5f5Qiang Lei Department of Mathematics, Harbin Institute of Technology Harbin , China. Email: leiqianghit.edu.cnSeptember, 2014)5f5 Outline15f5)5f5 5f5(Linear operator and Linear functional)1 Y5fk.5f?(X,1),(Y,2)kK?5m, T : X Y 5f, x0 X. T3x0:Y?, X Jx X, x x0, KTx Tx0. eT3Xz:Y, KT3XY?.?(X,1),(Y,2)kK?5m, T : X Y 5f. Tk.?, XJu(X,1)?k .8B, T(B)(Y,2)?k.8.)5f5 5f5(Linear operator and Linear functional)1 Y5fk.5f?(X,1),(Y,2)kK?5m, T : X Y 5f, x0 X. T3x0:Y?, X Jx X, x x0, KTx Tx0. eT3Xz:Y, KT3XY?.?(X,1),(Y,2)kK?5m, T : X Y 5f. Tk.?, XJu(X,1)?k .8B, T(B)(Y,2)?k.8.)5f5n?(X,1),(Y,2)kK?5m, T : X Y 5f, XJTY?, KTk.?.53?TVS, k.5f7Y?.e1,2mX?, 1 2, K(X,2) ?k.87(X,1)?k.8.n?(X,1),(Y,2)kK?5m, T : X Y 5f, x0 X. T3x0:Y?=?T3X Y.)5f5n?(X,1),(Y,2)kK?5m, T : X Y 5f, XJTY?, KTk.?.53?TVS, k.5f7Y?.e1,2mX?, 1 2, K(X,2) ?k.87(X,1)?k.8.n?(X,1),(Y,2)kK?5m, T : X Y 5f, x0 X. T3x0:Y?=?T3X Y.)5f5n?(X,1),(Y,2)kK?5m, T : X Y 5f, XJTY?, KTk.?.53?TVS, k.5f7Y?.e1,2mX?, 1 2, K(X,2) ?k.87(X,1)?k.8.n?(X,1),(Y,2)kK?5m, T : X Y 5f, x0 X. T3x0:Y?=?T3X Y.)5f5n?(X,1),(Y,2)kK?5m, T : X Y 5f, XJTY?, KTk.?.53?TVS, k.5f7Y?.e1,2mX?, 1 2, K(X,2) ?k.87(X,1)?k.8.n?(X,1),(Y,2)kK?5m, T : X Y 5f, x0 X. T3x0:Y?=?T3X Y.)5f52 Dm?5fn?X,Y D5m, TlX?Y ?5f, Tk. ?73M 0, ?x X, kkTxk Mkxk.n?X,Y D5m, TlX?Y ?5f, KT Y?=?Tk.?.)5f52 Dm?5fn?X,Y D5m, TlX?Y ?5f, Tk. ?73M 0, ?x X, kkTxk Mkxk.n?X,Y D5m, TlX?Y ?5f, KT Y?=?Tk.?.)5f5k.5f|?m?X,Y D5m, L(X,Y )L?NlX?Y ?k. 5f. XJY = X, rL(X,Y )PL(X). u? ?T1,T2 L(X,Y )9 K, (T1+ T2)(x) = T1x + T2x, (T1)(x) = T1x, KL(X,Y ) 5m. AO?, X= L(X,K)X?mm. ?f X?X ?Y5.?TlDmX?Y ?k.5f, kTk =sup xX,x6=0kTxk kxk,KkkL(X,Y ) ?, fT?. (L(X,Y ),kk) D5m.)5f5k.5f|?m?X,Y D5m, L(X,Y )L?NlX?Y ?k. 5f. XJY = X, rL(X,Y )PL(X). u? ?T1,T2 L(X,Y )9 K, (T1+ T2)(x) = T1x + T2x, (T1)(x) = T1x, KL(X,Y ) 5m. AO?, X= L(X,K)X?mm. ?f X?X ?Y5.?TlDmX?Y ?k.5f, kTk =sup xX,x6=0kTxk kxk,KkkL(X,Y ) ?, fT?. (L(X,Y ),kk) D5m.)5f5n?X,Y D5m,T L(X,Y ), KkTk = supxX,x6=0kTxk kxk= infM : kTxk Mkxk = supkxk1kTxk = supkxk=1kTxk.neY Banachm, KL(X,Y )Banachm.)5f5n?X,Y D5m,T L(X,Y ), KkTk = supxX,x6=0kTxk kxk= infM : kTxk Mkxk = supkxk1kTxk = supkxk=1kTxk.neY Banachm, KL(X,Y )Banachm.)5f5RieszLnn(F.RieszLn)?XHilbertm, f X, K73 ?y X, ?f(x) = (x,y), kfk = kyk.)5f5?nn(mN?n) ?X,Y Banach m, T : X Y k.5 f. XJT?, KTmN?.n(_fn) ?X,Y Banach m, T : X Y _?k. 5f, K_T1k.?.n(?dn) ?XBanachm. eXk#k k1? 3de, XBanachm. XJ3?C ?kxk Ckxk1, x X, K?d, =kC1, ?C1kxk1 kxk Ckxk1, x X.)5f5?nn(mN?n) ?X,Y Banach m, T : X Y k.5 f. XJT?, KTmN?.n(_fn) ?X,Y Banach m, T : X Y _?k. 5f, K_T1k.?.n(?dn) ?XBanachm. eXk#k k1? 3de, XBanachm. XJ3?C ?kxk Ckxk1, x X, K?d, =kC1, ?C1kxk1 kxk Ckxk1, x X.)5f5?nn(mN?n) ?X,Y Banach m, T : X Y k.5 f. XJT?, KTmN?.n(_fn) ?X,Y Banach m, T : X Y _?k. 5f, K_T1k.?.n(?dn) ?XBanachm. eXk#k k1? 3de, XBanachm. XJ3?C ?kxk Ckxk1, x X, K?d, =kC1, ?C1kxk1 kxk Ckxk1, x X.)5f5n(4”n) ?X,Y Banach m, T : X Y 45f, KT k.?.mN?nBanach_fn ?dn 4”n.n(k.?n?n) ?XBanachm, Y D5 m, M L(X,Y )xk.5f, zx M, supTMkTxk 0?f(x) ,x A;f(x) + ,x B.lA, l5LABO3L?.)5f5n(8ln)?XHausdorff5 m,A X9B X?8.1eAkS:, K3f X9 R, ?Ref(x) Ref(y), x A,y B.2eA;8, B48, Xm, K 3f X9a,b R, ?Ref(x) 0.d, Xwm, XwLX?f. y, Xw= X., xnfux(xnw x)?=?f(xn) f(x), f X.)5f5Dm?f ?XDm, X?fdXp?, fPw, Hausdorff, ?zf X Y?f?. 3fe, XTVS, “:?x X : |fk(x)| 0.d, Xwm, XwLX?f. y, Xw= X. , xnfux(xnw x)?=?f(xn) f(x), f X.)5f5u8(AOfm), f44.n?XDm, SX?8, K1S?f4Sw?u?4S;28f4?=?4?./, f7. X, enHilbertm?I O?, Kenw 0, ?kenk = 1. ?3l1