А.Н.Шерстнев - Математический анализ
..pdfdEJSTWITELXNO, W SILU OGRANI^ENNOSTI POSLEDOWATELXNOSTI (fn ) I USLO- WIQ kAn , Ak ! 0 PERWOE SLAGAEMOE W PRAWOJ ^ASTI (1) MOVET BYTX SDE- LANO MENX[E NAPERED ZADANNOGO ^ISLA PRI DOSTATO^NO BOLX[OM s. dLQ \TOGO s POSLEDOWATELXNOSTX (Asfns) SHODITSQ, A ZNA^IT, SHODITSQ POSLEDO- WATELXNOSTX Asfnn , POSKOLXKU PRI n > s (fnn) | PODPOSLEDOWATELXNOSTX POSLEDOWATELXNOSTI (fns). sLEDOWATELXNO, WTOROE SLAGAEMOE W PRAWOJ ^AS- TI (1) TAKVE MOVET BYTX SDELANO MENX[E NAPERED• ZADANNOGO ^ISLA PRI BOLX[IH n. oSTAETSQ• U^ESTX 244.2.
3.pUSTX PODPOSLEDOWATELXNOSTX An 2 C(H) FUNDAMENTALXNA. w SILU POLNOTY B(H) SU]ESTWUET A 2 B(H ), ^TO kAn , Ak ! 0. iZ P. 2 TEPERX SLEDUET, ^TO A 2 C(H ).
4.dOSTATO^NOSTX UVE USTANOWLENA (SM. P. 2 I 244.3). dOKAVEM NEOB-
HODIMOSTX |
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pUSTX |
fe1; e2; : : :g | |
ORTONORMIROWANNYJ BAZIS W |
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k=1h ; ekiek | |
KONE^NOMERNYE OPERATORY |
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ORTOPROEKTORY |
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k=1h ; ekiAek |
| TAKVE POSLEDOWATELXNOSTX KONE^NOMERNYH OPERATOROW I |
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DOSTATO^NO USTANOWITX, ^TO n kA , APnk ! 0 (n ! 1). rASSMOTRIM |
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RQD fURXE f = k=1hf; ekiek PROIZWOLXNOGO WEKTORA f 2 H I ZAMETIM, ^TO |
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(A , APn)f = |
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k=n+1hf; ekiAek; |
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kfk=1 k |
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kfk=1;f2(I,Pn)H k |
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TAK ^TO 1 |
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NAPROTIW |
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tOGDA NAJDETSQ TAKAQ POSLEDOWATELXNOSTX |
gn 2 (I , |
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Pn )H; kgnk |
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^TO |
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(2) |
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kAgnk > =2: |
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zAMETIM, ^TO DLQ L@BOGO f 2 H |
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(3) |
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hgn ; fi ! 0 (n ! 1): |
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421
dEJSTWITELXNO,
jhgn; fij |
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jhgn; k=1hf; ekiekij |
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= jhgn; k=n+1hf; ekiekij |
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= j k=n+1hf; ekihgn; ekij |
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f; ek |
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gn ; ek |
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k=n+1 jh |
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k=n+1 jh |
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! 0 (n ! 1): |
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k=n+1 jhf; ekij |
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w SILU KOMPAKTNOSTI A NAJDETSQ• |
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PODPOSLEDOWATELXNOSTX (gnk ) POSLEDO- |
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WATELXNOSTI (g ) TAKAQ, ^TO (Ag |
nk |
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SHODITSQ: |
Ag |
nk |
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h. w SILU |
(2) h = . |
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s DRUGOJ STORONY (S U^ETOM• |
(3), |
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h |
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= 0: |
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k k |
k h |
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|PROTIWORE^IE.
