• 2022-06-17
    求证:任何有限维线性赋范空间都是自反的.
  • 证 设 [tex=2.286x1.357]D5Kq4jAKpYjd6eeCfmMJ7D+dyb1ZhhUM2Bz4hJlLVOJ1zlYtLqzSBb1uEbW5C80D[/tex]是 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 维线性赋范空间 [tex=1.0x1.0]2vP91ZxLLBYpBJzmqRtNKw==[/tex] 中的[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 个线性无关的元素,  [tex=7.929x1.286]Bx2Bs51B6jNxfkvTiVd2wogVZQ8tC5/0W0lEfUjIGApl9vKrcCOnm/ItzFpWknJUdt9hgRB9Yzy+S9riTagXwg==[/tex]使得[tex=12.143x1.357]RSNm+L1HIJjLzggHNdusVybCwzrCk7zCRSQjGM07cEVLIoKb7hEvW4QYFNVhLsV7h1bKomo30uQXfv3LXxh/DEKVcGjpU+7rlGStd6Y242udhkkWSyyGcmcib4YogU01[/tex]从而对 [tex=7.357x2.714]ySjj1drHeJdv8LcRYNK9RlSzK/DAVOg2TW9tAtRqfavHm7WGLjjfEpLN5Av5w2yqIFZMDT2fsWRQuvVvelXBpw==[/tex][tex=17.571x3.286]vDoW8m1TJN7+SEMmV3qtQCJ3xCF1O3/Zf+6knmHGvdW2Kh3JvKbCKBdl0PvFmjPXEdujpYepgah9hawiLQuUGBtEp4C+BuE/LIW2RRwCZ85cQ1Evmm46bBf6/lMQujEZX79imf+I+85LcsV/hIcBZIVewRZsKp+NbLAkR15+yNE=[/tex]故对 [tex=6.357x1.286]/+bKx4kKI1Trkk66qF9bh/YJJJ5lUlrwxWz2ZZYEWEByf97ri+DkOZhx1C6rHJsEhuGN11F2fvlRUhbOQ12HEQ==[/tex]有[tex=24.5x3.357]7F6g81jeDkDOuwHHNpu+xUOdB1+S5mbhNMO2ISlwD+WaYlBg9cLRvn4qPDp+j0c7pnUS0bbpUY1kFfMs//eDrvs1R2vLzUKwIZa2RgFhWfbjn/ahGfVp2BNa9TlTRyMTWrifqtYO5z+/U/LMpHrnG5Xh1NSeW4Zynq5J46TVM6NfXJFTJaxP5uBo8KsZAqv36BqtykqNO4ulzCcSq7lvttWCxkrO/QmDkDQxCWJ6MiqcnMRoTpuiYHeWkvtroXA5Kch5rm0ikbcquK+Bx4WuI24zYoKFKGXOnA/FMuswbRE17VmSt+QQEH0yA84ysRtpmskd9U7kaQZYGaGLmXTEJQ==[/tex]由此可见[tex=6.5x3.286]jeAd26hoTeqO9J7nRu4w6nKHgcHTttc5AHjobq2J4TXhQPDjR+AJwVshMEBs0VYnESTewIc1Ri44bU/UX84MvQ==[/tex][tex=7.857x1.286]6mwaFytx+E0Tmw6+4rByszdnv7WCFipZWgPXR1fuEFCSnygx3hbeg4a5tKPXesZ1dzmHNeeFY3Lf1IflEbxA4ARFaq4KXxwCQRHF90LyQzo=[/tex] 我们有[tex=35.714x3.357]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[/tex][tex=12.571x3.357]WBohl8Wnk8YYk3Ifw1hoPKb40nGgmiOawTPXPkCly0aiGTAHKsSRa2aw9goCSJxdk96nbVJWNkgkKwJk86hVR6nCTbn7BqP0esDR+S4E2Zv/8kG0+ud67ljNzBziU/RVM5o3Ckc4/SqNzXcDYH2UGE9DnCD7YlzpzOWdFJoSbj71qQLdLmtgghrOAPR0Ke65R/2txjw1QYyOgDGFqyGCFA==[/tex]因此,若[tex=10.0x3.286]3kAVhvumjelkvrU0r+vL/RJhHB1Zp1Qck68/E0gQJKYImPpmdsX0Ij7hYPbsBeJfl77pkTngrungftls01rne3sdVFMWSr22PcNbAXn00bLx2iHasTKRcGGTYgbh5AUUwGmg5MQxh1vfwioQEXyayBVgdpPD9zcRuWpCOSDd78c=[/tex]则有[tex=10.286x1.357]vDoW8m1TJN7+SEMmV3qtQA/nTgfA0/ZzShIuvGeCmEkKmAJ3zmAM1BcZnxvJZiupznBY3stCxA3F7RPG8R6yC9/jkcUoGNSKW08bEEfGdQoO1MfyiAeLs7Uk5qwX4kndM9CiX3RAlAPpDV6CRB80CA==[/tex]即 [tex=3.429x1.286]e2Mlizc5yRswg6N87RyoGLDRiPHO1kckh/Pywa/lIjM=[/tex]其中[tex=0.5x0.786]xdTs2QHMXTpKzI7ZnwCRMQ==[/tex]是 [tex=3.643x1.071]6Uz9yfWuFpRxY0XBZeg6+CgPNq5xZ7OFPXCoPeaX5nhFfenb8+uwXCVJBqUeN625[/tex]的自然映射,从而 [tex=0.5x0.786]xdTs2QHMXTpKzI7ZnwCRMQ==[/tex] 是满射. 即证得[tex=1.0x1.0]2vP91ZxLLBYpBJzmqRtNKw==[/tex]$\mathscr{ 是自反的.

    内容

    • 0

       证明:赋范线性空间中的任何完备子空间是闭子空间. 

    • 1

      设[tex=0.857x1.0]N7iCrOsS+NNEUUlnsYCi1g==[/tex]为赋范线性空间,试证明(1)序列弱完备的赋范线性空间必是巴拿赫空间

    • 2

      距离空间,线性空间,赋范线性空间,Banach空间,内积空间,Hilbert空间的内在关系

    • 3

      如果[tex=0.786x1.0]XvHgf70VtK2FH5G93l0k3g==[/tex]是无穷维赋范线性空间,则在[tex=0.786x1.0]XvHgf70VtK2FH5G93l0k3g==[/tex]上存在不连续的线性泛函。

    • 4

      线性空间[img=13x19]17de8302a296a1a.png[/img],按范数[img=145x71]17de8302ad30bda.png[/img]成为赋范空间. 则该赋范空间是内积空间的充要条件是( ). 未知类型:{'options': ['', '', '', ''], 'type': 102}