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        "quote": "我们现在考虑一个复向量空间 $V$，它首先可能是无限维的。向量仍将用小写罗马字母表示，例如 $u$ 和 $v$，但复标量将用希腊字母表示，例如字母表前部的 $\\alpha$、$\\beta , \\ldots$。词语 <u id=\"concept-5-inner-product\" class=\"concept-term\" data-concept-identity=\"urn:uuid:9dc809a9-95f3-4c45-bb82-06dd391d1f6b\" data-concept-route=\"/concept-identities#concept-9dc809a9-95f3-4c45-bb82-06dd391d1f6b\">内积(inner product)</u>，或 <u id=\"concept-5-scalar-product\" class=\"concept-term\" data-concept-identity=\"urn:uuid:10861f1d-f72d-4e92-808c-bba46d7cd49c\" data-concept-route=\"/concept-identities#concept-10861f1d-f72d-4e92-808c-bba46d7cd49c\">标量积(scalar product)</u>，在复向量空间上",
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        "quote": "\n\n> [!definition] 定义 · <span class=\"definition-name\">复内积(complex inner product)</span>\n> <span id=\"source-block-081\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">定义（<u class=\"concept-term\" data-concept-id=\"concept-4-complex-inner-product\" data-concept-role=\"definition\" data-concept-identity=\"urn:uuid:9dc809a9-95f3-4c45-bb82-06dd391d1f6b\" data-concept-route=\"/concept-identities#concept-9dc809a9-95f3-4c45-bb82-06dd391d1f6b\">复内积(complex inner product)</u>）。</span>复向量空间 $V$ 上的厄米内积是一个映射\n>\n> <span id=\"source-block-082\" class=\"block-anchor\"></span>\n>\n> $$\n> \\langle \\cdot , \\cdot \\rangle : V \\times V \\to \\mathbf {C}\n> $$\n>\n> <span id=\"source-block-083\" class=\"block-anchor\"></span>\n>\n> 即共轭对称\n>\n> <span id=\"source-block-084\" class=\"block-anchor\"></span>\n>\n> $$\n> \\langle v, w \\rangle = \\overline {{\\langle w , v \\rangle}}\n> $$\n>\n> <span id=\"source-block-085\" class=\"block-anchor\"></span>\n>\n> 在两个变量上均非退化，在第二个变量上线性，在第一个变量上反线性：对于 $\\alpha \\in \\mathbf { C }$ 和 $u , v , w \\in V$。\n>\n> <span id=\"source-block-086\" class=\"block-anchor\"></span>\n>\n> $$\n> \\langle u + v, w \\rangle = \\langle u, w \\rangle + \\langle v, w \\rangle , \\quad \\langle \\alpha u, v \\rangle = \\overline {{\\alpha}} \\langle u, v \\rangle\n> $$\n",
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        "quote": "so they provide an [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/orthonormal-basis\" data-concept-identity=\"urn:uuid:dabe32db-8d84-4c17-9eeb-583f0609a265\" data-concept-route=\"/concept-identities#concept-dabe32db-8d84-4c17-9eeb-583f0609a265\" id=\"first-e134353abe687ae26a43-en\">orthonormal basis</u>](../../../../../concepts/linear-algebra/orthonormal-basis.md) for $\\mathbf{C}^2$ with respect to the Hermitian [<span id=\"first-7fbfc15c27cd8776f6cd-en\" class=\"block-anchor\"></span><u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/complex-inner-product\" data-concept-identity=\"urn:uuid:9dc809a9-95f3-4c45-bb82-06dd391d1f6b\" data-concept-route=\"/concept-identities#concept-9dc809a9-95f3-4c45-bb82-06dd391d1f6b\" id=\"first-7fbfc15c27cd8776f6cd-complex-en\">inner product</u>](../../../../../concepts/linear-algebra/complex-inner-product.md).",
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