5.pROWERIM UTWERVDENIE DLQ SEPARABELXNOGO PROSTRANSTWA. pUSTX (An ) | POSLEDOWATELXNOSTX KONE^NOMERNYH OPERATOROW, SHODQ]AQSQ K
A(2 C(H)) PO NORME (P. 4). w SILU 243.3{4 (An) | POSLEDOWATELXNOSTX
KONE^NOMERNYH OPERATOROW, PRI^•EM kA , A k = kAn , Ak ! 0. sNOWA W
n
SILU P. 4 A 2 C(H ).
6. sLEDUET NEPOSREDSTWENNO IZ OPREDELENIQ. nAPRIMER, IZ USLOWIJ
A 2 C(H); B |
2 B(H ) SLEDUET, ^TO DLQ KAVDOJ OGRANI^ENNOJ POSLEDO- |
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WATELXNOSTI (fn) W H POSLEDOWATELXNOSTX (Bfn) TAKVE OGRANI^ENA. w |
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SILU 244.2 POSLEDOWATELXNOSTX (ABfn) OBLADAET SHODQ]EJSQ PODPOSLEDO- |
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WATELXNOSTX@ (ABfnk ). sNOWA W SILU 244.2 AB |
2 C |
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7. pUSTX (I , A)fn ! g. mOVNO S^ITATX, ^TO fn 2 [Ker(I , A)]? . fiZ |
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PREDSTAWLENIQ (232.4) fn = f0 |
+ f00 |
(f0 |
Ker(I |
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A); f00 |
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[Ker(I |
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A)]?) |
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(fn ) |
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POLU^AEM |
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oSTAETSQ POMENQTX POSLEDOWATELXNOSTX |
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POSLEDOWATELXNOSTX (f00). |
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pOKAVEM TEPERX, ^TO POSLEDOWATELXNOSTX (fn ) OGRANI^ENA. eSLI (fn ), NAPROTIW, NE OGRANI^ENA, TO, PEREHODQ K PODPOSLEDOWATELXNOSTI, MOVNO S^ITATX, ^TO kfnk ! +1 I TOGDA (IZ SHODIMOSTI (I , A)fn ) SLEDUET,
^TO (I , A)kn ! , GDE kn kfnk (!!). pOSKOLXKU A | KOMPAKTNYJ fn
OPERATOR, MOVNO S^ITATX (PEREHODQ SNOWA K PODPOSLEDOWATELXNOSTI), ^TO
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Akn SHODITSQ. pO\TOMU SHODITSQ I POSLEDOWATELXNOSTX kn = (I , A)kn + |
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Akn. pUSTX kn ! h. tOGDA h 2 [Ker(I ,A)]? I khk = 1. s DRUGOJ STORONY, |
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A)kn |
= , I ZNA^IT, h |
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Ker(I |
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A). pO\TOMU h |
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[Ker(I,A)] |
[Ker(I,A)]? ) h = , ^TO PROTIWORE^IT RAWENSTWU khk = 1. |
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pUSTX CT> 0 TAKOWO, ^TO kfnk C (n |
2 N). tAK KAK A | KOMPAKTNYJ |
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OPERATOR, SU]ESTWUET PODPOSLEDOWATELXNOSTX (fnk ) TAKAQ, ^TO (Afnk ) SHO- |
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DITSQ, A ZNA^IT, SU]ESTWUET f |
lim fnk |
(= lim[(I |
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A)fnk + Afnk ]), TAK |
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^TO g = lim(I |
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8. u P R A V N E N I E. pUSTX A 2 C(H ) I P | ORTOPROEKTOR. pOKAVITE, ^TO R((I + A)P ) ZAMKNUTO.
x246. iNTEGRALXNYE KOMPAKTNYE OPERATORY
w TEORII LINEJNYH INTEGRALXNYH URAWNENIJ KL@^EWU@ ROLX IGRA@T INTEGRALXNYE OPERATORY T WIDA
( ) |
(T f )(t) = Z K(t; s)f(s) (ds); |
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GDE FUNKCIQ K(t; s) NAZYWAETSQ QDROM OPERATORA T , OPREDEL•ENNOGO NA PODHODQ]EM PROSTRANSTWE FUNKCIJ f , KOTORYE W SWO@ O^EREDX ZADANY NA NEKOTOROM PROSTRANSTWE S MEROJ (M; ).
