.",
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