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瑕佹眰锛堣│琛岋級銆嬶紝棣栭兘甯寖澶у(xu茅)鍑虹増绀�锛�1994銆� 娉涘嚱鍒嗘瀽 绗竴绔� 搴﹂噺绌洪枔鑸囩窔鎬ц肠鑼冪┖闁� 鑰冭│瑕侀粸(di菐n)锛� 搴﹂噺绌洪枔鐨勬蹇碉紝渚嬪瓙锛涘害閲忕┖闁撲腑鐨勬敹鏂傛€ц垏閫g簩(x霉)鎬э紱绋犲瘑鎬�锛涘彲鍒嗘€�锛汣auchy 鍒� 鑸囧害閲忕┖闁撶殑瀹屽倷鎬�锛涘绺槧鍍忓師鐞嗗強鍏舵噳(y墨ng)鐢紱绶氭€ц肠鑼冪┖闁撶殑姒傚康锛屼緥瀛�锛汢anach 绌洪枔 鐨勬蹇�銆� 鑰冭│鍏�(n猫i)瀹癸細 绗竴绡€(ji茅) 搴﹂噺绌洪枔鐨勬蹇佃垏渚嬪瓙 璺濋洟鍙婂害閲忕┖闁撶殑瀹氱京锛涗緥瀛愶紙姝愭皬绌洪枔 n R 锛涢€g簩(x霉)鍑芥暩(sh霉)绌洪枔 ],[ baC 锛涙暩(sh霉)鍒楃┖闁� p l 绛夛級銆� 绗簩绡€(ji茅) 搴﹂噺绌洪枔涓殑妤甸檺? 绋犲瘑鎬�? 鍙垎绌洪枔 闋�(l菒ng)鍩熺殑姒傚康锛涙敹鏂傞粸(di菐n)鍒�锛涙湁鐣岄泦锛涘叿楂旂┖闁撲腑鏀舵杺鎬х殑鎰忕京锛涚瀵嗘€ц垏鍙垎绌洪枔鐨勬蹇碉紱 涓嶅彲鍒嗙┖闁撶殑渚嬪瓙銆� 绗笁绡€(ji茅) 閫g簩(x霉)鏄犲皠 鏄犲皠閫g簩(x霉)鎬х殑鍚勭ó瀹氱京鍙婂叾绛夊児(ji脿)鎬�銆� 绗洓绡€(ji茅) Cauchy 榛�(di菐n)鍒楄垏瀹屽倷搴﹂噺绌洪枔 搴﹂噺绌洪枔涓� Cauchy 榛�(di菐n)鍒楃殑姒傚康锛涘畬鍌欏害閲忕┖闁撶殑瀹氱京锛涘畬鍌欏害閲忕┖闁撹垏涓嶅畬鍌欏害閲忕┖ 闁撶殑鍚勯渚嬪瓙锛涘害閲忕┖闁撻枆瀛愮┖闁撶殑瀹屽倷鎬�銆� 绗簲绡€(ji茅) 搴﹂噺绌洪枔鐨勫畬鍌欏寲 绛夎窛鍚屾(g貌u)锛涘害閲忕┖闁撶殑瀹屽倷鍖栧畾鐞嗭紱 绗叚绡€(ji茅) 澹撶府鏄犲儚鍘熺悊鍙婂叾鎳�(y墨ng)鐢� 澹撶府鏄犲儚鐨勫畾缇�锛涘绺槧鍍忓師鐞�锛涘湪闅卞嚱鏁�(sh霉)瀹氱悊鍙婂父寰垎鏂圭▼涓殑鎳�(y墨ng)鐢�銆� 绗竷绡€(ji茅) 绶氭€х┖闁� 鏈瘈(ji茅)鍏�(n猫i)瀹圭偤绶氭€х┖闁撶殑鍩烘湰姒傚康銆傚洜?q奴)W鐢熷凡鍦ㄩ珮绛変唬鏁�(sh霉)瑾茬▼涓(xu茅)閬庢湁闄愮董绌洪枔鐨勬湁闂�(gu膩n) 