Материалы к экзамену по функану (1135187)

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XZeºÒ¹^g_k¹¼iYr]XǼ—¯ ´ÜÎ YSjYrc\peH²i^e_±_Y\^imi]l®³h¹·i]le±Hcrmij®_°Yr^i]le±_e[²i^ieXZ]l^g_kXZ]_±_Y A∗∗ ≡ A ´ÛC²i^iYZf³YZj³YrkimiYÓ*]le_·ip¹gkglq\¿R±glY\]_XǼÂ^iYrc ­ ji¼i^ikienœÌÝ^iYrq\elj®±_Yrk]lkie_n϶]le·ipÊe_nuÌÍjmikiY\nikiecreÏe_²Yr^g_]le_^g A h»f³Y\nXZ]_± ­ Æ Î λYrcr∈e¶mCq;°g_kg»Òe_±e_creC²i^ieXZ]l^g_kXZ]_±gS±¸°glkglÒe_±edÌÝkgºf CϽhY\XZjmËX ­ÊÎ Y\XZ]_± ­ Y\]m”e_cr^³glkimi·YrkuÌÍjmikiY\niki¿Rn³Ïe²iYr^³gl]le^ Rλ = (A − λI)−1 h+k³glqr¿R±glY[*¿RnÅ^iY\qrelj³®_±_Y\ki]len›e_²iY\^gl]_e_^g´Þ kieº¾Y\Xr]l±_e›^Yrc ­ ji¼i^ki¿OҊ]le_·iY\pŽe°_e_q\kgº·g_Yr]_XǼ ρ(A) hÐf³e²ieljkYrkimiY=por]_e[ ­ [*kieº¾Y\XZ]_± ­ σ(A) =kglq\¿R±g_Yr]_XǼ1Xr²iYrp]l^ie[e_²Yr^g_]le_^g ´C \ ρ(A)Õ Yrc\pÊed±mÊf³Yr]_®h¹·i]le]_e_·ip¹g x ∈ σ(A) YX½jmFA±¿R²ieljkYrkieef³kieHmiq¶X½j³Y½f ­ Æ Î m¹Ò ­ X½je±_min)Óß_´àSárâ]ºg_pimiYV]le_·pimÅkg_qr¿R±glÆ;]=XZe_°XZ]l±Yrkiki¿[*mqrkgº·Yrkim¹¼[*mÅe_²iY\^gl]le^g A(AhXr−e_e_λI)]_±_Yr]6=XZ]l{0}± ­ Æ Î ¯ Y\Y²ef[*kieº¾dYXZ]l±e7XZ²iY\pi]l^g ¯ f³mXrpi^iYr]_ki¿[UXZ²iY\pi]l^ie[Oh±Yrpi]_e_^g¯ hjYr¾g Î miYH±7àSárâ ¯ Xre_°XZ]l±Yrkiki¿[*m%±_Y\pi]le^g_[*mJe_²Yr^g_]le_^g A ´)ãBj³g_X\XZmi·Y\XZpminJ²i^im[*Y\^ ¯e_²Yr^g_]le_^ ­ [*kieº¾dY\kim¹¼FkgH·imXZjeÓ A : f → af ÓY½f³mik³XZ]l±Yrkikgl¼V]_e_·ip¹gY\creËXZ²Yrpi]l^³gihp¹g_pkiY\]l^ ­ f³kieH²i^ie±_Y\^imi]l® ¯ o\]leXZe°XZ]_±_Yrkkie_Yq\kgº·iY\kimiY a ´³ÛC°e_qrk³gº·iYrkimY_Ó σp (A) ´iÑÚkim[+glkmiY_Óß`¹´íi´kiYXr[*e_]_^¼kg7]le³h*·i]le%or] ­ ·gXZ]l®JXZ²Yrpi]l^³gDkg_qr¿R±glÆ;]Väåf³m³XZpi^iY\]lkienÊæçh*ekg7e·iYrki®7fg»¾Y[*eº¾dY\]›XZeXZ]leº¼i]l®Åmiq7pÊe_ki]_mik ­Ê­ [+gÂ]le_·Yrp—´ ¬ ^im[*Y\^—Óб l2(C) e_²Yr^g_]le_^ÉjY\±_ecre›Xèf³±micZgm[*Y\Yr]Ëf³mXZp^iYr]lk¿Rn=XZ²Yrpi]l^Ó³±1p¹gº·iY\Xr]l±_Y(x1 , x2 , ...) → (x2 , ...)XZe°Xr]l±_Y\kiki¿OÒ8±Yrpi]le^ie_±¶±¿XZ] ­ ²g_Æ;] (1, λ, λ2 , ...) ÌÍ]_σ^iYrp°=e_±{λg_kimi∈Y C|λ|: <|λ|1 ≤±e_qr1}kmip¹glYr]Smiq;]le ¯cre³hÊ·i]leHor]lmV±_Y\pi]le^g¶felji¾ki¿ŽjYZ¾Hgl]l®± l2 é f³e²ielj³k¹¼iÆ Î miYfedqg_[*pik ­ ]lecreH[*kieº¾dYXZ]l±g·imXZj³g = 1 ]leº¾dY1±_¿Rk ­ ¾f³Yrkkie3jYr¾gl]D±=XZ²Yrpi]l^Y ¯ kieDo\]le ­ ¾Yf³^ ­ c\e_nÅXZj ­ ·gln ¯±_]_e_^ien³|λ|Ͻ´êìë (A − λI) 6= X Ì X ¯ ²i^eXZ]_^glkXr]l±_e³hRkgÉpÊe_]le^ie[Äf³Yrn³XZ]l± ­ Yr]”e_²Yr^g_]le_^³ÏZhef³kg_pÊeêìë − λI) = X ÌÍe_°^glqÃjmikiY\nikiecre%e_²iY\^gl]le^g7±%°_YXZpÊe_kYr·ikie[*Yr^ike[¨²i^ieXZ]l^³glkXr]l±_YkiYC(Ae_°º¼iq\g_kH°_¿R]l®¸qg_[*pik ­ ]_¿[Ͻ´ ¬ elj ­ ·iY\kikgl¼·gXZ]l®XZ²iY\pi]l^g8kglq\¿R±g_Yr]_XǼHkiYr²^iYr^i¿R±ki¿[XZ²Yrpi]l^e[ze_²iY\^gl]le^g”ÌÍmikiecÇfg%±Je_²i^iYZf³YZj³YrkimiY3kYr²i^iY\^i¿R±_ke_cre%±_pijÆ;·g_Æ;]JmÉeXr]ºg_]le_· ¯ki¿RnœXZ²iY\pi]l^—hRXr[O´íϽ´;ÛC°_eqrkgº·YrkimiYÓ σc (A) ´ ¬ ^m[*Yr^—ÓOe²iYr^g_]le_^ ­ [*keº¾dYrkm¹¼± l2 k³ge_c\^glkmi·iYrkk ­ Ɗ²ieXZjY½f³e±g_]lYZj®kieXr]l® λn ∈ l∞ ´ Õ Y\crpÊe¶²i^ie±_Y\^imi]l®¶]»glpi¾dYm]le³h_·i]_e¸pi^ie ¯[*Y λn ± σpA °_e_j®_ÈdYkimi·iY\cre7kYËjYr¾Hm]´îdgljYrYh)±7eXZ]»g»j®ki¿OÒÂX½j ­ ·gl¼¹Ò%^iY\qre_j®_±Yrki]»gXZee_]l±Yr]_Xr]l± ­ Yr]Ve_²iY\^gl]le^ ­ h#e°_^g_]lkie[ ­ p ­ [*kieº¾YrkimÆÄkgV²ieXZjY½f³e±g_]lYZj®(A − νI)−1kieXZ]l® λn − ν h—]le3Y\Xr]l®h)e_·iY\±_mÊf³keh#e_²Yr^g_]le_^ ­D­ [*keº¾dYrkm¹¼%k³g (λn − ν)−1 ´#î8eXr]»gl]le· ¯¯kie˱¿R°_^g_]l®or] ­ X\g_[ ­ ƲieX½jYZf³e_±gl]lYrj®_kieXZ]l®¸äìf³eXr]»gl]le·ikie²¹jeºÒe_nÊæ½hi·i]_e_°_¿¨e°_^g_]lki¿Rne_²Yr^g_]le_^V± lp