Luận văn thạc sĩ phương trình tích phân

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

▲■❊◆P❍❖◆❊ ❈❍❊❯❈❍❖❯❚❍❖❘

P❍×❒◆● ❚❘➐◆❍ ❚➑❈❍ P❍❹◆

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✻


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
▲■❊◆P❍❖◆❊ ❈❍❊❯❈❍❖❯❚❍❖❘

P❍×❒◆● ❚❘➐◆❍ ❚➑❈❍ P❍❹◆

❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷
▲❯❾◆ ❱❿◆ ❚❍❸❈
ữớ ữợ ồ


◆❣✉②➯♥ ✲ ✷✵✶✻


▲í✐ ❝❛♠ ✤♦❛♥
❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣

t❤ü❝ ✈➔ ❦❤ỉ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ✤➲ t➔✐ ❦❤→❝✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♠å✐
sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷đ❝ ❝↔♠ ì♥ ✈➔ ❝→❝ t❤ỉ♥❣
t✐♥ tr➼❝❤ ❞➝♥ tr ữủ ró ỗ ố
♥❣➔② ✳✳✳ t❤→♥❣ ✳✳✳ ♥➠♠ ✷✵✶✻

◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥

▲■❊◆P❍❖◆❊ ❈❍❊❯❈❍❖❯❚❍❖❘

✐


▼ư❝ ❧ư❝
▲í✐ ❝❛♠ ✤♦❛♥
▼ư❝ ❧ư❝
▼ð ✤➛✉
✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ Lp ✳ ✳ ✳
✶✳✷ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ổ rt
ổ ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✷ ❑❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✸ ❑❤æ♥❣ ❣✐❛♥ L2ρ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸ ❚♦→♥ tû ✤è✐ ①ù♥❣ ❤♦➔♥ t♦➔♥ ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✶ P❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✷ ❚♦→♥ tû ❧✐➯♥ ❤ñ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✸ ❚♦→♥ tû ✤è✐ ①ù♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✹ ❚♦→♥ tû ❤♦➔♥ t♦➔♥ ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✺ ❚♦→♥ tû ✤è✐ ①ù♥❣ ❤♦➔♥ t♦➔♥ ❧✐➯♥ tư❝ ✳ ✳

✷ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥

✷✳✶ ❚♦→♥ tû t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷ P❤➙♥ ❧♦↕✐ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✶ ❑❤→✐ ♥✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥
✷✳✷✳✷ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦♠ ✳
✷✳✷✳✸ P❤÷ì♥❣ tr➻♥❤ ❱♦❧t❡rr❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✐✐

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✐
✐✐
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✹

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✾
✶✵
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✶✸
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✷✵
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✷✻

✷✻
✸✵

✸✵
✸✶
✸✶


✷✳✸ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✸✳✶ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ❧♦↕✐ ♠ët
✷✳✸✳✷ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à
Pữỡ tr t ợ ố ự
Pữỡ tr t ợ s
ỵ r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✻✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✻✳✷ ❈→❝ ỵ r ✳ ✳ ✳
✷✳✼ P❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❑➳t ❧✉➟♥

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✐✐✐


▼ð ✤➛✉
◆❤✐➲✉ ✈➜♥ ✤➲ ❝õ❛ t♦→♥ ❤å❝✱ ❝ì ❤å❝✱ ✈➟t ỵ ỳ ữỡ
tr tr õ ữ t ữợ t ỳ ữỡ tr
ồ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❧➔ ♠ët tr♦♥❣
♥❤ú♥❣ ❝ỉ♥❣ ❝ư t♦→♥ ❤å❝ ❤ú✉ ➼❝❤ ✤÷đ❝ ❞ị♥❣ tr t ồ ỵ tt
t ự ử
Pữỡ tr t➼❝❤ ♣❤➙♥ ❤♦➦❝ ♣❤÷ì♥❣ tr➻♥❤ ❋r❡❞❤♦♠ ❧♦↕✐ ♠ët ❧➔ ♣❤÷ì♥❣
tr➻♥❤ ❝â ❞↕♥❣✿
b

f (x) =

K (x, y)φ (y) dy,

a < x < b,

a

tr♦♥❣ ✤â f (x)✱K (x, y) ❧➔ ♥❤ú♥❣ ❤➔♠ trữợ (x) ữ
t õ t ð ❝↔ tr♦♥❣ ✈➔ ♥❣♦➔✐ ❞➜✉ t➼❝❤ ♣❤➙♥ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥
➜② ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❋r❡❞❤♦♠ ❧♦↕✐ ❤❛✐✿
b

φ (x) =

K (x, y)φ (y) dy + f (x) ,

a < x < b.

a

ữợ ừ t ❧➔ ❤ú✉ ❤↕♥ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ➜② ❣å✐ ❧➔ ♣❤÷ì♥❣
tr➻♥❤ ❱♦❧t❡rr❛ ❧♦↕✐ ♠ët ✈➔ ❧♦↕✐ ❤❛✐ t÷ì♥❣ ù♥❣ ❝â ❞↕♥❣✿
x

f (x) =

K (x, y)φ (y) dy,

a < x < b.

a
x

φ (x) =

K (x, y)φ (y) dy + f (x) ,
a

a < x < b.

P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❝ỉ♥❣ ❝ư t♦→♥ ❤å❝ ❤ú✉ ➼❝❤
♥❤➜t ✤÷đ❝ sû ❞ư♥❣ t ỵ tt t ựs ử ❜✐➺t ♥â
✶


❝á♥ ❣✐ó♣ ➼❝❤ ❝❤♦ ✈✐➺❝ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❣✐↔♥❣ ❞↕② ð ❝→❝ tr÷í♥❣ ❝❛♦
✤➥♥❣ ✈➔ ✤↕✐ ❤å❝✳
❚r♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ❚♦→♥ ð ❜➟❝ ✤↕✐ ❤å❝✱ tỉ✐ ✤➣ ✤÷đ❝ ❝→❝ t ổ
ợ t ữỡ tr t ✈❛✐ trá ❝õ❛ ♥â ✤è✐ ✈ỵ✐ ❜ë ♠ỉ♥
t♦→♥ ❤å❝✳ ❙❛✉ ữủ t ổ ợ t tổ t ữỡ
tr t rt q trồ ợ t q trồ õ ũ ợ sỹ ữợ
ú ù t t➻♥❤ ❝õ❛ ❝→❝ t❤➛②✱ ❝æ ❣✐→♦ tr♦♥❣ ❇ë ♠æ♥ ❣✐↔✐ t➼❝❤ tỉ✐
✤➣ ❝❤å♥ ✤➲ t➔✐✿ ✧P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✧ ❧➔♠ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣✳
◗✉❛ ❧✉➟♥ ✈➠♥ ♥➔② tæ✐ ♠✉è♥ ♥❣❤✐➯♥ ự ởt số ỵ tt ỡ ừ
ữỡ tr t ♣❤➙♥✳
◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠ ỗ
ữỡ
ữỡ r ởt số ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ Lp✱ ❦❤ỉ♥❣
❣✐❛♥ ❍✐❧❜❡rt ✈➔ ❝→❝ t♦→♥ tû tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ♥❤÷✿ t♦→♥ tû ❧✐➯♥
❤ñ♣✱ t♦→♥ tû ✤è✐ ①ù♥❣✱ t♦→♥ tû ❤♦➔♥ t♦➔♥ ❧✐➯♥ tö❝✱ t♦→♥ tû ✤è✐ ①ù♥❣ ❤♦➔♥
t♦➔♥ ❧✐➯♥ tö❝✳ ✣➙② ❧➔ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝➛♥ t❤✐➳t ❝❤✉➞♥ ❜à ❝❤♦ ❝❤÷ì♥❣ ✷
❝õ❛ ❧✉➟♥ ✈➠♥✳
❈❤÷ì♥❣ ✷✿ ✣➙② ❧➔ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②
❝❤ó♥❣ tỉ✐ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥
♥❤÷✿ t♦→♥ tû t➼❝❤ ♣❤➙♥✱ ♣❤➙♥ ❧♦↕✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ữỡ
tr t ợ ố ự ữỡ tr t ợ s
ữỡ tr t ợ t ý ỵ r