1. pUSTX SNA^ALA |
K(t; s) | NEPRERYWNAQ FUNKCIQ NA KWADRATE |
0 t; s 1. tOGDA T |
KORREKTNO OPREDEL•EN NA PROSTRANSTWE NEPRERYW- |
NYH FUNKCIJ C[0; 1]. pRI \TOM T | OGRANI^ENNYJ LINEJNYJ OPERATOR. (w \TOM SLU^AE M = [0; 1]; | LINEJNAQ MERA lEBEGA.) dEJSTWITELXNO, OGRANI^ENNOSTX T SLEDUET IZ OCENKI
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(T f )(t) |
max |
K(t; s) |
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[0; 1]): |
(zDESX kfk = max jf (t)j | IZWESTNAQ NORMA W C[0; 1].)
0 t 1
2.w USLOWIQH P. 1 T | KOMPAKTNYJ OPERATOR.
w SILU 219.10 DOSTATO^NO UBEDITXSQ, ^TO T B1[ ] | RAWNOSTEPENNO NE- PRERYWNOE SEMEJSTWO FUNKCIJ (SM. 219.9). tAK KAK K RAWNOMERNO NEPRE- RYWNA NA KWADRATE M M,
423
8" > 0 9 > 0 8t; s; t0; s0 2 M (k(t; s) , (t0; s0)k < ) |
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w ^ASTNOSTI, |
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jK (t; s) , K(t0; s0)j < "). |
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8" > 0 9 > 0 8t; t0; s 2 M (jt , t0j < ) jK(t; s) , K(t0; s)j < "): |
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sLEDOWATELXNO, |
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8" > 0 9 > 0 8f |
2 B1 [ ] 8t; t0 2 M (jt , t0j < ) |
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K (t; s) |
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^TO I TREBOWALOSX. >
mY PEREJDEM• TEPERX K USLOWIQM KOMPAKTNOSTI OPERATORA T W GILX- BERTOWOM PROSTRANSTWE FUNKCIJ L2(M; ). pREDWARITELXNO USTANOWIM LEMMU.
3. pUSTX ffj (t)gj2N; fgk (s)gk2N | ORTONORMIROWANNYE BAZISY W SE- PARABELXNYH GILXBERTOWYH PROSTRANSTWAH L2 (M1; 1) I L2 (M2; 2) SO- OTWETSTWENNO. tOGDA SISTEMA FUNKCIJ ffj (t)gk (s)g QWLQETSQ ORTONOR- MIROWANNYM BAZISOM W L2 (M1 M2; 1 2 ).
dLQ UDOBSTWA MY PROWED•EM DOKAZATELXSTWO PRI PREDPOLOVENII, ^TO MERY 1; 2 KONE^NY. pREVDE WSEGO, ffj (t)gk (s)g | ORTONORMIROWANNAQ SISTEMA W L2 (M1 M2; 1 2). oSTAETSQ• LI[X UBEDITXSQ, ^TO ONA ZAMK- NUTA. pUSTX
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f(t; s)fj (t)gk (s) 1(dt) 2(ds) = 0: |
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M1 M2 |
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pO TEOREME fUBINI 214.2 |
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f (t; s)fj (t)gk(s) 1 (dt) 2 (ds) |
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M1 M2 |
Z f (t; s)fj (t) 1(dt) gk (s) 2(ds) = 0 (j; k 2 N). |
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= Z |
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M2 |
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s U^ETOM ZAMKNUTOSTI SISTEMY |
fgk(s)g |
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(M2; 2) |
SLEDUET |
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^TO DLQ |
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PROIZWOLXNOGO FIKSIROWANNOGO j SU]ESTWUET Sj M2 TAKOE, ^TO |
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62 |
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f (t; s)fj (t) 1(dt) = 0 (s Sj ); 2(Sj ) = 0: |
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M1 |
