O Principio de Ekeland e o Teorema de Nash-Moser

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❘❖❇▲✃❉❖ ▼❆❑✬❙ ▼■❘❆◆❉❆ ❙❊❚❚❊

❖ P❘■◆❈❮P■❖ ❉❊ ❊❑❊▲❆◆❉ ❊ ❖ ❚❊❖❘❊▼❆ ❉❊ ◆❆❙❍✲▼❖❙❊❘

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐✲ ❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳

❱■➬❖❙❆

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Ficha catalográfica preparada pela Seção de Catalogação e Classificação da Biblioteca Central da UFV

T

Sette, Roblêdo Mak's Miranda, 1985-

S495p O princípio de Ekeland e o teorema de Nash-Moser : 2013 Roblêdo Mak's Miranda Sette. – Viçosa, MG, 2013.

iii, 78f. : il. ; 29cm.

Orientador: Sandro Vieira Romero.

Dissertação (mestrado) - Universidade Federal de Viçosa. Referências bibliográficas: f. 77-78.

1. Banach, Espaços de. 2. Espaços lineares normalizados. 3. Análise funcional. 4. Convergência. 5. Teoria das medidas. I. Universidade Federal de Viçosa. Departamento de

Matemática. Programa de Pós-Graduação em Matemática.

II. Título.

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❚r❛♥s♣♦rt❛✐ ✉♠ ♣✉♥❤❛❞♦ ❞❡ t❡rr❛ t♦❞♦s ♦s ❞✐❛s ❡ ❢❛r❡✐s ✉♠❛ ♠♦♥t❛♥❤❛✳ ❈♦♥❢ú❝✐♦

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❙❛♥❞r♦ ❱✐❡✐r❛ ❘♦♠❡r♦ ♣❡❧❛ s❡r✐❡❞❛❞❡ ❡ ❝♦♥st❛♥t❡ ❞✐s✲ ♣♦s✐çã♦ ❡♠ ♠❡ ❛❥✉❞❛r✱ ♣❡❧❛s ❞✐❝❛s ❞❡ ❧✐✈r♦s✱ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ❡♠ ♠❡✉s ♠♦♠❡♥t♦s ❞❡ ❞✐✜❝✉❧❞❛❞❡ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣♦r t♦❞♦ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ ❛❞q✉✐r✐ ❡st✉❞❛♥❞♦ ❝♦♠ ❡❧❡❀ ❛❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♦✉tr♦s ♠❡str❡s q✉❡ ♣❛rt✐❝✐♣❛r❛♠ ❞❡ t♦❞❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❞❡s❞❡ ♦ ❡♥s✐♥♦ ♠❛✐s ❜ás✐❝♦❀ ❛❣r❛❞❡ç♦ à ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ à ♠✐♥❤❛ ♠ã❡ ❆♥❣é❧✐❝❛ ❡ ♠✐♥❤❛ ✐r♠ã ❚❤❛❧✐❛♥❡ ♣♦r t♦❞♦ ♦ s✉♣♦rt❡ ❛♦ ❧♦♥❣♦ ❞❡ss❡ t❡♠♣♦❀ ❛❣r❛❞❡ç♦✱ ❡ ♠✉✐t♦✱ ❛♦s ❝♦❧❡❣❛s ❞♦ ♠❡str❛❞♦ q✉❡ ❢♦r❛♠ ❣r❛♥❞❡s ❝♦♠♣❛♥❤❡✐r♦s ❛♦ ❧♦♥❣♦ ❞❡ss❡ ❝r❡s❝✐♠❡♥t♦✱ ❡♠ ♣❛rt✐❝✉❧❛r à ❛♠✐❣❛ ❆♥♥❛ P❛✉❧❛ ▼❛❝❤❛❞♦ q✉❡ s❡♠♣r❡ ❡st✉❞♦✉ ❝♦♠✐❣♦ ❡ q✉❡ ❛❝r❡❞✐t♦ s❡r ✉♠❛ ❞❛s ♠❛✐s ❜r✐❧❤❛♥t❡s ♣❡ss♦❛s q✉❡ ❝♦♥❤❡ç♦❀ ❛❣r❛❞❡ç♦ ✐♠❡♥s❛♠❡♥t❡ ❛♦ ♠❡✉ ✏✐r♠ã♦✑ P❛✉❧♦ ❩❛♥✲ ❝❤❡tt❛✭P✉❧✐♠✮ q✉❡ s❡♠♣r❡ ❢♦✐ ❡ s❡♠♣r❡ s❡rá ♠❡✉ ♣♦rt♦ s❡❣✉r♦❀ ❛❣r❛❞❡ç♦ à ❈❛♣❡s ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦ q✉❡ r❡❝❡❜✐ ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦ ❡ss❡ t❡♠♣♦ ❡♠ ♠✐♥❤❛ ♣❡sq✉✐s❛❀ ♣♦r ú❧t✐♠♦ ❡ ♥ã♦ ♠❡♥♦s ✐♠♣♦rt❛♥t❡✱ ❛❣r❛❞❡ç♦ ❛♦s ❛♠✐❣♦s q✉❡ ❞❡ ✉♠ ♠♦❞♦ ♦✉ ❞❡ ♦✉tr♦ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ q✉❡ ❡✉ t✐✈❡ss❡ ê①✐t♦ ❡♠ t✉❞♦ q✉❡ ❛❧♠❡❥❡✐✳