ữỡ t
ữủ t ữợ sỹ ữợ ❞➝♥ t➟♥ t➻♥❤ ❝õ❛ ❚❙✳ ◆❣✉②➵♥
❚❤à ◆❣➙♥✳ ◆❤➙♥ ❞à♣ ♥➔② ❝❤♦ ♣❤➨♣ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tợ ổ
ữớ t t ữợ ú ù tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ♥❣❤✐➯♥
❝ù✉ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳
❚r♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✱ tæ✐ ♥❤➟♥ ✤÷đ❝ r➜t ♥❤✐➲✉ sü ❣✐ó♣ ✤ï
✤ë♥❣ ✈✐➯♥ ❝õ❛ ❝→❝ t❤➛②✱ ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲
✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✈➔ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝✳ ❚ỉ✐ ①✐♥ ✤÷đ❝ ❜➔② tä
✷


❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥✳
❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï
tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ r➧♥ ❧✉②➺♥ t↕✐ ❑❤♦❛✱ ❚r÷í♥❣✳
❈✉è✐ ❝ị♥❣ ❞♦ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥
✈➠♥ ❝❤➢❝ ❝❤➢♥ s➩ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sõt tổ rt
ữủ ỳ ỵ ❝❤➾ ❜↔♦✱ ✤â♥❣ ❣â♣ ❝õ❛ ❝→❝ t❤➛②✱ ❝æ ❣✐→♦ ✈➔ ❝→❝
❜↕♥ ❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳

✸


❈❤÷ì♥❣ ✶

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❦❤æ♥❣ ❣✐❛♥ Lp
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❬✷❪✱❬✸❪ ❈❤♦ (X, M, µ) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✤ë ✤♦✱ tr♦♥❣

✤â ❳ ❧➔ ❦❤æ♥❣ ❣✐❛♥✱ ▼ ❧➔ ♠ët σ✲✤↕✐ sè ❝→❝ t➟♣ ❝♦♥ ❝õ❛ ❳✱ µ ❧➔ ♠ët ✤ë ✤♦
tr➯♥ ▼✳ ❈❤♦ ♣∈ [1; +∞) ❧➔ ♠ët sè t❤ü❝✳ ❍å t➜t ❝↔ ❝→❝ ❤➔♠ sè ❢✭①✮ ❝â ❧ô②

t❤ø❛ ❜➟❝ ♣ ❦❤↔ t➼❝❤ tr ồ ổ Lp(X, à).
ữ
Lp (X, µ) = {f : X −→ R :

|f |p dµ < ∞}.
x

❑❤✐ ❳ ❧➔ t➟♣ ✤♦ ✤÷đ❝ t❤❡♦ ♥❣❤➽❛ ▲❡❜❡s❣✉❡ tr
s t t t Lp(X) t Lp(X, à).
ợ p = ỵ

Rk

à ở

L (X) = {f : X −→ R|ess sup|f (x)| < +∞}.

tr♦♥❣ ✤â
ess sup |f (x)| = inf {M > 0|µ{x ∈ X||f (x)| > M } = 0}.
x∈X

▼➺♥❤ ✤➲ ✶✳✶✳ [2] , [3] ủ Lp(X, à) ợ t tổ tữớ
tr số ợ

f (x)

Lp (X,à)

|f |p dà

=

1
p

ợ ộ f Lp (X, à)

X

ởt ổ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✳
✹


❈❤ù♥❣ ♠✐♥❤✳

❉➵ t❤➜② r➡♥❣✱ ✈ỵ✐ ♠å✐ f, g ∈ Lp(X, µ), ✈ỵ✐ ♠å✐ k ∈ K, t❛

❝â |f + g| ≤ 2max{|f |, |g|}.
❚ø ✤â✱ s✉② r❛

|f + g|p ≤ 2p max{|f |p , |g|p ≤ 2p (|f |p + |g|p ) .

❱➟② f + g ∈ Lp(X, µ).◆❣♦➔✐ r❛ kf Lp(X, à). ữ Lp(X, à) õ
ố ợ t tổ tữớ tr số ♥â ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥
t✉②➳♥ t➼♥❤✳
❚❛ ❜✐➳t r➡♥❣✱ |f |pdµ = 0 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f = 0 ❤➛✉ ❦❤➢♣ ♥ì✐ tr➯♥ ❳
X
♥➯♥ ✤✐➲✉ ❦✐➺♥ t❤ù ♥❤➜t ❝õ❛ ❝❤✉➞♥ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ✣✐➲✉ ❦✐➺♥ t❤ù ❤❛✐ ❧➔
❤✐➸♥ ♥❤✐➯♥✱ ✤✐➲✉ ❦✐➺♥ t❛♠ ❣✐→❝ ✤÷đ❝ s✉② r❛ tø ❜➜t ✤➥♥❣ t❤ù❝ s
ữủ ự

ỵ [2] , [3] Lp(X, µ) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳

❈❤ù♥❣ ♠✐♥❤✳

●✐↔ sû {fn} ❧➔ ♠ët ❞➣② ❈❛✉❝❤② tr♦♥❣ Lp(X, µ)✱ tù❝ ❧➔
fn − fm = 0.

lim

m,n→∞

❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐
m, n ≥ nk ✱

k ∈ N

tỗ t ởt số

||fm fn || <

t

nk ∈ N∗

s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐

1
.
2k

✈ỵ✐ ♠å✐ n ≥ nk .
❑❤ỉ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t n1 < n2 < ... < nk < ... ❑❤✐
✤â
||fm − fn || <

1
2k

||fn+1 − fn || <

❱ỵ✐ ♠å✐ s ∈ N∗✱ ✤➦t

1
.
2k

s

|fnk+1 (x) − fnk (x)| ∈ Lp (X, µ).

gs (x) = |fn1 (x)| +
k=1

✺


❑❤✐ ✤â {gs}s∈N ❧➔ ♠ët ❞➣② ✤ì♥ ✤✐➺✉ t➠♥❣ ❝→❝ ❤➔♠ sè ✤♦ ✤÷đ❝✱ ❦❤ỉ♥❣ ➙♠✳
⑩♣ ❞ư♥❣ ❜ê ✤➲ ❋❛t♦✉ {gs} t s r tỗ t s
lim gs (x) ✈➔

∗

lim (gs (x))p dµ ≤ lim inf

s→∞

(gs (x))p dµ = lim inf gs

s→∞

X

s→∞

p

.