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424
pOLAGAQ S S Sj , POLU^AEM OTS@DA
j
Z |
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62 |
2 |
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f (t; s)fj (t) 1 |
(dt) = 0 (s S; j |
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N); 2 |
(S) = 0: |
M1 |
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tAK KAK SISTEMA ffj (t)g ZAMKNUTA, s 62S ) f (t; s) = 0 P. W. OTNOSITELXNO1 . pUSTX A = f(t; s) 2 M1 M2jf (t; s) 6= 0g. wOSPOLXZUEMSQ TEOREMOJ fUBINI W FORME 214.5. tAK KAK
At fs 2 M2 j (t; s) 2 Ag = fs 2 M2 j f (t; s) 6= 0g S
P. W. OTNOSITELXNO 1, IMEEM
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1 2 (A) = Z |
Z 2(At) 1(dt) 2 (ds) = 0; |
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M2 |
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OTKUDA f(t; s) = 0 P. W. OTNOSITELXNO 1 2: |
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4. |
pUSTX L2(M; ) | SEPARABELXNOE GILXBERTOWO PROSTRANSTWO, |
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K 2 |
L2 (M M; ). tOGDA W USLOWIQH P. 1 T | KOMPAKTNYJ OPE- |
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RATOR. |
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sLEDUET PROWERITX SLEDU@]IE TRI FAKTA:
(1)f 2 L2(M; ) ) T f 2 L2(M; ) (KORREKTNOSTX OPREDELENIQ T),
(2)kT fk Ckfk (f 2 L2 (M; )) (OGRANI^ENNOSTX T),
(3)SU]ESTWUET POSLEDOWATELXNOSTX Tn KONE^NOMERNYH OPERATOROW W L2 (M; ) TAKAQ, ^TO Tn ! T PO NORME (KOMPAKTNOSTX T).
uTWERVDENIE (1) SLEDUET IZ OCENKI
Z |
j(T f )(t)j2 (dt) = Z |
jZ K(t; s)f (s) (ds)j2 (dt) |
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M M |
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Z |
Z jK (t; s)j2 (ds) Z jf (s)j2 (ds) (dt) |
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= Z Z jK(t; s)j2 (ds) (dt) kfk2 < +1: |
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M M |
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425
oTS@DA VE SLEDUET (2) S |
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C = kKk = [Z Z |
jK (t; s)j2 (ds) (dt)]1=2: |
M M |
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dLQ POSTROENIQ POSLEDOWATELXNOSTI Tn RASSMOTRIM ORTONORMIROWANNYJ BAZIS ffj(t)g W L2(M; ). w SILU P. 3 ffj (t)fk (s)g | ORTONORMIROWANNYJ BAZIS W L2 (M M; ). pO\TOMU
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K(t; s) = |
jkfj (t)fk (s); |
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K OTNOSITELXNO |
BAZISA |
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I RQD SHODITSQ PO NORME |
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(Tnf )(t) Z Kn (t; s)f(s) (ds) |
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KONE^NOMERNYJ OPERATOR T |
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= j;k=1 jkh ; fkifj ). |
pRI \TOM SM |
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PUNKT (2) NASTOQ]EGO DOKAZATELXSTWA) |
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^TO I TREBOWALOSX. >
5. u P R A V N E N I E. pOKAVITE, ^TO W USLOWIQH P. 4 (T f )(t) =
Z K (s; t)f (s) (ds) (f 2 L2(M; )).