鍏�(n猫i)瀹癸紝鏁呭彧闇€绨¤鍥為¨骞跺挤(qi谩ng)瑾�(di脿o)鐒¢檺缍窔鎬х┖闁撶殑鐗瑰緛鍗冲彲銆� 绗叓绡€(ji茅) 绶氭€ц肠鑼冪┖闁撳拰 Banach 绌洪枔 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L 绌洪枔锛� p l 绌洪枔锛� ],[ baL p 绌� 闁�锛涢€忓竟鐞嗚В澹撶府鏄犲儚鍘熺悊鍙婂叾绨″柈鎳�(y墨ng)鐢�銆傝兘鐛�(d煤)绔嬭В绛斿熀鏈殑缈�(x铆)椤�銆� 绗簩绔� 绶氭€ф湁鐣岀畻瀛愬拰绶氭€ч€g簩(x霉)娉涘嚱 鑰冭│瑕侀粸(di菐n)锛� 绶氭€ф湁鐣岀畻瀛�锛岀窔鎬ч€g簩(x霉)娉涘嚱锛岀窔鎬х畻瀛愮┖闁�锛屽叡杌涚┖闁撱€� 鑰冭│鍏�(n猫i)瀹癸細 绗竴绡€(ji茅) 绶氭€ф湁鐣岀畻瀛愯垏绶氭€ч€g簩(x霉)娉涘嚱 绶氭€ф湁鐣岀畻瀛愯垏绶氭€ч€g簩(x霉)娉涘嚱鐨勬蹇�锛屼緥瀛愶紝鏈夌晫鑸囬€g簩(x霉)鐨勭瓑鍍�(ji脿)鎬�锛岀窔鎬ф湁鐣岀畻瀛愰浂绌� 闁撶殑鎬ц唱(zh矛)锛岀畻瀛愯寖鏁�(sh霉)銆� 绗簩绡€(ji茅) 绶氭€х畻瀛愮┖闁撳拰鍏辫粵绌洪枔 绶氭€х畻瀛愮┖闁撶殑绲�(ji茅)妲�(g貌u)鍙婂叾瀹屽倷鎬�锛屽叡杌涚┖闁�锛屼繚璺濈畻瀛�锛屽悓妲�(g貌u)鏄犵収锛屽悓妲�(g貌u)锛屼竴浜涘叿楂旂┖ 闁撶殑鍏辫粵绌洪枔銆� 鑰冩牳瑕佹眰锛� 鎺屾彙绶氭€ф湁鐣岀畻瀛�锛岀窔鎬ч€g簩(x霉)娉涘嚱锛屾湁鐣屾€э紝閫g簩(x霉)鎬�锛岀畻瀛愯寖鏁�(sh霉)锛屽叡杌涚┖闁擄紝淇濊窛绠楀瓙锛� 鍚屾(g貌u)鏄犵収锛屽悓妲�(g貌u)绛夊熀鏈蹇碉紱鎺屾彙鏈夌晫鑸囬€g簩(x霉)鐨勭瓑鍍�(ji脿)鎬у畾鐞�锛屽熀鏈畾鐞�锛涜兘澶犺▓(j矛)绠楃啊鍠殑绠� 瀛愯寖鏁�(sh霉)鍜屼竴浜涘叿楂旂┖闁撶殑鍏辫粵绌洪枔銆傝兘鐛�(d煤)绔嬭В绛斿熀鏈殑缈�(x铆)椤�銆� 绗笁绔� 鍏�(n猫i)绌嶇┖闁撳拰 Hilbert 绌洪枔 鑰冭│瑕侀粸(di菐n)锛� 鍏�(n猫i)绌嶇┖闁�锛屾姇褰卞畾鐞�锛孒ilbert 绌洪枔锛屽氨鑼冪洿浜ょ郴锛孒ilbert 绌洪枔涓婄窔鎬ч€g簩(x霉)娉涘嚱鐨勮〃绀�銆� 鑰冭│鍏�(n猫i)瀹癸細 绗竴绡€(ji茅) 鍏�(n猫i)绌嶇┖闁撶殑鍩烘湰姒傚康 鍏�(n猫i)绌嶇┖闁撹垏 