X

▼➦t ❦❤→❝✱ t❛ ❝â
s

s

gs ≤ fn1 +

fn+1 − fnk < fn1 +
k=1

k=1

1
< fn1 + 1,
2k

♥➯♥
lim inf gs

p

s→∞

❉♦ ✤â

< +∞.

lim (gs (x))p dµ < +∞.

s→∞
X

✣✐➲✉ ✤â ❦➨♦ t❤❡♦s→∞
lim (gs (x))p dµ ❤ú✉ ❤↕♥ ❤➛✉ ❦❤➢♣ ♥ì✐✱ s✉② r lim gs (x)
s
tỗ t ỳ ♥ì✐✳ ◆❤÷ ✈➟② ❝❤✉é✐
∞

fn1 (x) +
k=1

(f n+1 (x) − fnk (x)) ,

❤ë✐ tư t✉②➺t ✤è✐ ❤➛✉ ❦❤➢♣ ♥ì✐✱ ❞♦ ✤â ❤ë✐ tư ❤➛✉ ❦❤➢♣ ♥ì✐✱ tù❝ ❧➔ ❦❤✐ s →
s tỗ t ợ ỳ ỡ ❝õ❛ ❤➔♠
s

(f nk+1 (x) − fnk (x)) .

fns+1 (x) = fn1 (x) +
k=1

●å✐ f0 (x) ❧➔ ❣✐ỵ✐ ❤↕♥ ❤➛✉ ❦❤➢♣ ♥ì✐ ❝õ❛ ❤➔♠ fn

k+1

(x)✱

❦❤✐ s → ∞. ❱➻

fnk+1 (x) ≤ lim gs (x) ∈ Lp (X, µ) ,
s→∞

♥➯♥ t❤❡♦ ✣à♥❤ ỵ ở tử t õ
p

|f0 (x)p | dà = lim

fns+1 (x) dµ,

s→∞

X

X

✻


tù❝ ❧➔ f0 ∈ Lp (X, µ) . ⑩♣ ❞ư♥❣ ❜ê ✤➲ ❋❛t♦✉ ♠ët ❧➛♥ ♥ú❛✱ t❛ ❝â
f0 − fnk =
≤ lim

s→∞ X

p

lim fnk+1 (x) − fnk (x) dµ

X s→∞

fnt+1 − fn1

s→∞

t=k

s
s→∞ t=k

1
2t

s→∞
s

p

s

= lim inf
≤ lim

p

fnk+1 (x) − fnk (x) dµ = lim inf fnk+1 − fnk

∞

=
t=k

≤ lim

fnt+1 − fn1

s→∞ t=k

p

p

1
2t .

❚ø ✤➙② s✉② r❛ k→∞
lim f0 − fn = 0. õ ợ ồ > 0, tỗ t m1 ∈ N∗
s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ nk ≥ m1 t❛ ❝â
k

ε
f0 = fnk < .
2

▼➦t ❦❤→❝✱ ✈➻ {fn} ❧➔ ❞➣② tỗ t m2 s n, nk ≥ m2
t❤➻ fn − fn < 2ε ✳ ❈❤å♥ n0 = max {m1, m2} ❦❤✐ ✤â ✈ỵ✐ ♠å✐ n ≥ n0✱ ❧➜②
nk ≥ n0 ✱ t❛ ❝â
k

f0 − fn ≤ f0 − fnk + fnk − fn < ε.

◆❤÷ t❤➳ fn ❤ë✐ tư tỵ✐ f0✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ Lp (X, à) ổ
ỵ ữủ ự

ỵ [2] , [3] à ở ❤ú✉ ❤↕♥ ✈➔ 1 ≤ p ≤ q < +∞ t ợ
ộ số f Lp (X, à)

f

p

f

1



1

pq ,
p .µ (X)

tø ✤â s✉② r❛ Lq (X, µ) ⊂ Lp (X, µ) ⊂ Ll (X, µ) .

❚❛ t❤➜② ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✮ ❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣ tr♦♥❣ tr÷í♥❣
q
❤đ♣ q = p✳ ❳➨t tr÷í♥❣ ❤đ♣ p < q✳ ❱➻ pq > 1 ✈➔ (q−p)
> 1 ✈➔ pq + q−p
q = 1✱
♥➯♥ t❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❍♦❧❞❡r t❛ ❝â
❈❤ù♥❣ ♠✐♥❤✳

X



 pq 

|f.g|dµ ≤ 

|f | p dµ 

q

X

X

✼

 q−p
q
q

|g| q−p dµ

.


ợ ồ f, g ữủ tr X ❈❤å♥ g ≡ 1✱ t❤❛② f ❜ð✐ |f |p✱ t❛ t
ữủ



p
q

|f |p dà
X

q

(|f |p ) p dà

dà

,

X

X

tữỡ ữỡ ợ

qp
q



p1



|f |p dà

1q



|f |q dµ µ (X)

q−p 1
q ·p

.

X

X

❚ø ✤â s✉② r❛ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✮✳
❉♦ µ (X) < ∞ ✱ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✮ s✉② r❛ r➡♥❣✿ ♥➳✉ f ∈ Lq (X, µ) t
f Lp (X, à) ợ ồ 1 p ≤ q < +∞. ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ ❜❛♦ ❤➔♠ tự
ự
ỵ ữủ ự

ỵ [2] , [3] ▼é✐ ❤å ❤➔♠ s❛✉ ✤➙② ❧➔ trò ♠➟t tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Lp
✶✳ ❍å ❝→❝ ❤➔♠ sè ✤ì♥ ❣✐↔♥ tr➯♥ [0; 1]✱
✷✳ ❍å ❝→❝ ❤➔♠ sè ❧✐➯♥ tö❝ tr➯♥ [0; 1]

ỵ [2] , [3] ổ Lp ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❦❤↔ ❧②✳
✶✳✷ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ổ rt
ổ ữợ

[1] , [2] , [3] ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ tr➯♥
K✳

❍➔♠ sè

, :X ×X →K

(x, y) → x, y ,

t❤ä❛ ♠➣♥ ✈ỵ✐ ♠å✐ x, y, z ∈ X, λ ∈ K
(1)
(2)
(3)
(4)

y, x = x, y ;
x + y, z = x, z + y, z ;
λx, y = λ x, y ;
x, x ≥ 0; x, x = 0 ⇔ x = 0.
✽


ữủ ồ t ổ ữợ tr X ố
♣❤➛♥ tû x, y tr♦♥❣ X ✳

▼➺♥❤ ✤➲ ✶✳✷✳ [1] , [2] , [3] ❈❤♦
✶✳ ❱ỵ✐ ♠å✐ x, y ∈ X t õ

x, y

ữủ ồ t ổ ữợ ừ

, ởt t ổ ữợ tr X õ
| x, y |2 ≤ x, x y, y .

