泛函分析7.pdfVIP

{ } 璁綳 鏄竴涓泦鍚堬紝蟿锛漊 | U 鈯俋 . 褰撓 s.t. 1)鈭? X 鈭埾凔 2) 浠绘剰骞垛垐蟿锛? 锛変换鎰忎簩涓氦鈭埾 鏃讹紝绉跋 鏄疿 涓婄殑鎷撴墤銆 { } 璁綳 鏄竴涓泦鍚堬紝饊剠锛滲| B 鈯俋 . 褰擆€剠 s.t. 1) 鈭€x 鈭圶, 鈭 B锟 鈭堭€剠 s. t. x 鈭圔锟 ; 2) B 鈭〣 : 鈭€x 鈭圔 鈭〣 , 鈭 B 鈭堭€剠 s. t. x 鈭圔 鈯侭 鈭〣 ; 锟 锟 锟 锟 锟 锟 锟 锟 绉 饊剠鏄 a basis. { } 杩欐椂锛屾槗璇 U: 鈭€x 鈭圲, 鈭 B锟 鈭堭€剠 s. t. x 鈭圔锟 鈯俇 鏄嫇鎵戯紝绉颁綔鐢 饊剠 鐢熸垚鐨勬嫇鎵戙€ { ( ) } 渚嬪锛氳 (X, d) 鏄窛绂荤┖闂达紝饊剠锛漁 x, r : 鈭€x 鈭圶, 鈭€r 0 is a basis. 鐢别€剠 鐢熸垚鐨勬嫇鎵戝氨鏄窛绂绘嫇鎵戙€ ( ) 娉細璁綳鏄嚎鎬х┖闂达紝p x , q(x) 鏄疿涓婄殑seminorms. { ( ) } { ( ) } ( ) ( ) 鍒 x 鈭圶 : q x 1 鈯 x 鈭圶 : p x 1 iff p x 鈮 x 鈭€x 鈭圶. 璇佹槑锛氣噽鏄剧劧 ( ) ( ) 鈬掑浐瀹歺锟 鈭圶, 涓嬭瘉p x锟 鈮 x锟 锛氣垁蔚 0 1 1 ( ) ( ) q 锟 ( ) x锟斤拷 1 鈬掟€仢锟 ( ) x锟斤拷 1 鈬掟€仢 x锟 饊仦 x锟 + 蔚 q x锟 + 蔚 q x锟 + 蔚 鈥栤€ 鈥栤€ Def. If 鈭 and 鈭 are two norms on X, they are said to be 锟 锟 equivalent norm if they define the same topology on X. 鈥栤€ 鈥栤€ Th. 1.5 璁 X 鏄嚎鎬х┖闂达紝 鈭 鍜 鈭 閮芥槸X 涓婄殑鑼冩暟銆 锟 锟 鈥 鈥 鈥 鈥 鍒 these two norms are equivalent iff 鈭僣, C 0 饊仩. 饊仭. 饊亹 x 锟 鈮 x 锟 鈮 鈥 鈥 C x 锟? 鈭€x 鈭圶. Th.3.1 璁綳 鏄湁闄愮淮鐨勭嚎鎬х┖闂淬€ 鈥栤€ 鈥栤€ 鍒橷 涓婁换鎰忎簩涓寖鏁 鈭 鍜 鈭 鏄瓑浠风殑銆 锟 { }