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        "quote": "> [!exercise] <a class=\"object-permalink\" href=\"#problem-10-34\">习题 10.34</a> \n>\n> 设向量空间 $V$ 上的两个范数 $\\| u \\| _ { 1 }$ 与 $\\| u \\| _ { 2 }$ 称为 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:11c7fa5b-d5f8-40eb-ae0b-7e37ff4bebb1\" data-concept-route=\"/concept-identities#concept-11c7fa5b-d5f8-40eb-ae0b-7e37ff4bebb1\" id=\"first-9551e9906b8359a8622e-zh\">等价范数(equivalent norms)</u>，如果存在常数 $A$ 和 $B$，使得\n>\n> $$\n> \\| u \\| _ {1} \\leq A \\| u \\| _ {2} \\quad \\text { and } \\quad \\| u \\| _ {2} \\leq B \\| u \\| _ {1}\n> $$\n>\n> 对于所有 $u \\in V$。如果两个范数等价，则证明以下结论：\n>\n> (a) 若关于一种范数 $u _ { n } \\to u$，则关于另一种范数这也成立。\n>\n> (b) 关于一种范数连续的每个线性泛函，关于另一种范数也连续。\n>\n> (c) 设 $V = \\mathcal { C } [ 0 , 1 ]$ 为区间 [0, 1] 上连续复函数的向量空间。通过考虑函数序列\n>\n> $$\n> f _ {n} (x) = \\frac {n}{\\sqrt {\\pi}} \\mathrm{e} ^ {- n ^ {2} x ^ {2}}\n> $$\n>\n> 证明这些范数\n>\n> $$\n> \\| f \\| _ {1} = \\sqrt {\\int_ {0} ^ {1} | f | ^ {2} \\mathrm{d} x} \\quad \\text { and } \\quad \\| f \\| _ {2} = \\max \\{| f (x) | \\mid 0 \\leq x \\leq 1 \\}\n> $$\n>\n> 并不等价。\n>\n> (d) 证明由 $F ( f ) = f ( 1 )$ 定义的线性泛函关于 $\\lVert \\cdot \\rVert _ { 2 }$ 连续，但关于 $\\lVert \\cdot \\rVert _ { 1 }$ 不连续",
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