kiY¸°_¿Oj3°¿uecr^g_kimi·iY\kiki¿[œÌÍkiY8X ­ÊÎ Y\Xr]l±_e±gljϽӹ²ieX½jYZf³e_±gl]lYrj®_keXZ]_® λne_c\^glkmi·iYrk³gF± hXr]»g»je3°¿R]l®hm[*YrY\]7Òe]¼%°¿ïefk ­ ²^iY½f³Yrj®_k ­ Æð]le_·p ­ hpÊe_]le^g»¼—hp¹glpj³YrcrpÊe¸qg_[*Y\C]lmi]l®³h° ­ f³Yr]8jYZ¾Hgl]l®dm[*Yrkike8±8kYr²i^iY\^i¿R±_ke[›Xr²iYrpi]_^iYÌÝY\XZjmÃν ]le_j®_pÊe¸kYe_p¹gl¾dYr]XǼJqglef³ke7±=Xr²iYrpi]_^iYËfmXZpi^Yr]lkie[Ͻ´ ÜÎ Y1ef³mk›²i^im[*Y\^ ¯ ± L2[a, b] e_²iY\^gl]le^e_°ljgºfg_Yr]kiYr²i^Yr^i¿R±ki¿[ÉXZ²iY\pi]l^ie[Oh³XZe±_²g»fglÆ Î m[”XZe±XZY[Åe]l^iYrqA : f (x) → xf (x)pÊe[Ohik³gHpe]le_^e[²i^ieXr]l^g_kXZ]l±eËXZ]l^e_mi]_XǼ—Ó σcA = [a, b] ÌÝ·g_Xr]leËmXr²ielj®q ­ Yr]_XǼ—h¹p¹XZ]»g_]lm—¯ h±›qgºfgº·glÒ²ieXZ]l^e_Yrkm¹¼Še_²iY\^gl]le^g›X3qgºfg_kiki¿[ÙXZ²Yrpi]l^e[Ͻ´Rµ¶g_pekiYrÔ)h±_eqr±_^³g Î gl¼XZ®pŽXZj ­ ·glÆ/ß_h ­ ²^imi±_YZf³Yrkkie_c\e›]»g_[Äe²iYr^g_]le_^³gJjYr±e_c\e›Xèf³±_micZg=äñe°_ef³epÊæFYZf³mikim·ikie_c\epi^ ­ cZg%p¹glpÉ^g_q3jYZ¾mi]±ÂkYr²i^iY\^i¿R±_ke[ïXZ²iY\pi]l^iY”Ìòf³Y\nXZ]l±mi]lYrj®_kie³h²iY\^iYr±ef³mi] (x1 , x2 , ...) → (x2 − λx1 , x3 − λx2 , ...) = (y1 , y2, ...) mJp¹glp%kiYr]l^ (A­ fki−e7λI)²i^ie±_Y\^imi]l®³¯ h∞PÌÍ]º´ p—´ ­ p¹g_q\glkkg»¼7X ­ [+[+gÃm[*YrY\]1·gXZ]lm·iki¿RY = −x1 + λ−n xn → −x1x1 = −λ−n yn²i^im |λ| ≤n=11ÏZ´ Ü X½jm |λ| = 1 hÊ]le λ ∈/ σp h¹ef³k³glpÊe ­ p¹glq\g_kiki¿Rn1e_²iY\^gl]le^ËkiYSkiY\²i^iYr^¿R±_Yrk1mfgl¾dY8kY8±_Y\q½f³YX ­ÊÎ YXZ]l± ­ Y\]iÓifeXZ]»g_]le_·ikeñ_¿R°^gl]_® yn ∈ l2 \ l1 6= ∅ ¯ k ­ Òe_]l® yn = 1 hn·i]le°_¿ukgkiY\[e_°^gl]lk¿Rn1e²iYr^g_]le_^V^g_qre_ÈdYZj³XǼ—´µ¶glpÊekiYrԗh²ieX½jYZf³kimin3X½j ­ ·³gln ¯ êìë (A − λI) 6= X h¹²i^imi·Y\[m1qg_[*¿Rp¹glkimY êìë (A − λI) 6=´óÐ]»g·g_Xr]l®XZ²iY\pi]l^gHkg_qr¿R±glYr]XǼËeXZ]»g_]le_·ike_n—h¹²i^m[*Yr^F]le·ipimVeXZ]»g_]le_·ike_creHXZ²Yrpi]l^³gXe¯ ²iYr^g_]le_^J²i^³gl±_ecreFXèf³±_mcZg1±3²^ieXr]l^glk³XZ]l±Y lp, 1 ≤ p ≤ ∞ : (a1 , ...