Hilbert 绌洪枔鐨勫畾缇�锛屽钩琛屽洓閭婂舰鍏紡锛屽収(n猫i)绌嶇┖闁撶殑鍒ゅ畾銆� 绗簩绡€(ji茅) 鎶曞奖瀹氱悊 榛�(di菐n)鍒伴泦鍚堢殑璺濋洟锛屽嚫闆�锛屾サ灏忓寲鍚戦噺瀹氱悊锛岄泦鍚堢殑姝d氦锛孒ilbert 绌洪枔鐨勬浜ゅ垎瑙�锛屾姇 褰辩畻瀛愬強鍏舵€ц唱(zh矛)銆� 绗笁绡€(ji茅) Hilbert 绌洪枔涓殑灏辫寖鐩翠氦绯� 灏辫寖鐩翠氦绯�锛孎(xi脿n)ourier 绯绘暩(sh霉)闆�锛孊essel 涓嶇瓑寮忥紝Parseval 鎭掔瓑寮�锛屽畬鍏ㄥ氨鑼冪洿浜ょ郴鐨勫畾缇� 鑸囧垽瀹�, Fourier 灞曞紡锛孏ram-Schmidt 姝d氦鍖栭亷绋�锛孒ilbert 绌洪枔鐨勫悓妲�(g貌u)銆� 绗洓绡€(ji茅) Hilbert 绌洪枔涓婄殑绶氭€ч€g簩(x霉)娉涘嚱 Riesz 琛ㄧず瀹氱悊锛屽叡杌涚畻瀛愬強鍏舵€ц唱(zh矛)銆� 绗簲绡€(ji茅) 鑷即绠楀瓙銆� 閰夌畻瀛愬拰姝e父绠楀瓙 鑷即绠楀瓙銆� 閰夌畻瀛愬拰姝e父绠楀瓙鐨勫熀鏈蹇佃垏绨″柈鎬ц唱(zh矛)銆� 鑰冩牳瑕佹眰锛� 鎺屾彙鍏�(n猫i)绌嶇┖闁擄紝Hilbert 绌洪枔锛屽钩琛屽洓閭婂舰鍏紡锛屽氨鑼冪洿浜ょ郴锛孊essel 涓嶇瓑寮�锛孭arseval 鎭掔瓑寮忥紝F(xi脿n)ourier 灞曞紡锛屾姇褰辩畻瀛愶紝鍏辫粵绠楀瓙锛岃嚜浼寸畻瀛愶紝閰夌畻瀛愬拰姝e父绠楀瓙绛夊熀鏈蹇�锛� 鎺屾彙妤靛皬鍖栧悜閲忓畾鐞�锛屾姇褰卞畾鐞�锛屽畬鍏ㄥ氨鑼冪洿浜ょ郴鐨勫垽瀹氬畾鐞�, Riesz 琛ㄧず瀹氱悊绛夊熀鏈畾鐞� 鐨勫収(n猫i)瀹硅垏璀夋槑锛涜兘鐛�(d煤)绔嬭В绛斿熀鏈殑缈�(x铆)椤�銆� 绗洓绔� Banach 绌洪枔涓殑鍩烘湰瀹氱悊 鑰冭│瑕侀粸(di菐n)锛� Hahn-Banach 寤舵嫇瀹氱悊锛孯iesz 琛ㄧず瀹氱悊锛岀窔鎬ц肠鑼冪┖闁撲腑鐨勫叡杌涚畻瀛愶紝 绗竴绡€(ji茅) 娉涘嚱寤舵嫇瀹氱悊 娆$窔鎬ф硾鍑�锛孒ahn-Banach 娉涘嚱寤舵嫇瀹氱悊鐨勫(sh铆)褰㈠紡銆佸京(f霉)褰㈠紡鍙婂叾鎺ㄨ珫銆� 绗簩绡€(ji茅) ],[ baC 鐨勫叡杌涚┖闁�銆丷iesz 琛ㄧず瀹氱悊 绗笁绡€(ji茅) 鍏辫粵绠楀瓙 绗洓绡€(ji茅) 绶氭€ц肠鑼冪┖闁撲腑鍏辫粵绠楀瓙鐨勫畾缇╁強鎬ц唱(zh矛)銆� 