✷✳ ❍➔♠ sè · : X → R+ ①→❝ ✤à♥❤ ❜ð✐

✭✶✳✷✮

x, x , x ∈ X,

x =
❧➔ ♠ët ❝❤✉➞♥ tr➯♥ X ✳

▼➺♥❤ ✤➲ ✶✳✸✳ [1] , [2] , [3] ❈❤♦
x+y

2

, ❧➔ t➼❝❤ ổ ữợ tr X õ

+ xy

2

=2

x

2

+ y

2

,

tr õ Ã s t ổ ữợ

ổ ❍✐❧❜❡rt

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ [1] , [2] , [3] ❈➦♣ (X,

, )✱

tr♦♥❣ ✤â X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥
t✉②➳♥ t➼♥❤✱ , ❧➔ t ổ ữợ tr X ữủ ồ ổ ❣✐❛♥ t✐➲♥
❍✐❧❜❡rt✳ ❑❤ỉ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt ✤➛② ✤õ ✈ỵ✐ ♠❡tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ ①→❝
✤à♥❤ ❜ð✐ ✭✶✳✷✮ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ rt

ỵ [1] , [2] , [3] X ởt ổ t rt õ
t ổ ữợ ❧➔ ♠ët ❤➔♠ sè ❧✐➯♥ tư❝ tr➯♥ X × X ✳

●✐↔ sû { xn, yn } ❧➔ ♠ët ❞➣② ❝→❝ ♣❤➛♥ tû ❤ë✐ tư ✈➲ x0, y0
tr♦♥❣ X × X tù❝ ❧➔ n→∞
lim xn = x0 ∈ X, lim yn = y0 ∈ X. ❑❤✐ ✤â ✈ỵ✐ ♠é✐
n→∞

❈❤ù♥❣ ♠✐♥❤✳

n ∈ N∗ ,

| xn , yn − x0 , y0 | ≤ | xn , yn − xn , y0 | + | xn , y0 − x0 , y0 |
= | xn , yn − y0 | + | xn − x0 , y0 |
≤ xn yn − y0 + y0 xn − x0 .

❉♦ xn → x0 ♥➯♥ ❞➣② {xn} ợ ở tự tỗ t k > 0 s❛♦ ❝❤♦

✈ỵ✐ ♠å✐ n ∈ N∗✳ ❚ø ✭✶✳✸✮ t❛ ❝â
lim xn , yn = x0 , y0 .

n→∞

✾

✭✶✳✸✮

xn ≤ K


t t ổ ữợ số tử
ỵ ữủ ự
ử ổ L2 (X, µ)✿ ❈❤♦ (X, M, µ) ❧➔ ♠ët ❦❤ỉ♥❣ ở

ợ f, g L2(X, à) t

f, g = f gdà.
X

t r t ổ tỗ t↕✐ ✈➔ ❤ú✉ ❤↕♥ ✈➻ t❤❡♦ ❜➜t ✤➥♥❣
t❤ù❝ ❍♦❧❞❡r✿


 12 

|f g|dµ ≤ 

|f |2 dµ 

X

 12

|g|2 dµ < +∞.

X

X

❉➵ t❤➜② ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤➔♠ sè ①→❝ ✤à♥❤ ởt t ổ
ữợ tr L2(X, à) r t ợ ộ f L2(X, à)
12



f =

f, f = 

f gdµ = 

X

 21



|f |2 dµ .

X

✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ L2(X, µ) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳
✶✳✷✳✸ ❑❤æ♥❣ ❣✐❛♥ L2ρ

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ [1] , [2] , [3] ❱ỵ✐ a < x < b ①➨t ❤➔♠ trå♥❣
ρ (x) = (x − a)α (b − x)β , α, β > 1

ỵ L2(a, b) t ừ tt u(x) ữỡ t ợ
trồ ♥❣❤➽❛ ❧➔
 21

b



ρ (x) |u (x)|2 dx < ∞.

u :=
a

ổ ữợ tr L2 (a, b)

ữủ ❝æ♥❣ t❤ù❝
b

u, v

ρ

✭✶✳✺✮

ρ (x) u (x) v(x)dx.

:=
a

✶✵

✭✶✳✻✮


ó r ợ t ổ ữợ ✭✶✳✻✮ t❤➻ L2ρ(a, b) ❧➔ ♠ët ❦❤æ♥❣
❣✐❛♥ ❍✐❧❜❡rt✳

✶✳✸ ❚♦→♥ tû ✤è✐ ①ù♥❣ ❤♦➔♥ t♦➔♥ ❧✐➯♥ tö❝
✶✳✸✳✶ P❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝

❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ t❛ ❝❤➾ ①❡♠ ①➨t ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ ❝õ❛ ❦❤ỉ♥❣
❣✐❛♥ ❍✐❧❜❡rt✳ ❈❤♦ X ❧➔ ♠ët ổ rt ỵ x, y t ổ
ữợ ❝õ❛ ❤❛✐ ♣❤➛♥ tû x, y tr♦♥❣ X ✱ t❛ ❜✐➳t r➡♥❣✱ ✈➔ ✈ỵ✐ ♠é✐ ❝è ✤à♥❤
y ∈ X ✱ ♣❤✐➳♠ ❤➔♠
x∗ (x) = x, y

❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t tử tr X

ỵ s [1] , [2] , [3] ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ◆➳✉
x∗ ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tr X t tỗ t t ởt
tỷ y ∈ X s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x ∈ X,

x∗ (x) = x, y
◆❣♦➔✐ r❛ x∗ = y .