, an, ... ) →´î¸ji¼DjÆ;°e_creh#p¹glpD[*eº¾kie ­ °_Y½fmi]l®XǼ—hàSárâ − λI) = {0} h(0, a1 , ... an , ... )Î Y\Xr]l± ­ Yr]ºh—]º´ Y´ 0ef³kg_pÊehe_·iY\±_mÊfkieh êìë (A) 6= X mÂλ^i∈Y\qre_Cj®_±Yrki]»gV±3]le_·pY λ = 0 kiYËX ­(A²i^imkgºfj³YZ¾mi]HkiYr²^iYr^i¿R±kie[ ­ XZ²iY\pi]l^ ­ A ´ÛC°e_qrk³gº·iYrkimY_Ó σr (A) ´ÚÑ e_e° Î YCcre±_e_^¼—h ­ XZje_±miYHÌñßÏÚ[*eº¾Yr]±_¿R²ie_jk¹¼i]l®XǼÃef³kie±_^iY[*YrkikieHXHÌô`ÏÚm¹jmJÌÝíÏZhief³kg_pÊe λ ±Hor]_e[X½j ­ ·³glY¶±XrYrcÇfge]lkieXǼi]Hp1f³mXrpi^iY\]lkie[ ­ XZ²Yrpi]l^ ­ ÌÝmFXZjY½fe_±g_]lYZj³®_kieH±XrY¶]l^im3²ief[*kieº¾dYXZ]l±gdXr²iYrp]l^gFe²iYr^³gl]le^gòe_²g_^ikieVkiYr²iY\^iY\XrYrp¹glÆC]_XǼ³Ï½´—Ñ pÊe_kiY\·ikie[*Y\^ikie[²i^ieXZ]l^³glkXr]l±_Y3Ìô`ÏCm¹jmŠÌÝíÏB±XZY\cÇfg ¯±lj³Yrp ­ ]ÃÌõßϽhʲieor]le[ ­ ±8]»glpimiÒ˲i^ieXr]l^g_kXZ]l±g»ÒHe_²iY\^gl]_e_^Ëe°lj³g»fglY\]¶]lelj®pÊe¶f³mXrpi^iY\]lki¿[›Xr²iYrpi]_^ie[Oh²i^im·iY\[ïXZeXZ]leº¼ Î m[Umiq3pÊe_kYr·ikiecre=·imX½jg%Xre_°XZ]l±Yrkiki¿Oқq\kgº·iY\kimin—´ÑÄÔYZje[U¾dY λ ¯ ]_e_·ip¹g7f³mX ¯pi^iY\]lkiecre3XZ²iY\pi]l^gh#Y\XZjm A − λI kiY\mikiö—Y\pi]lmi±YrkÉÌ÷äèXZp¹jY\mi±g_Yr]ºækiYrpÊe_]_e_^i¿RY]le_·pim³Ï½h)kiYr²i^Yr^i¿R±kie_c\e−1 kiYrecr^g_kimi·iY\kuÌôX ­ÊÎ YXZ]l± ­ Yr]=kiYFkg%±XrY\[²^ieXr]l^glk³XZ]l±YϽh+m”eXr]»gl]le·ikie_c\ehY\XZjm¯ Y\XZjm (A−1−kλI)rYe³fkieqrkgº·Yrk—´(A − λI)ø glq\e_±_¿RY*]lY\e_^iY[*¿Je;XZ²iY\pi]l^iYвi^ieȸjecreXZY[*Y\Xr]l^giÓ\^iYrq\elj®±_Y\ki]lkieY[*kieº¾dYXZ]l±eOjÆ;°e_creRe²iYr^³gl]le ¯^g;e_]lpi^¿R]lehºgR²ie]le[ ­ XZ²Yrpi]l^8jÆ;°e_c\eRe_²iY\^gl]le^gRqg_[*pik ­ ]º´»ÑDc\m¹j®°_Yr^]le_±e[V²i^ieXZ]l^g_kXZ]_±_YÖXr²iYrpi]_^kiY\²i^iYr^¿R±_kiecre jmkiYrnike_cree_²Yr^g_]le_^g¸kiYr² ­ XZ]º´ùвiYrp]l^Ãj³mikiYrnkie_c\e8e_²Yr^g_]le_^g A ÔiYrjmipÊe[ÂjYZ¾mi]d±pi^ ­ c\Y¶^gºf³m ­ Xrg kAk himÂÌÍ°eljY\YSXrm¹j®kie_Y ­ ]l±_Y\^¾dfYrkimiYºÏ½hi±Hpi^ ­ c\Y¶^gºfm ­ X\g ρ(A) = limn→∞ pkAnk h²i^im·iY\[Šo\]»gËe_ÔYrkip¹gÃXr]l^iecZg»¼—hgËY\XZjm7e²iYr^g_]le_^DkiYr²i^Yr^i¿R±Yrk ¯ ]le1Y Î YmFf³eXZ]lmcZglÆ Î g»¼XǼ›Ì ρ(A) =kg_qr¿R±glÆ;]ÂXr²iYrpi]_^g»j®ki¿[Ù^g»f³m ­ Xre[ze_²iY\^gl]_e_^g AÏZ´Ðú*Yrq\elj®±_Y\ki]»g=kYr²i^iY ¯sup kxk : x ∈ σ(A)`n^i¿R±kie_c\eÃjmkiYrnike_creVe_²Yr^g_]le_^g1Y\Xr]l®FkiYr²^iYr^i¿R±kg»¼%m=c\elje[*e_^ÀSk³g»¼ÌݱV]_e_²ie_je_c\mimD^g_±_kie[*Yr^ike_ne_²Yr^g_]le_^ike_nFXÇÒef³m[*eXZ]lmÏe²iYr^g_]le_^ ¯ q\kgº·ikgl¼1À ­ kipiÔim¹¼—´ûO’ü=ý+£O§ x ¥3•#þzÿÄ£ÐÿS™C˜¥ ªBŸ;¢C•;¢C˜«³˜Ä¢C•—¡;™C•—™;šÅý¢;˜«˜Ä˜¡C•—™ t*w ˜™ t ý«³Ÿ;ª x •)™ w ˜ý˜¥ ¡;™C˜ vw ™ t ¢ vw ý•uŸ×˜™ t §—˜Ö¥ •—«˜ v ˜¡;™Ú£Oþ •—¢C¢;˜«˜ ]»glp—hº² ­ XZ]l®Bq\g»fglkd²i^iemiqr±elj®ki¿Rn¸jmkiYrnik¿RndkiY\²i^iY\^i¿R±_ki¿Rne_²Yr^g_]le_^ ±Bcrmij®_°Yr^i]le±_e[3²i^ieXZ]_^glkXr]l±_Y H ´ú+g_X\Xr[*e]l^im[ A∗ ´ ¬S­ XZ]_® x ∈ Ker(A), y ∈ Im(A∗ ) h]le_cÇfgF²A^¼[*e=miqËe²i^iY½fYZjY\kim¹¼ ¯jY\crpÊed²elj ­ ·imi]_®h¹·i]le (x, y) = (x, A∗ y0) = (Ax, y0 ) = (0, y0) = 0 h]_eY\XZ]_® KerA ⊥ ImA∗ ´ ¬ e_p¹gl¾dY\[°_e_jYrY¶Xrm¹j®_ke_Y ­ ]l±Yr^¾f³YrkimY_Ó KerA = (ImA)⊥ h³]º´ Y_´³jÆ;°e_n3±Yrpi]le^—he_^i]lecrekg»j®kie_n ImA∗ jYr¾Hm]± KerA ´î8YrnXr]l±_m]lYZj®kieh² ­ XZ]l® y ∈ (ImA∗ )⊥ h#]lecÇfg ∀x ∈ ImA∗ m³[*YrY\[ (x, y) = 0 = (A∗ x0, y) =´ Õ Y\crpÊe˱_m¹f³Yr]l®³h·i]leÃ]_eº¾dfY\XZ]_±_e (x0 , Ay) = 0 ±¿R²ieljkYrkieËfji¼3jÆC°_e_c\e x0 he]lp ­ fgòi^¼[*e(x0 , Ay)X½j³Y½f ­ Yr] Ay = 0 ⇒ y ∈ KerA ´ ]»g_p KerAh+m”p¹glp”X½jYZfXZ]l±miY ImA∗ = (KerA)⊥ h= (ImA∗ )⊥h´⊥KerA∗ = (ImA) ImA = (KerA)⊥O’ t ¥3–;¢ w ˜ vwx ˜™ t § t ˜¡C•—™ t*w ˜™ t £Rý+ªC£Éžz•—«³˜ v £ v R¥3¥3˜ •ÿ¶Ÿ;¢CŸ;¢C˜«˜Ÿ –Ú˜Ö¥3¡ t – w ¢;˜Ö«³˜×µ¶gl²e[*kim[Oh_·i]le¶e_²Yr^g_]le_^ kg_qr¿R±g_Yr]_XǼpÊe[*²g_pi]lki¿[OhY\X½j³mdj³Æ;°_eYRe_c\^glkim·iYrkike_YR[*kieº¾dYXZ]l±ee_kd²iYr^Yr±_efmi];±C²i^iY½fpe[*²glp]lkiAe_Yhºm¹jm—h»·i]_eC]leB¾YÚX\g_[*e_YhZf³ji¼¸jÆC°_e_nde_c\^glkim·iYrkike_n {xn }k³glnÊf³Y\]_XǼ]»glp¹g»¼V²ief³²ieX½jYZf³e_±gl]lYrj®_keXZ]_® xn hi·i]le Axn XÇÒefmi]_XǼ—´Õ Y\[+[+g´ ¬S­ XZ]_®8e_²Yr^g_]le_^pÊe[*²g_pi]lki¿Rn—h¹e²iYr^³gl]le^h Y½f³mkimi·iki¿Rn)´—e_cÇfg ImAq\g[*pik ­ ]º´³î8YrnXr]l±_mi]_YZj®kieh² ­ KXr]l® ¯ yn ∈ ImA h yn → y ´)ù ­ÊÎ AY\Xr=]l± I­ −ÆC] KxnI :¯ Axn = yn = xn − Ax´¬ ^ieYrÔimi^ ­ ¼%ÌݲieHkiYre°ºÒef³m³[*eXZ]_m³Ï x kg KerA h±XrYrcÇfg[*eº¾kieH±_¿R°^gl]l® x Y Î YSmFe_^i]_e_crekg»j³®_ki¿n[àSárâ A Ì xn = x1 + x2 h x1 ∈ KerA h nx2 ∈ (KerA)⊥ ⇒ Axn = Ax1 + Axn2 = Ax2 h—]º´ Y_´e_] x1 ±X^g_±_kiekimi·iY\creHkinYSq\g_±_nmXrmin]m1e]²elj³g_cZnglkim¹¼1Yrcrek ­ jY[Åkim·iYrc\ekYBmnq\[*Yrkmi]_nXǼ³Ï½´ ¬ ^inY½f³²eljeº¾m[Onh¹·i]_ekiYrecr^g_kimi·iY\kgiÓº]lecÇfgBkglnÊfYr]_XǼ^g_XÇÒef¼ Î gl¼XǼS²ef³²ieXZjY½fe_±g_]lYZj³®_kieXZ]l® n } : kxn k → ∞{xn }m A x = 1 Axn → 0 ]º´ p)´ Axn → y h kxn k → ∞ m=±Ãfglj®_kiY\niÈ8Y[{x[*¿ ^³g_XrX\[+gl]l^mi±g_Y\[kx kkx k±[*YXZ]le {xn} ecr^g_kimi·iY\kik ­ Æ x ´+îdg»jY\Y_hmXr²ielj³®_q ­ ¼pÊe[*²³glpi]lkeXZ]_®Âe_²iY\^gl]le^g K hÖXrkie_±gkx k[*eº¾kie›±¿R°_^g_]l®Â²ief³²ieX½jYZf³e_±gl]lYrj®_keXZ]_®h+]»glp ­ ÆdhÖ·i]le ‚Kx ‚ XÇÒef³mi]_XǼ—´ )e_cÇfgÅmiqDXÇÒef³m[*eXr]lm‚‚‚x‚­­i²_eji·mO[»hi·_]Bem°³f\YC]ÇXÒefiml]®ÇXüÌϽ´ ¬S­ XZ]l®SXZee_]l±Yr]_Xr]l± ­ Æ Î min8²^iY½f³YrjAxnxnxn = Axn +Kxn­—Óei·rY±Êm³fik3e·l]e)hFg7±ZX¹mjik\Yi²i^rY^R¿_±ikeZXl]mh´Ûf³kg_pe xn ∈ (KerA)⊥ h¯ x0kx0 k = 1A Ax0 = 0X ]len¾dYkYr²i^iY\^i¿R±_keXZ]_m A qg_[*pik ­ ]le8m˲ie_]_e[ ­ x0 ∈ (KerA)⊥ ¯ ²i^ie]lmi±_e^iYr·miY⊥ ±dXZm¹j ­ ±(KerA)´ ¬ e_or]_e[ ­ ±XZYrcÇfgDkg_nÊf³Y\]_XǼ²ieX½jYZf³e_±gl]lYrj®_keXZ]_®XF]lY[Oh·i]le x0 6= 0 Ì kx0k = 1Ïm x0 ∈ KerA­_e\c^lgkimi·rYkikO¿DÒF±ZX_e±_epi²keZX_]—mhSà\áâm Axn → y ´ qHpÊe[*²³glpi]lkeXZ]_m K X½j³Y½f ­ Yr]ºh—·i]lexnkg_nÊf³Yr]XǼòief³²ieX½jYZf³e_±gl]lYrj®_keXZ]_® xnxn ]»⊥g_pÊgl¼—hA·]le KxXÇÒef³mi]XǼ]leº¾YXÇÒefmi]_XǼ—´ ¬ elj ­ ·g_Y\[Oh*·i]leJ±=jÆ;°e[XZj ­ ·glYF[*eº¾kieJ±_n¿R°_^³gl]l®%XÇÒef⇒¼ Îd­ xÆBnXǼÂ=²ieAxXZjnY½f³e+±Kxg_]lYZjn ®kieXr]l®]»glp ­ Ædh³·]le n → y ´—ÛC·iY\±_mÊf³keh·i]le Ax0 = y h³²ieor]le[ ­ y ∈ êìë A ´ Þ ¿œfe_p¹glqg»jm—h³·i]lefxjin ¼›→²^ixe_0miq\±_e_j®_kien {yAxêìëêìë h·]le7±7]le·ikieXZ]lm›mÅe_qrk³gº·glY\]h)·i]leDêìë Aq\g[*pik ­ ]le³´ S]lem1]_^iYr°ne_}±gl⊆jeXr®dA²e_p¹:glqgly]ln®³´ → y y ∈ Akkkknknknkkkknknknknkklllklklkkkkk’Ž•—™Cý t £ w •—˜™C•—¥ t ™;•ÿ¶«³˜ª x ¥ t¬S­ Xr]l®H±crm¹j³®_°_Y\^i]le±_e[²i^ieXZ]l^³glkXr]l±_Y¶qgºfg_kie ­ ^³gl±_kYrkimiYhicÇf³YpÊe[*²glpi]_ki¿Rn—´Ñ”·g_Xr]lkieXZ]lm—hʱd]ºg_pÊe[J±_m¹f³Y;qgl²imXr¿R±g_Æ;]_XǼmiki]_Yrcr^³g»j®ki¿RY ­ ^ϕg_±_=kiY\kiKϕm¹¼+8^ifY½fcrelj³K®[+g¶¯ ±]le_^iecre¸^efgRbh#q\g»fglkik¿RY¸fj¼=À ­ kipiÔimin%miq L2(a, b) ´³fY\XZ® K(x, t) ¯ ¼Êf³^ieVmiki]_Y ¯ϕ(x) = K(x, t)ϕ(t)dt + f (x)aícr^³g»j®kie_c\e ­ ^gl±kiYrkimi¼ ¯ ZX m³[+[*Yr]l^im·ikieFmD²i^mikgºf³jYZ¾mi] L2([a, b]2) h]º´ Y´ ­ fe_±lj³Yr]l±e_^¼iYr] ­ X½je±_m¹¼[Rb Rbm´ ¬ e_jeº¾mi± A = I − K h²Yr^iY\²imiÈ8Y[Éor]le ­ ^³gl±_kYrkimiY¶±2|K(x, t)| dxdt < ∞ K(x, t) = K(t, x)±_a m¹af³Y Aϕ = f ´¬ Yr^±g»¼Ë]_Yre_^Y\[+g 8^iY½f³c\elj®[+giÓÊkiY\ef³kie_^ef³kie_Y ­ ^gl±kiYrkimY = f ^glq\^iYrÈdm[*ed]_e_cÇfgdm1]_elj®pe­ Yr]1miqd²^iY½f³¿Úf ­Î m¹ÒFf³± ­ Ò7°m¹jY\]le_±³Ó]lecÇfgihpÊe_cÇfg f ⊥ àSárâ A∗ ´î8e_p¹glqgl]lYrj®Xr]l±_e]_^imi±_m³g»j®kie1XZjY½f AϕXZecçj³g_XrkieF²iYr^±_e[ ­ êìë A = (àSá\â A∗ )⊥ h)XZecçj³g_XrkieF±_]le^ie[ ­ êìë A = êìë A ´ÛC·Yr±_m¹f³kieh)·i]le ­ ^gl±kiY ¯kimiY Aϕ = f ^³glqr^YrÈ8m³[*eJ]_e_cÇfg%mŠ]le_j®_pÊeJ]le_cÇfgihpÊe_cÇfg f ∈ êìë A h+fgljYrYFjecrmi·Y\XZp¹gl¼ÔiY\²ie_·ip¹g]l^im±_mglj®_kgVÓ ¯ ÏZ´O’ w ˜Ö™ t £ w •—˜™C•—¥ t ™;•ÿ¶«³˜+ª x ¥ t ˜ ¢ t þ • t ª x w —• ™C¢ *t w ;Ÿ ý t ™C•ÿS«˜+ª x ¥ tØRj³®»]lYr^ik³gl]lmi±g 8^iYZf³cre_j®[+gÓjm°_e ­ ^³gl±_kYrkimiYÌôXr[O´±_¿RÈdYÏ;m[*YrY\]1²^imDj³Æ;°_e[ef³kieHm3]le_j®_pÊeHef³kie^YrÈ8Y\kimiYh¹jmi°eef³kie_^ef³kie_Y ­ Aϕ^gl±ki=YrkfmiY Aϕ0 = 0 m[*Y\Yr]kiY\k ­ jYr±e_Y¶^iY\È8f Yr∈kmiHY_´¬ ^¼[*eX½jYZf ­ Y\]dmiqÑÚ]le_^e_nŠ]lY\e_^iY[*¿ 8^iY½f³c\elj®[+giÓÐêìëàSárâ´ ¬ ep¹g»¾dY[ï±_kgº·gljY_hÖ·i]leàSárâ A ⊕ êìë A∗ ´ÑÚe ¯ ²iY\^i±_¿OÒ#hlmq²i^iYZf³¿Úf ­ÊÎ AY\cr=e¸°Hm¹jYr⇔]»g¶êìë AA∗ q=g_[*{0}pik ­ ]le³h_qg_[*pik ­ ]leXZ]l®dàSárâ A X½jYZHf ­ Y\=]miqSkiY\²i^iYr^¿R±_kieXZ]lm A ´iµ¶g_pÊe_kiY\ԗh¹XZkie±gdmiq¶²i^iYZf³¿Úf ­ÊÎ m¹Ò1°_m¹jY\]le_±³hiàSárâ A = (êìë A∗)⊥ h¹·i]_e±HX ­ [+[*Ym1fglY\]dk³g_[]l^iYr° ­ Y[*e_Y ­ ]l±_Y\^¾f³Y\kimiY´îdg»j³YrY_h_²ieljeº¾m[ ∈ N H k = êìë Ak ´ÊÛC·iY\±_mÊf³keh H ⊇ H 1 ⊇ H 2 ⊇ ...

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