绗簲绡€(ji茅) 缍卞畾鐞嗗拰涓€鑷存湁鐣屾€у畾鐞� 绗竴缍遍泦锛岀浜岀侗闆�锛孊aire 缍卞畾鐞�, 涓€鑷存湁鐣屾€у畾鐞嗗挤(qi谩ng)鏀舵杺銆佸急鏀舵杺鍜屼竴鑷存敹鏂� 寮�(qi谩ng)鏀舵杺銆佸急鏀舵杺銆佸急*鏀舵杺鍜屼竴鑷存敹鏂傜殑瀹氱京锛屼緥瀛�锛岀浉浜掗棞(gu膩n)绯�锛屽挤(qi谩ng)鏀舵杺鐨勫厖瑕佹浠躲€� 绗叚绡€(ji茅) 閫嗙畻瀛愬畾鐞� 閫嗙畻瀛愬畾鐞嗗強鍏惰瓑鏄�銆� 绗竷绡€(ji茅) 闁夊湒璞″畾鐞� 绶氭€х畻瀛愮殑鍦栬薄锛岄枆绠楀瓙锛岄枆鍦栬薄瀹氱悊銆� 鑰冩牳瑕佹眰锛� 鎺屾彙鏈珷娑夊強鍒扮殑鎵€鏈夊熀鏈蹇�锛屽熀鏈畾鐞�锛涚敱浜� Hahn-Banach 寤舵嫇瀹氱悊锛孯iesz 琛ㄧず 瀹氱悊锛孊aire 缍卞畾鐞�锛岄€嗙畻瀛愬畾鐞�锛岄枆鍦栬薄瀹氱悊鏄硾鍑藉垎鏋愬熀绀�(ch菙)鐞嗚珫鐨勪富瑕佹(g貌u)鎴愰儴鍒嗭紝瑕佹眰 鐔熺反鎺屾彙閫欎簺鍏�(n猫i)瀹�锛涜兘鐛�(d煤)绔嬭В绛斿熀鏈殑缈�(x铆)椤屻€� 绗簲绔� 绶氭€х畻瀛愮殑璀� 鑰冭│瑕侀粸(di菐n)锛� 绨¤浠嬬垂绶氭€х畻瀛愮殑璀滅殑姒傚康锛屽熀鏈€ц唱(zh矛)銆� 璀滅殑姒傚康 姝e墖绠楀瓙锛屾鍓囬粸(di菐n)锛屾鍓囬泦锛岃瓬榛�(di菐n)锛岀壒寰佸€�锛岀壒寰佸悜閲忥紝榛�(di菐n)璀�锛岄€g簩(x霉)璀滐紝渚嬪瓙銆� 绗竴绡€(ji茅) 绶氭€ф湁鐣岀畻瀛愯瓬鐨勫熀鏈€ц唱(zh矛) 璀滈泦鐨勯枆鎬�銆� 鑰冩牳瑕佹眰锛� 浜嗚В绶氭€х畻瀛愮殑璀滅殑姒傚康锛屽熀鏈€ц唱(zh矛)銆� 涓�銆佸弮鑰冩浉鐩� 1銆� 绋嬪叾瑗勭瓑锛屻€婂(sh铆)璁婂嚱鏁�(sh霉)鑸囨硾鍑藉垎鏋愬熀绀�(ch菙)銆嬶紝楂樼瓑鏁欒偛鍑虹増绀�锛� 1983锛� 绗竴鐗堛€� 2銆� 鐜嬭伈鏈�锛� 閯董琛�锛屻€婂(sh铆)璁婂嚱鏁�(sh霉)鑸囨硾鍑藉垎鏋愭瑕併€�锛岀浜屽唺锛岄珮绛夋暀鑲插嚭鐗堢ぞ锛�1992锛� 绗簩鐗�銆� 3銆� 澶忛亾琛岀瓑锛屻€婂(sh铆)璁婂嚱鏁�(sh霉)璜栬垏娉涘嚱鍒嗘瀽銆�锛屼笅鍐婏紝楂樼瓑鏁欒偛鍑虹増绀�锛� 1985锛岀浜岀増銆�
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