M
426
|lementy teorii neograni~ennyh operatorow
x247. pONQTIE ZAMKNUTOGO OPERATORA
1. lINEJNYM OPERATOROM (W DALXNEJ[EM PROSTO OPERATOROM) T W GILXBERTOWOM PROSTRANSTWE H NAZYWAETSQ LINEJNOE OTOBRAVENIE T : D(T ) ! H , GDE D(T) | LINEAL W H (ON NAZYWAETSQ OBLASTX@ OPRE- DELENIQ T). oTMETIM, ^TO DLQ L@BOGO LINEJNOGO OPERATORA T = . oPE- RATOR T NAZYWAETSQ PLOTNO ZADANNYM, ESLI LINEAL D(T ) PLOTEN W H. lINEALY Ker T ff 2 D(T)j T f = g I R(T ) fT fj f 2 D(T )g NAZYWA- @TSQ SOOTWETSTWENNO QDROM I OBRAZOM OPERATORA T .
p R I M E R Y. 2. w GILXBERTOWOM PROSTRANSTWE H = L2 [0; 1] OPREDELIM OPERATOR M : (Mf)(t) tf(t) (0 t 1); M OPREDELEN• WS@DU W H I OGRANI^EN.
3. w GILXBERTOWOM PROSTRANSTWE H = L2(R) SNOWA POLOVIM (Mf )(t) tf(t) (t 2 R), GDE D(M) = ff 2 L2(R)j tf(t) 2 L2 (R)g; M PLOTNO ZADAN, NO
2 D(M ), kfnk = 1, NO kMfnk2 = Zn
n2 (n 2 N). oPERATORY W PRIMERAH 2,3 NAZYWA@TSQ OPERATORAMI UMNOVE- NIQ NA NEZAWISIMU@ PEREMENNU@.
u P R A V N E N I Q. 4. pOKAVITE, ^TO T (fn) (nfn ) | NEOGRANI^ENNYJ PLOTNO ZADANNYJ LINEJNYJ OPERATOR W `2 .
5.w GILXBERTOWOM PROSTRANSTWE L2 (R) POLOVIM (T f )(t) = f0(t) (f 2 D(T ) D), GDE PROSTRANSTWO D OPREDELENO W 170.1. uBEDITESX, ^TO T | NEOGRANI^ENNYJ PLOTNO ZADANNYJ LINEJNYJ OPERATOR.
6.aLGEBRAI^ESKIE OPERACII NAD LINEJNYMI OPERATORAMI W GILXBER- TOWOM PROSTRANSTWE H OPREDELQ@TSQ SOGLA[ENIQMI:
D(A + B) |
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D(A) \ D(B); |
(A + B)f |
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D( A) |
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( A)f |
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D(AB) ff 2 D(B )j Bf 2 D(A)g; |
(AB)f A(Bf ): |
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427
eSLI Ker A = f g, TO OPREDELEN• OPERATOR A,1, OBRATNYJ K A : D(A,1)
R(A); A,1(Af ) f . pRI \TOM R(A,1) = D(A) I AA,1 = iR(A); A,1A = iD(A) (SM. 1.2). lINEJNYJ OPERATOR A NAZOW•EM OBRATIMYM, ESLI OPERATOR
A,1 OPREDELEN• WS@DU I OGRANI^EN.
7. gRAFIKOM LINEJNOGO OPERATORA T W GILXBERTOWOM PROSTRANSTWE
H NAZYWAETSQ MNOVESTWO ,(T) fff; T fgj f 2 D(T )g( H H ) | POD- MNOVESTWO ORTOGONALXNOJ SUMMY GILXBERTOWYH PROSTRANSTW (SM. 233.2).
8. z A M E ^ A N I E. mNOVESTWO , H H QWLQETSQ GRAFIKOM NEKOTOROGO OPERATORA W H TTOGDA , | LINEAL W H H, NE SODERVA]IJ PAR WIDA f ; gg; g 6= (!!).
9. lINEJNYJ OPERATOR S : D(S) ! H NAZYWAETSQ RAS[IRENIEM OPE- RATORA T (PI[EM T S), ESLI D(T) D(S) I T f = Sf (f 2 D(T )). oTMETIM, ^TO T S TTOGDA ,(T ) ,(S).
