三维形式的柯西不等式的证明如下:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/dc54564e9258d1099a68f259df58ccbf6d814d55?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/b151f8198618367a4bdc243620738bd4b21ce580?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/8718367adab44aedbd1be50fbd1c8701a08bfb80?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/5fdf8db1cb1349542dafb38f584e9258d0094aa3?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/ac345982b2b7d0a2f88a5f76c5ef76094a369a9e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
两边开平方得:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/35a85edf8db1cb136ad867f8d354564e93584ba3?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
柯西不等式是由大数学家柯西(Cauchy)在研究数学分析中的“流数”问题时得到的。但从历史的角度讲,该不等式应称作Cauchy-Buniakowsky-Schwarz不等式【柯西-布尼亚科夫斯基-施瓦茨不等式】因为,正是后两位数学家彼此独立地在积分学中推而广之,才将这一不等式应用到近乎完善的地步。
柯西不等式在解决不等式证明的有关问题中有着十分广泛的应用,所以在高等数学提升中与研究中非常重要,是高等数学研究内容之一。
扩展资料:
1、向量形式的柯西不等式:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/3801213fb80e7bec5789580a212eb9389a506b8d?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
2、向量形式推广:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/962bd40735fae6cd1c0df08a01b30f2443a70f9b?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
3、概率论形式的柯西不等式:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/d4628535e5dde7119d51181da9efce1b9c16615d?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
4、积分形式的柯西不等式:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/79f0f736afc3793177c1ddc2e5c4b74542a911ab?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)