❍✐➸♥ ♥❤✐➯♥✱ ♥➳✉ x∗ = 0 t❤➻ ♣❤➛♥ tû y = 0 ❧➔ tỷ
t tọ ỵ sỷ x = 0✳ ❑❤✐ ✤â L = ker x∗ ❧➔ ♠ët ổ
õ tỹ sỹ ừ X tỗ t↕✐ ♠ët ♣❤➛♥ tû ❦❤→❝ ❦❤æ♥❣
z ❝õ❛ X trü❝ ❣✐❛♦ ✈ỵ✐ L✱ tù❝ ❧➔ z⊥x ✈ỵ✐ ♠å✐ x ∈ L✳ ❚❛ ①➨t ♣❤✐➳♠ ❤➔♠
z ∗ : X → K ①→❝ ✤à♥❤ z ∗ (x) = x, z ✈ỵ✐ ♠å✐ x ∈ X ✳ ❉➵ t❤➜② z ∗ ❧➔ ♠ët
♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤✱ ❧✐➯♥ tö❝ tr➯♥ X ✈➔
❈❤ù♥❣ ♠✐♥❤✳

Ker x∗ ⊂ Kerz

õ tỗ t K s z∗ = αx∗. ❉♦ z∗(z) = x, z = z 2 = 0 ♥➯♥
z ∗ = 0✱ ❦➨♦ t❤❡♦ α = 0✳ ❚ø ✤â s✉② r❛ x∗ = α1 z ∗ . ✣➦t λ = α1 t❛ ❝â x∗ = λz ∗ .
❉♦ ✈➟② t❛ ❝â
x∗ (x) = λz ( x)∗ = λ x, z = x, λz

✈ỵ✐ ♠å✐ x ∈ X ✳ ✣➦t y = λz t❛ ❝â
✶✶


✱ ✈ỵ✐ ♠å✐ x ∈ X.
❚❛ ❝á♥ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ tỷ y t sỷ tỗ t y tọ
ỵ õ
x, y = x, y ✱ ✈ỵ✐ ♠å✐ x ∈ X.
❉♦ ✤â x, y − y = 0 ✈ỵ✐ ♠å✐ x ∈ X ✳ ❚ø ✤â s✉② r❛ y − y = 0 ❤❛②
y = y . ❉➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ x∗ = y .
ỵ ữủ ự

t →♥❤ ①↕ A : X → X ∗ ♥❤÷ s❛✉✿ ợ ộ
tỷ y X t tữỡ ù♥❣ ✈ỵ✐ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ x∗ ①→❝
✤à♥❤ tr ỵ t A ởt ✤➥♥❣ ❝ü tø X ❧➯♥ X ∗.
x∗ (x) = x, y

❍➺ q✉↔ ✶✳✶✳ [1] , [2] , [3] ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ ❞➣② {xn} ❤ë✐ tö ②➳✉
✤➳♥ ♣❤➛♥ tû x0 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ lim xn , x = x0 , x ✱ ✈ỵ✐ ♠å✐ x ∈ X.
n→∞

●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ X ∗ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧✐➯♥ ❤ñ♣ ❝õ❛ X.
❚❛ tr❛♥❣ ❜à ❝❤♦ X ∗ ♠ët t ổ ữợ ữ s
x , y = A1 x∗ , A−1 y ∗ , x∗ , y ∗ ∈ X ∗ ,

tr♦♥❣ ✤â A ❧➔ ♣❤➨♣ ✤➥♥❣ ❝ü t✉②➳♥ t➼♥❤ tø X ✈➔♦ X ∗. ❚❛ ❝â ❦➳t q s

ỵ [1] , [2] , [3] ổ ❣✐❛♥ ❧✐➯♥ ❤đ♣ X ∗ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ X ❧➔
♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳

❚❤➟t ✈➟②✱ ✈➻ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ♥➯♥ ❤✐➸✉ ♥❤✐➯♥ X ∗
❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❚❛ ❝â
❈❤ù♥❣ ♠✐♥❤✳

x∗ , x∗ =

A−1 x∗ , A−1 x∗ = A−1 x∗ = x∗

A−1 x∗ , A−1 x∗ =

✈ỵ✐ ♠å✐ x∗ ∈ X ∗, A ❧➔ ♠ët s♦♥❣ →♥❤ ✤➥♥❣ ❝ü tø X ❧➯♥ X ∗. ♥❤÷ ✈➟②✱ ❝❤✉➞♥
tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X ∗ trị♥❣ ✈ỵ✐ ❝❤✉➞♥ s✐♥❤ ❜ð✐ t➼❝❤ ✈ỉ ữợ tr

X X ổ rt
ỵ ữủ ự




✶✳✸✳✷ ❚♦→♥ tû ❧✐➯♥ ❤đ♣

●✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ A ∈ L(X). ❱ỵ✐ ♠é✐ ♣❤➛♥ tû ❝è ✤à♥❤
y ∈ X ✱ ①➨t ❤➔♠ sè
x∗y : X → K,

①→❝ ✤à♥❤ ❜ð✐ x∗y (x) =
t✉②➳♥ t➼♥❤ ✈➔

Ax, y

✈ỵ✐ ♠å✐ x ∈ X ✳ ❚❛ ❝â x∗y ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠

x∗y (x) = | Ax, y | ≤ Ax

y ≤ A

y

x ,

ợ ồ x X. ữ xy ởt ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❣✐ỵ✐ ♥ë✐ tr➯♥
❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt X
xy A

y .

ỵ tỗ t ♥❤➜t ♠ët ♣❤➛♥ tû z ∈ X s❛♦ ❝❤♦
x∗y (x) = x, z , ∀x ∈ X.

✣➦t A∗y = z t❛ ✤÷đ❝ ♠ët →♥❤ ①↕ A∗ : X → X. ❚❛ ❝â A∗ ❧➔ ♠ët →♥❤ ①↕
t✉②➳♥ t➼♥❤✳ ❚ø ✣à♥❤ ỵ t ủ ợ t tự t ❝â
A∗ y = z = x∗y ≤ A

y ,

✈ỵ✐ ♠å✐ y ∈ X. ❱➟② A∗ ❧➔ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤ ❣✐ỵ✐ ♥ë✐ ✈➔
A∗ ≤ A .

❚♦→♥ tû A ∈ L(X) ①→❝ ✤à♥❤ ♥❤÷ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ t♦→♥ tû ❧✐➯♥ ❤đ♣ ❝õ❛
t♦→♥ tû t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ A.

❚ø ✤à♥❤ ♥❣❤➽❛ t♦→♥ tû A∗, t❛ s✉② r❛ t♦→♥ tû A∗ ✤÷đ❝ ①→❝ ✤à♥❤ ✤➥♥❣
t❤ù❝
Ax, y = x, A∗ y , x, y ∈ X.

❚♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ t♦→♥ tû A∗ ❣å✐ ❧➔ t♦→♥ tû ❧✐➯♥ ❤ñ♣ t❤ù ❤❛✐ ❝õ❛ A
ữủ ỵ A.

