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        "quote": "设 $V$ 是一个实有限维向量空间，且 $\\dim$ $V = n$。向量空间 $V$ 上的一个 <u id=\"concept-5-real-inner-product\" class=\"concept-term\" data-concept-identity=\"urn:uuid:1438172c-4529-4171-9338-e797f954f1fb\" data-concept-route=\"/concept-identities#concept-1438172c-4529-4171-9338-e797f954f1fb\">实内积(real inner product)</u>，当不会引起混淆时通常简称为内积，是一个映射 $V \\times V \\to \\mathbb { R }$，它对每一对向量 $u$、$v \\in V$ 都指定一个实数 $u \\cdot v \\in \\mathbb { R }$，并满足以下三个条件：",
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        "quote": "向量空间 $V$ 上的[<span id=\"first-7fbfc15c27cd8776f6cd-zh\" class=\"block-anchor\"></span><u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/real-inner-product\" data-concept-identity=\"urn:uuid:1438172c-4529-4171-9338-e797f954f1fb\" data-concept-route=\"/concept-identities#concept-1438172c-4529-4171-9338-e797f954f1fb\" id=\"first-7fbfc15c27cd8776f6cd-real-zh\">内积(inner product)</u>](../../../../../concepts/linear-algebra/real-inner-product)是一种附加结构：它赋予向量长度和夹角的概念，并给出 $V^*\\simeq V$ 的识别。在实数域上：\n<span id=\"first-5061937789f0fe3448b6-zh\" class=\"block-anchor\"></span>",
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