10. lINEJNYJ OPERATOR T W GILXBERTOWOM PROSTRANSTWE H NAZYWAETSQ ZAMKNUTYM, ESLI ,(T) ZAMKNUTO W H H (T. E. ,(T) | PODPROSTRANSTWO GILXBERTOWA PROSTRANSTWA H H); OPERATOR T NAZYWAETSQ ZAMYKAEMYM, ESLI ON OBLADAET ZAMKNUTYM RAS[IRENIEM.
11. z A M E ^ A N I E. kLASS ZAMYKAEMYH OPERATOROW WKL@^AET W SE-
BQ KLASS OGRANI^ENNYH LINEJNYH OPERATOROW. fuBEDIMSQ, ^TO DLQ OGRA- |
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NI^ENNOGO LINEJNOGO OPERATORA T EGO PRODOLVENIE PO NEPRERYWNOSTI |
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S (x227) QWLQETSQ ZAMKNUTYM OPERATOROM. pUSTX ffn; Sfng ! ff; hg W |
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H H. |
tOGDA |
fn ! f; Sfn ! h, |
I W SILU KONSTRUKCII OPERATORA |
SM |
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x227) f |
2 D(S). iZ NEPRERYWNOSTI S OTS@DA SLEDUET, ^TO h = Sf, T. E. |
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12. wS@DU OPREDEL•ENNYJ OPERATOR ZAMKNUT TTOGDA ON OGRANI^EN.
dOSTATO^NOSTX USTANOWLENA W PREDYDU]EM PUNKTE. nEOBHODIMOSTX QW- LQETSQ SLEDSTWIEM TEOREMY O ZAMKNUTOM GRAFIKE 231.5. >
~ASTO UDOBNOJ BYWAET \POKOORDINATNAQ" FORMA SWOJSTWA ZAMKNUTOS- TI OPERATORA:
13. (i) oPERATOR T ZAMKNUT TTOGDA |
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fn 2 D(T ); fn ! f; T fn ! g |
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(ii) oPERATOR T ZAMYKAEM TTOGDA |
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fn 2 D(T); fn ! ; T fn ! g |
WLE^ET |
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• g = : |
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428
(i) | PROSTAQ PEREFORMULIROWKA OPREDELENIQ IZ P. 10. iZ ZAMYKAEMOSTI T NEMEDLENNO SLEDUET USLOWIE W (ii). oBRATNO, PUSTX WYPOLNENO USLOWIE W (ii). oPREDELIM OPERATOR S:
D(S) |
ff 2 Hj 9(fn) |
D(T ) 9g 2 H (fn ! f; T fn ! g)g; |
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S OPREDEL•EN KORREKTNO: DEJSTWITELXNO, PUSTX f0 |
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POSLEDOWATELXNOSTX TAKAQ, ^TO f0 |
f; T f0 |
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f0 ) |
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iZ (ii) SLEDUET, ^TO g = g0. dALEE PO POSTROENI@ S |
T; ,(S) = ,(T), , |
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TAK ^TO S ZAMKNUT. |
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14. kAVDYJ ZAMYKAEMYJ OPERATOR T OBLADAET NAIMENX[IM ZAMK- NUTYM RAS[IRENIEM (ONO OBOZNA^AETSQ T I NAZYWAETSQ ZAMYKANIEM OPERATORA T). pRI \TOM ,(T ) = ,(T),.
iSPOLXZUEM KONSTRUKCI@ OPERATORA S IZ PREDYDU]EGO PUNKTA I POLO- VIM T S. tOGDA (KAK OTME^ENO WY[E) SPRAWEDLIWO RAWENSTWO ,(T ) = ,(T), . eSLI R | E]E• ODNO ZAMKNUTOE RAS[IRENIE OPERATORA T , TO
,(T) ,(R), A ZNA^IT, ,(T ) = ,(T ), ,(R), = ,(R). sLEDOWATELX-
T:> R, TAK ^TO T | NAIMENX[EE ZAMKNUTOE RAS[IRENIE OPERATORA
15.p R I M E R [NEZAMYKAEMOGO OPERATORA]. oPREDELIM W GILXBERTOWOM PROSTRANSTWE L2[0; 1] OPERATOR T : D(T ) C[0; 1]; (T f )( ) f(1) . rASSMOTRIM POSLEDOWATELXNOSTX NEPRERYWNYH FUNKCIJ fn , SHODQ]U@SQ
W L2[0; 1] K , I W TO VE WREMQ TAKIH, ^TO fn(1) = 1. tOGDA T fn ! , GDE ( ) = (0 1). iZ 13(ii) SLEDUET, ^TO T NE ZAMYKAEM.