ỵ [2] . ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ A ∈ L(X). ❦❤✐ ✤â
✶✸

A∗∗ = A, ✈➔ A∗ = A .
❈❤ù♥❣ ♠✐♥❤✳

❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ✈➲ t♦→♥ tû ❧✐➯♥ ❤ñ♣ t❛ ❝â

Ax, y = x, A∗ y = A∗ y, x = y, A∗∗ x = A∗∗ x, y ,

✈ỵ✐ ♠å✐ x, y ∈ X. ❉♦ ✤â Ax = A∗∗x,✈ỵ✐ ♠å✐ x ∈ X, ❦➨♦ t❤❡♦ A = A∗∗.
▼➠t ❦❤→❝✱ A∗∗ ≤ A∗ ≤ A s r A = A = A .
ỵ ữủ ự

ỵ [2] , [3] sỷ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ A, B ∈ L(X), λ ∈
K. ❦❤✐ ✤â t❛ ❝â
✶✮ (A + B)∗ = A∗ + B ∗ ;
✷✮ (λA)∗ = λA∗ ;
✸✮ (BA)∗ = A∗ B ∗ ;
✹✮ IX∗ = IX .

❈❤ù♥❣ ♠✐♥❤✳

❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ x, y ∈ X. ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ t♦→♥ tû ❧✐➯♥

tö❝ t❛ ❝â✿
✶✮ x, (A + B)∗ y

= (A + B) (x) , y = Ax + Bx, y
= x, A∗ y + B ∗ y = x, (A∗ + B ∗ ) y .

❙✉② r❛ (A + B)∗ y = (A∗ + B ∗) y ✈ỵ✐ ♠å✐ y ∈ X, ❦➨♦ t❤❡♦

(A + B)∗ = A∗ + B ∗ .

✷✮

x, (λA)∗ y = (λA) x, y = λ (Ax) , y = Ax, λy
= x, A∗ λy

❙✉② r❛ (λA∗) y = A∗

λy

= x, λA∗ y .

✈ỵ✐ ♠å✐ y ∈ X ✱ ❦➨♦ t❤❡♦
(λA)∗ = λA∗ .

x, (B · A)∗ y = (B · A) x, y = B (Ax) , y = Ax, B ∗ y
✸✮
= x, A∗ (B ∗ y)
❙✉② r❛ (B · A)∗ y = A∗ (B ∗y) ✈ỵ✐ ♠å✐ y ∈ X, ❦➨♦ t❤❡♦
(B · A)∗ = A∗ · B ∗ .

✹✮

x, IX∗ y = IX x, y .
✶✹


❙✉② r❛ x = IX x = IX = 1 ✈ỵ✐ ♠å✐ x ∈ X
IX∗ y = y = IX∗ = 1 ✈ỵ✐ ♠å✐ y ∈ X.

❉♦ ✤â s✉② r❛ IX = IX .
ỵ ữủ ự
õ ởt số t q ữủ tr tr ỵ s

ỵ [2] , [3] sỷ X ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ A ∈ L(X).
❑❤✐ ✤â A ❧➔ ỗ ổ A ỗ ổ
ự

sỷ A ởt ỗ ổ õ
A Ã A1 = A1 A = IX .

ỵ t❛ ❝â
∗

A−1 · A∗ = A · A−1
∗
A∗ · A−1 = A−1 · A

∗
∗

= IX∗ = IX ;
= IX∗ = IX .

A ởt ỗ ổ (A)1 = A−1 ∗ .
◆❣÷đ❝ ❧↕✐✱ ♥➳✉ A∗ ❧➔ ♠ët ♣❤➨♣ ỗ ổ t t ự tr A
ởt ỗ ổ A = A A ụ ởt ỗ ổ
ỵ ữủ ự

ỵ [2] , [3] ◆➳✉ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ A ∈ L(X) t❤➻

X = kerA ⊕ A∗ (X) ✈➔ X = kerA∗ ⊕ A (X),
tr♦♥❣ ✤â ⊕ ❧➔ ỵ ừ tờ trỹ t trỹ

ú ỵ r kerA, A∗ (X) ❧➔ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣
♥➯♥ ✤➸ ❝❤ù♥❣ ♠✐♥❤ X = kerA ⊕ A∗ (X) t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤

❈❤ù♥❣ ♠✐♥❤✳

❝õ❛ X

⊥

kerA =A∗ (X) .

❚❤➟t ✈➟② ✈ỵ✐ ♠é✐ x ∈ kerA t❤➻ Ax = 0✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦
✈ỵ✐ ♠å✐ y ∈ X ✳ ❉♦

Ax, y = 0

x, A∗ y = Ax, y = 0,

♥➯♥ x⊥A∗y ✈ỵ✐ ♠å✐ y ∈ X, tù❝ ❧➔ x⊥A∗(X). ❚ø t➼♥❤ ❝❤➜t tử ừ
t ổ ữợ t s r xA (X). ❚ù❝ x ∈ A∗ (X)⊥✳
✶✺


◆❣÷đ❝ ❧↕✐✱ ♥➳✉ x ∈ A∗ (X)⊥ t❤➻ x⊥A∗ (X)✱ s✉② r❛ x⊥A∗ (X), ❦➨♦ t❤❡♦
x, A∗ y = 0 ✈ỵ✐ ♠å✐ y ∈ X. ❉♦
Ax, y = x, A∗ y = 0,

♥➯♥ Ax⊥y ✈ỵ✐ ♠å✐ y ∈ X, ✤✐➲✉ ♥➔② ❝❤➾ ①↔② r❛ ❦❤✐ Ax = 0, tù❝ ❧➔ x ∈ kerA.
◆❤÷ ✈➟② kerA = A∗ (X)⊥, ♥➯♥ t❛ ❝â ✤➥♥❣ t❤ù❝ t❤ù ♥❤➜t✳ ✣➥♥❣ t❤ù❝
❝á♥ ❧↕✐ tr♦♥❣ ✤à♥❤ ỵ ữủ tứ tự ự
t A A ợ ú ỵ A.
ỵ ữủ ❝❤ù♥❣ ♠✐♥❤✳
✶✳✸✳✸ ❚♦→♥ tû ✤è✐ ①ù♥❣

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ [2] , [3] ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ t♦→♥ tû A

✤÷đ❝ ❣å✐ ❧➔ t♦→♥ tû ✤è✐ ①ù♥❣ ✭t♦→♥ tû tü ❧✐➯♥ ❤ñ♣✮ ♥➳✉ A = A∗.
◆❤➟♥ ①➨t ✶✳✷✳ ❚❛ ❝â t♦→♥ tû A ❧➔ t♦→♥ tû ✤è✐ ①ù♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
Ax, y = x, Ay ✈ỵ✐ ♠å✐ x, y ∈ X.
❱➼ ❞ö ✶✳✷✳ A ∈ L (Cn) ❧➔ t♦→♥ tû ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ♠❛ tr➟♥ (aij ), i, j =
1, 2, .., n. ❑❤✐ ✤â A ✤è✐ ①ù♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
aij = aij ✱ ✈ỵ✐ ♠å✐ i, j = 1, 2, ..., n.

