Конспект лекций по математическому анализу. Шерстнев А.Н. - 424 стр.

   8" > 0 9 > 0 8t; s; t0; s0 2 M (k(t; s) , (t0; s0)k <  )
                                                         jK (t; s) , K (t0; s0)j < ").
 w ^ASTNOSTI,
    8" > 0 9 > 0 8t; t0; s 2 M (jt , t0j <  ) jK (t; s) , K (t0; s)j < "):
sLEDOWATELXNO,
   8" > 0 9 > 0 8f 2 B1[] 8t; t0 2 M (jt , t0j <  )
                   j(Tf )(t) , (Tf )(t0)j  0max
                                              s1
                                                     jK (t; s) , K (t0; s)j kf k < ").
 ^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)gj 2N; 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 PROWEDEM 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        Z
                       f (t; s)fj (t)gk (s)1(dt)2(ds) = 0:
                     M1 M2
pO TEOREME fUBINI 214.2
   Z
       f (t; s)fj (t)gk (s)1(dt)2(ds)
M1 M2
      Z Z                        
    =         f (t; s)fj (t)1(dt) gk (s)2(ds) = 0 (j; k 2 N).
      M2 M1
 s U^ETOM ZAMKNUTOSTI SISTEMY fgk (s)g W L2(M2; 2) SLEDUET, ^TO DLQ
PROIZWOLXNOGO FIKSIROWANNOGO j SU]ESTWUET Sj  M2 TAKOE, ^TO
              Z
                f (t; s)fj (t)1(dt) = 0 (s 62 Sj ); 2(Sj ) = 0:
                M1

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