u P R A V N E N I Q. pUSTX T | ZAMKNUTYJ OPERATOR W GILXBERTOWOM PROSTRANSTWE H.
16. Ker T | ZAMKNUTOE PODPROSTRANSTWO W H . 17. eSLI OPREDELEN• T ,1 , TO ON TAKVE ZAMKNUT.
18. eSLI A 2 B(H), TO A + T I TA ZAMKNUTY. wERNO LI ANALOGI^NOE UTWERVDENIE, ESLI T | ZAMYKAEMYJ OPERATOR?
429
x248. sOPRQVENNYJ• OPERATOR
1. pUSTX T | PLOTNO ZADANNYJ OPERATOR W GILXBERTOWOM PROSTRAN- STWE H. oPREDELIM SOPRQV•ENNYJ OPERATOR T :
D(T ) |
fg 2 H j 9g 2 H |
8f 2 D(T ) (hT f; gi = hf; g i)g; |
T g |
g (g 2 D(T )): |
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uBEDIMSQ W KORREKTNOSTI DANNOGO OPREDELENIQ. sLEDUET PROWERITX, ^TO (A) \LEMENT g 2 H OPREDEL•EN ODNOZNA^NO, (B) POLU^ENNYJ OPERATOR T LINEEN (!!). pROWERIM (A). pUSTX, NAPROTIW, ESTX E]E• ODIN \LEMENT h TAKOJ, ^TO WYPOLNENO RAWENSTWO
hT f; gi = hf; h i (f 2 D(T)):
wY^ITAQ IZ NEGO PODOBNOE RAWENSTWO DLQ \LEMENTA g , IMEEM:
0 = hf; h i , hf; g i = hf; h , g i (f 2 D(T)):
tAK KAK LINEAL D(T) PLOTEN W H, POLU^AEM, ^TO h = g : >
oTMETIM, ^TO DLQ T 2 B(H) DANNOE OPREDELENIE SOGLASUETSQ S PREV- NIM OPREDELENIEM SOPRQVENNOGO• OPERATORA (239.1).
2. z A M E ^ A N I E. oPERATOR T MOVET I NE BYTX PLOTNO ZADANNYM. rASSMOTRIM W KA^ESTWE ILL@STRACII OPERATOR T IZ PRIMERA 247.15. eSLI
g 2 D(T ), TO DLQ WEKTORA g W P. 1: |
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f(1)Z01 |
g( ) d = hT f; gi = hf; g i = Z01f ( )g ( ) d : |
tAK KAK LINEAL ff 2 C [0; 1] : f (1) = 0g PLOTEN W L2[0; 1], OTS@DA SLEDUET, ^TO g = . pO\TOMU DLQ f1 ( ) 1 (0 1) IMEEM
1 1
h ; gi = Z0 g( ) d = f1(1)Z0 g( ) d = hf1; i = 0:
tAKIM OBRAZOM, LINEAL D(T ) f g?, I ZNA^IT, NE PLOTEN W L2 [0; 1].
3. pOLU^IM WYRAVENIE DLQ GRAFIKA OPERATORA T ^EREZ GRAFIK OPE- RATORA T . oPREDELIM DLQ \TOGO OPERATOR U W PROSTRANSTWE H H RA- WENSTWOM
Uff; gg fg; ,fg (f; g 2 H):
430