ỵ [2] , [3] ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ A, B ∈ L(X)

❧➔ ❝→❝ t♦→♥ tû ✤è✐ ①ù♥❣✳ ❑❤✐ ✤â
✶✮ A + B ❧➔ t♦→♥ tû ✤è✐ ①ù♥❣❀
✷✮ ❱ỵ✐ ♠å✐ λ ∈ R, λA ❧➔ t♦→♥ tû ✤è✐ ①ù♥❣❀
✸✮ ◆➳✉ A.B = B.A t❤➻ B.A ❧➔ t♦→♥ tû ✤è✐ ự
IX t tỷ ố ự
ú ỵ r λ = x + iy ∈ C tr♦♥❣ ✤â y = 0 ✈➔ A ❧➔ ♠ët t♦→♥ tû
✤è✐ ①ù♥❣ ❦❤æ♥❣ t➛♠ t❤÷í♥❣ t❤➻ λA ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ t♦→♥ tû ✤è✐ ự

ỵ [1] , [2] , [3] X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ A ❧➔ t♦→♥
tû ✤è✐ ự ỗ ổ õ A1 t tỷ ố ①ù♥❣✳
✶✻

ỵ t tỷ ủ A ừ t tỷ A
ởt ỗ ổ r (A)1 = (A−1)∗✳ ❉♦ A ❧➔ t♦→♥ tû ✤è✐ ①ù♥❣
♥➯♥ A∗ = A✱ ❦➨♦ t❤❡♦ A−1 = (A∗)−1 = (A−1)∗✳ ◆➯♥ A−1 t tỷ ố
ự ỵ ữủ ự
ự

ỵ [2] , [3] sỷ X ởt ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ A ∈ L(X)

❧➔ t♦→♥ tû ✤è✐ ①ù♥❣✳ ❑❤✐ ✤â

A = sup {| Ax, x | : x ≤ 1} = sup {| Ax, x | : x = 1} .
❈❤ù♥❣ ♠✐♥❤✳

✣➦t µ = sup {| Ax, x | :
| Ax, x | ≤ A

x ≤ 1} .
x

2

❚❛ ❝â

,

✈ỵ✐ ♠å✐ x ∈ X ✳ ◆➯♥
| Ax, x | A

ợ

x 1.




à = sup | Ax, x | ≤ A .
x≤1

▼➦t ❦❤→❝✱ ✈ỵ✐ ♠å✐ x, y ∈ X, t❛ ✤➲✉ ❝â
A (x + y) , x + y = Ax, x + Ax, y + Ay, x + Ay, y ;
A (x − y) , x − y = Ax, x − Ax, y − Ay, x + Ay, y .

❚rø ✤➥♥❣ t❤ù❝ tr➯♥ ❝❤♦ ✤➥♥❣ tự ữợ t ữủ
A (x + y) , x + y − A (x − y) , x − y = 2 Ax, y + 2 Ay, x .

❱➻

Ax, y = x, Ay = Ay, x ,

♥➯♥ tø ✤➥♥❣ t❤ù❝ tr➯♥ s✉② r❛
✭✶✳✾✮

A (x + y) , x + y − A (x − y) , x − y = 4Re Ax, y .

❉➵ ❞➔♥❣ t❤➜② r➡♥❣✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ µ,
| Az, z | ≤ µ z 2 , ✈ỵ✐ ♠å✐ z ∈ X.
❉♦ ✤â ✤➥♥❣ t❤ù❝ ✭✶✳✾✮ s✉② r❛ r➡♥❣
1
|Re Ax, y | ≤ µ

4

x+y

2

+ x−y
✶✼

2

2

1
= µ
2

x

2

+ y

2

,

✭✶✳✶✵✮

✤ó♥❣ ✈ỵ✐ ♠å✐ x, y ∈ X ✳ ❈❤å♥ x ∈ X s❛♦ ❝❤♦ x = 1✳ ◆➳✉ Ax = 0✱ t❛
❝❤å♥ y = Ax
Ax ✱ ❦❤✐ ✤â y = 1 ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✵✮ trð t❤➔♥❤
✭✶✳✶✶✮

Ax ≤ µ.

❍✐➸♥ ♥❤✐➯♥ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✶✮ ✤ó♥❣ tr♦♥❣ ❝↔ tr÷í♥❣ ❤đ♣ Ax = 0. ữ



A = sup Ax à.
x 1

t ủ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✽✮✈➔ ✭✶✳✶✷✮ t❛ ❝â
❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ❦❤✐ ✤➦t

A = µ.

µ = sup {| Ax, x | : x = 1} .
x ≤1

t❛ ❝ô♥❣ ❝â A = à.
ỵ ữủ ự

q [2] , [3] ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ A ∈ L(X) ❧➔
♠ët t♦→♥ tû ✤è✐ ①ù♥❣ ❦❤æ♥❣ t➛♠ tữớ õ tỗ t (A) s ❝❤♦

|λ| = A , tø ✤â s✉② r❛ σ (A) = .

ỵ [2, 3] . X ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ A ∈ L(X) ❧➔ ♠ët t♦→♥
tû ✤è✐ ①ù♥❣✱ λ1 , λ2 ❧➔ ❤❛✐ ❣✐→ trà r✐➯♥❣ ❦❤→❝ ♥❤❛✉ ❝õ❛ A✳ ❑❤✐ ✤â ❝→❝ ❦❤ỉ♥❣
❣✐❛♥ ❝♦♥ ù♥❣ ✈ỵ✐ ❝→❝ ❣✐→ trà r✐➯♥❣ λ1 , λ2 trü❝ ❣✐❛♦ ✈ỵ✐ ♥❤❛✉✳

❚❤➟t ✈➟②✱ ❝❤♦ λ1, λ2 ❧➔ ❤❛✐ ❣✐→ trà r✐➯♥❣ ❦❤→❝ ♥❤❛✉✱ x, y
❧➔ ❤❛✐ ✈❡❝tì r✐➯♥❣ ù♥❣ ✈ỵ✐ ❝❤ó♥❣✿ Ax = λx, Ay = µy. ❱➻ A ❧➔ ✤è✐ ①ù♥❣✿
Ax, y = x, Ay ♥➯♥λ x, y = µ x, y ❤❛② (λ − µ) x, y = 0, õ

ự

x, y = 0.

