Лекции по функциональному анализу для начинающих специалистов по математической физике. Арсеньев А.А. - 54 стр.

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МГУ | Москва

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f
n
(x) , g
n
(x)
f
n
(x) → f(x) , g
n
(x) → g(x) , n → ∞.
|f
n
(x) + g
n
(x)|
p
∈ L(X) |f
n
(x) + g
n
(x)|
p
→ |f(x) + g(x)|
p
.
|f(x) + g(x)|
p
≤ 2
p−1
(|f(x)|
p
+ |g(x)|
p
) ∈ L(X),
|f(x) + g(x)|
p
∈ L(X).
∀(f ∈ L
p
(X)) : kf | L
p
(X)k = I(|f|
p
)
1/p
f 7→ kf | L
p
(X)k
L
p
f
f(x) ∈ L
p
(X) , g(x) ∈ L
q
(X) f(x)g(x) ∈ L(X)
I(|fg|) ≤ kf | L
p
(X)k · kg | L
q
(X)k.
f
n
(x) , g
n
(x)
|f
n
(x)g
n
(x)| → |f(x)g(x)| ≤
|f(x)|
p
p
+
|g(x)
q
q
∈ L(X).
|f(x)g(x)| ∈ L(X)
f(x) 7→ f(x)/kf | L
p
(X)k , g(x) 7→ g(x)/kg |
L
q
(X)k kf | L
p
(X)kkg |
L
q
(X)k
f ∈ L
p
(X) , g ∈ L
p
(X),
kf + g | L
p
(X)k ≤ kf | L
p
(X)k + kg | L
p
(X)k.
1.1.12 ñóùåñòâóþò òàêèå ïîñëåäîâàòåëüíîñòè ýëåìåíòàðíûõ ôóíêöèé fn (x) , gn (x),
÷òî
               ï.â. fn (x) → f (x) , gn (x) → g(x) , n → ∞.   (1.89)
Èç 1.1.9 è (1.89) ñëåäóåò, ÷òî

     |fn (x) + gn (x)|p ∈ L(X) è ï.â. |fn (x) + gn (x)|p → |f (x) + g(x)|p .

Íî â ñèëó íåðàâåíñòâà (1.88):

              |f (x) + g(x)|p ≤ 2p−1 (|f (x)|p + |g(x)|p ) ∈ L(X),

ïîýòîìó â ñèëó ëåììû 1.1.13

                           |f (x) + g(x)|p ∈ L(X).

Ëåììà äîêàçàíà.
  Ïîëîæèì ïî îïðåäåëåíèþ

                  ∀ (f ∈ Lp (X)) : kf | Lp (X)k = I(|f |p )1/p             (1.90)

Îïðåäåëåííûé ðàâåíñòâîì (1.90) ôóíêöèîíàë f 7→ kf | Lp (X)k íàçûâà-
åòñÿ Lp -íîðìîé ôóíêöèè f .

Òåîðåìà 1.1.4. Åñëè f (x) ∈ L (X) , g(x) ∈ L (X), òî f (x)g(x) ∈ L(X)
                                   p                q

è ñïðàâåäëèâî íåðàâåíñòâî Ãåëüäåðà

                    I(|f g|) ≤ kf | Lp (X)k · kg | Lq (X)k.                (1.91)

    Äîêàçàòåëüñòâî. Ïóñòü ïîñëåäîâàòåëüíîñòè ýëåìåíòàðíûõ ôóíêöèé
fn (x) , gn (x) óäîâëåòâîðÿþò óñëîâèþ (1.89). Òîãäà

                                           |f (x)|p |g(x)q
                                                          
     ï.â. |fn (x)gn (x)| → |f (x)g(x)| ≤           +         ∈ L(X). (1.92)
                                               p       q

 ñèëó ëåììû 1.1.13 îòñþäà ñëåäóåò, ÷òî |f (x)g(x)| ∈ L(X).  íåðàâåí-
ñòâå (1.92) ñäåëàåì çàìåíó f (x) 7→ f (x)/kf | Lp (X)k , g(x) 7→ g(x)/kg |
Lq (X)k è ïîòîì ïðîèíòåãðèðóåì. Ïîñëå óìíîæåíèÿ íà kf | Lp (X)kkg |
Lq (X)k ïîëó÷èì (1.91). Òåîðåìà äîêàçàíà.

Òåîðåìà 1.1.5. Åñëè f ∈ L (X) , g
                               p
                                            ∈ Lp (X),   òî ñïðàâåäëèâî íåðàâåí-
ñòâî Ìèíêîâñêîãî:

              kf + g | Lp (X)k ≤ kf | Lp (X)k + kg | Lp (X)k.              (1.93)

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