Лекции по функциональному анализу для начинающих специалистов по математической физике. Арсеньев А.А. - 27 стр.

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МГУ | Москва

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{f
n
} ⊂
L
0
(X)
∀x: 0 ≤ f
n+1
(x) ≤ f
n
(x) lim
n→∞
f
n
(x) = 0,
lim
n→∞
I
0
(f
n
) = 0.
f
n
(x) I
0
(f
n
)
lim
n→∞
I
0
(f
n
)
M = sup{f
1
(x) | x ∈ X} , Z = {x | lim
n→∞
f
n
(x) 6= 0}.
mes(Z) = 0
g
n
,
∀(n , x ∈ X) : 0 ≤ g
n
(x) ≤ g
n+1
(x) ,
∀(x ∈ Z) : sup{g
n
(x) | 1 ≤ n < ∞} ≥ 1.
h
n
(x) = f
n
(x) − Mg
n
(x).
h
n
(x)
−∞
{h
n
(x)}
∀(n , x) : f
n
(x) ≤ M,
∀(x ∈ Z) : lim
n→∞
h
n
(x) ≤ lim
n→∞
f
n
(x) − M ≤ 0,
x ∈ C(Z) lim
n→∞
h
n
(x) ≤ −M ≤ 0
∀x : lim
n→∞
h
n
(x) ≤ 0 ,
Ëåììà 1.1.2. Åñëè ïîñëåäîâàòåëüíîñòü ýëåìåíòàðíûõ ôóíêöèé {f } ⊂    n
L0 (X)   óäîâëåòâîðÿåò óñëîâèÿì:

               ∀ x : 0 ≤ fn+1 (x) ≤ fn (x) è ï.â. lim fn (x) = 0,
                                                    n→∞

òî
                                lim I0 (fn ) = 0.
                               n→∞

    Äîêàçàòåëüñòâî. Òàê êàê ïîñëåäîâàòåëüíîñòü ýëåìåíòàðíûõ ôóíêöèé
fn (x) ìîíîòîííî íå âîçðàñòàåò, òî ÷èñëîâàÿ ïîñëåäîâàòåëüíîñòü I0 (fn )
ìîíîòîííî íå âîçðàñòàåò è ïðåäåë limn→∞ I0 (fn ) ñóùåñòâóåò. Íóæíî äî-
êàçàòü, ÷òî ýòîò ïðåäåë ðàâåí íóëþ.
    Ïóñòü

             M = sup{f1 (x) | x ∈ X} , Z = {x | lim fn (x) 6= 0}.
                                                    n→∞

Òàê êàê mes(Z) = 0, òî ñóùåñòâóåò òàêàÿ ïîñëåäîâàòåëüíîñòü ýëåìåí-
òàðíûõ ôóíêöèé gn , ÷òî

                  ∀(n , x ∈ X) : 0 ≤ gn (x) ≤ gn+1
                                                
                                                    (x) ,

è

                  ∀(x ∈ Z) : sup{gn (x) | 1 ≤ n < ∞} ≥ 1.

Ïîëîæèì
                          hn (x) = fn (x) − M gn (x).
Ïîñëåäîâàòåëüíîñòü hn (x) ñîñòîèò èç ýëåìåíòàðíûõ ôóíêöèé è ìîíîòîí-
íî íå âîçðàñòàåò, ïîýòîìó ó íåå â êàæäîé òî÷êå ñóùåñòâóåò ïðåäåë (íî â
íåêîòðûõ òî÷êàõ îí ìîæåò áûòü ðàâåí −∞).
   Òàê êàê ïîñëåäîâàòåëüíîñòü {hn (x)} ìîíîòîííî íå âîçðàñòàåò è

                ∀(n , x) : fn (x) ≤ M,

òî

                ∀(x ∈ Z) : lim hn (x) ≤ lim fn (x) − M ≤ 0,
                            n→∞               n→∞

     Åñëè x ∈ C(Z) òî limn→∞ hn (x) ≤ −M ≤ 0.
     Ïîýòîìó
                           ∀x : lim hn (x) ≤ 0 ,
                                  n→∞


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