ỵ ữủ ự

♥❣❤➽❛ ✶✳✻✳ [2] , [3] ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ A ∈ L(X) ❧➔
♠ët t♦→♥ tû ✤è✐ ①ù♥❣✳ ❑❤✐ ✤â ♣❤✐➳♠ ❤➔♠ h : X × X → C ①→❝ ✤à♥❤ ❜ð✐
❝æ♥❣ t❤ù❝

h(x, y) = Ax, y ,
✶✽


❝â ❝→❝ t➼♥❤ ❝❤➜t ♥❤÷ s❛✉✿
✶✮ h(x, y) = h(x, y) ✈ỵ✐ ♠å✐ x, y ∈ X.
✷✮ h(λ1 x1 + λ2 x2 , y) = λ1 h(x1 , y) + λ2 h(x2 , y) ✈ỵ✐ ♠å✐ x1 , y2 ∈ X ✈➔ ✈ỵ✐
♠å✐ λ1 , λ2 ∈ C.
✸✮ |h(x, y)| ≤ K x

y ✈ỵ✐ ♠å✐ x, y ∈ X ✱ tr♦♥❣ ✤â K ❧➔ ♠ët ❤➡♥❣ sè✳

P❤✐➳♠ ❤➔♠ h : X × X → C t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✶✮✱✷✮✱✸✮ ♥➯♥ tr➯♥
❣å✐ ❧➔ ♠ët ❞↕♥❣ s♦♥❣ t✉②➳♥ t➼♥❤ ❍❡r♠✐t❡✳

◆❤÷ ✈➟②✱ ♠é✐ t♦→♥ tû ✤è✐ ①ù♥❣ ①→❝ ✤à♥❤ ♠ët ❞↕♥❣ s t t ợ
ở rt ữủ ữủ tr ỵ s

ỵ [2] , [3] sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ h : X × X → C ❧➔

♠ët ❞↕♥❣ s♦♥❣ t✉②➳♥ t➼♥❤ ❣✐ỵ✐ ♥ë✐ rt õ tỗ t t ởt
t tỷ ố ①ù♥❣ A ∈ L(X) s❛♦ ❝❤♦

✭✶✳✶✸✮

h(x, y) = x, y ,
✈ỵ✐ ♠å✐ x, y ∈ X.
❈❤ù♥❣ ♠✐♥❤✳

❱ỵ✐ ♠é✐ ♣❤➛♥ tû ❝è ✤à♥❤ y ∈ X ✱ ✤➦t
fy (x) = h x, y ,

x ∈ X.

❑❤✐ ✤â fy ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤✳ ◆❣♦➔✐ r❛✱
|fy (x)| = |h(x, y)| ≤ K x

y

✈ỵ✐ ♠å✐ x ∈ X,

♥➯♥ fy ❧➔ ♠ët ♣❤✐➳♠ ợ ở

fy K y .

ỵ tỗ t ởt tỷ t z X s❛♦ ❝❤♦
fy (x) = x, z ,

✭✶✳✶✹✮

✈ỵ✐ ♠å✐ x ∈ X, ✈➔ fy = z .
❳➨t t♦→♥ tû A : X → X, ①→❝ ✤à♥❤ ❜ð✐ Ay = z ✈ỵ✐ ♠é✐ y ∈ X, tr♦♥❣ ✤â
z ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✶✹✮✳ ❍✐➸♥ ♥❤✐➯♥
✶✾


ợ ồ x X.
ự ữủ A ❧➔ t♦→♥ tû t✉②➳♥ t➼♥❤✳ ◆❣♦➔✐ r❛ ❞♦ Ay =
fy ≤ K y ✈ỵ✐ ♠å✐ x ∈ X ♥➯♥ A ❣✐ỵ✐ ♥ë✐✳ ❚✐➳♣ t❤❡♦ ❝❤ù♥❣ ♠✐♥❤ A ❧➔
t♦→♥ tû ✤è✐ ①ù♥❣✳ ❱ỵ✐ ♠å✐ x, y ∈ X,
h(x, y) = x, Ay

x, Ay = h(x, y) = h(x, y) = y, Ax = Ax, y .

❱➟② A∗ = A ✈➔ A t❤ä❛ ♠➣♥ ✤➥♥❣ t❤ù❝ ✭✶✳✶✸✮✳ ❱➔ ❝ô♥❣ tø ✤➥♥❣ t❤ù❝
♥➔② s r t t ừ A
ỵ ữủ ự ♠✐♥❤✳
◆❤➟♥ ①➨t ✶✳✸✳ ❈❤♦ h : X × X → C ❧➔ ♠ët ❞↕♥❣ s♦♥❣ t✉②➳♥ t➼♥❤ ❣✐ỵ✐ ♥ë✐
❍❡r♠✐t❡✳ ❑❤✐ ✤â ♣❤✐➳♠ ❤➔♠ k ①→❝ ✤à♥❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt X ❝❤♦
❜ð✐ ❝æ♥❣ t❤ù❝
k(x) = h(x, y),

x ∈ X,

❝❤➾ ♥❤➟♥ ❝→❝ ❣✐→ trà t❤ü❝✱ ❣å✐ ❧➔ ❞↕♥❣ t♦➔♥ ♣❤÷ì♥❣ ❍❡r♠✐t❡ ù♥❣ ✈ỵ✐ ❞↕♥❣
s♦♥❣ t✉②➳♥ t➼♥❤ ❣✐ỵ✐ ♥ë✐ ❍✐❧❜❡rt h. ◆❣♦➔✐ r❛ t❛ ❝á♥ ❝â |k (x)| ≤ K x 2 ✈ỵ✐
♠å✐ x ∈ X.
✶✳✸✳✹ ❚♦→♥ tû ❤♦➔♥ t♦➔♥ ❧✐➯♥ tư❝

◆➳✉ A ❧➔ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt X
t❤➻ tø x ≤ K s✉② r❛ Ax ≤ K A , ♥❣❤➽❛ ❧➔ A ❜✐➳♥ ♠é✐ t➟♣ ❜à ❝❤➦♥
t❤➔♥❤ ♠ët t➟♣ ❜à ❝❤➦♥✳
✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ [2] , [3] ❚❛ ♥â✐ r➡♥❣ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤ A tr♦♥❣
❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt X ❧➔ t♦→♥ tû ❤♦➔♥ t♦➔♥ ❧✐➯♥ tö❝ ♥➳✉ ♥â ❜✐➳♥ ♠é✐ t➟♣
❜à ❝❤➦♥ t❤➔♥❤ ♠ët t➟♣ ❤♦➔♥ t♦➔♥ ❜à ❝❤➦♥✳
◆❤➟♥ ①➨t ✶✳✹✳ ▼ët t♦→♥ tû t✉②➳♥ t➼♥❤ A ❧➔ ❤♦➔♥ t♦➔♥ ❧✐➯♥ tö❝ ♥➳✉ ♥â
❜✐➳♥ ♠é✐ t➟♣ ❜à ❝❤➦♥✱ ✤â♥❣✱ t❤➔♥❤ t➟♣ ❝♦♠♣❛❝t✳
▼ët t♦→♥ tû ❤♦➔♥ t♦➔♥ ❧✐➯♥ tö❝ t❤➻ ❧✐➯♥ tö❝✳ ▼➦t ❦❤→❝✱ ♠ët t♦→♥ tû A
❧✐➯♥ tö❝ ♠➔ ♠✐➲♥ ❣✐→ trà ImA ❝õ❛ ♥â ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉
❝õ❛ X t❤➻ ❝ơ♥❣ ❤♦➔♥ t♦➔♥ ❧✐➯♥ tư❝✳
✷✵