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        "quote": "> [!exercise] <a class=\"object-permalink\" href=\"#problem-12-4\">习题 12.4</a> \n>\n> 我们说一列分布 $T _ { n }$ 收敛到分布 $T$，记作 $T _ { n } \\to T$，如果对所有试验函数 $\\phi \\in \\mathcal { D }$ 都有 $T _ { n } ( \\phi ) \\to T ( \\phi )$（这有时称为 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:6a25e9a4-78c0-4b7a-bfe0-2b96d45c1ef2\" data-concept-route=\"/concept-identities#concept-6a25e9a4-78c0-4b7a-bfe0-2b96d45c1ef2\" id=\"first-11b996d2e4613b9a177a-zh\">弱收敛(weak convergence)</u>）。如果一列连续函数 $f _ { n }$ 在 $\\mathbb R$ 的每个紧子集上一致收敛到函数 $f ( x )$，证明相应的正则分布 $T _ { f _ { n } } \\to T _ { f }$\n>\n> 在分布意义下，证明我们有以下收敛：\n>\n> $$\n> f _ {n} (x) = \\frac {n}{\\pi (1 + n ^ {2} x ^ {2})} \\rightarrow \\delta (x),\n> $$\n>\n> $$\n> g _ {n} (x) = \\frac {n}{\\sqrt {\\pi}} \\mathrm{e} ^ {- n ^ {2} x ^ {2}} \\rightarrow \\delta (x).\n> $$",
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