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❙✉♠ár✐♦

❘❡s✉♠♦ ✸

❆❜str❛❝t ✹

■♥tr♦❞✉çã♦ ✺

✶ ❖ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞ ✼

✶✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ❋♦r♠❛s ❋r❛❝❛ ❡ ❋♦rt❡ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ ❈♦♥s❡q✉ê♥❝✐❛s ■♠❡❞✐❛t❛s ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✹ ❈❛r❛❝t❡r✐③❛çã♦ ❞❡ ✉♠ ❊s♣❛ç♦ ▼étr✐❝♦ ❈♦♠♣❧❡t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✷ ❊❧❡♠❡♥t♦s ❞❡ ❆♥á❧✐s❡ ❈♦♥✈❡①❛ ✶✺

✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❆ ❉❡r✐✈❛❞❛ ❞❡ ●ât❡❛✉① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✸ ❈♦♥❥✉♥t♦s ❈♦♥✈❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✹ ❋✉♥❝✐♦♥❛✐s ❈♦♥✈❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✺ ❖ ❙✉❜❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠ ❋✉♥❝✐♦♥❛❧ ❈♦♥✈❡①♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✻ ❆ ❉❡r✐✈❛❞❛ ❉✐r❡❝✐♦♥❛❧ ▲❛t❡r❛❧ ❞❡ ✉♠ ❋✉♥❝✐♦♥❛❧ ❈♦♥✈❡①♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✼ ❖ ❙✉❜❞✐❢❡r❡♥❝✐❛❧ ❞❛ ◆♦r♠❛ ❞❡ ✉♠ ❊s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✸ ❊s♣❛ç♦s P♦❧✐♥♦r♠❛❞♦s ✷✽

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✷ ❙❯▼➪❘■❖

✸✳✷ ❙❡♠✐♥♦r♠❛s ❡ ❊s♣❛ç♦s P♦❧✐♥♦r♠❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✸ ❚♦♣♦❧♦❣✐❛ ❞❡ ✉♠ ❊s♣❛ç♦ P♦❧✐♥♦r♠❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✹ ❊s♣❛ç♦s ▲♦❝❛❧♠❡♥t❡ ❈♦♥✈❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✺ ❈♦♥t✐♥✉✐❞❛❞❡ ❞❡ ❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✻ ❋❛♠í❧✐❛s ❞❡ ❙❡♠✐♥♦r♠❛s ❊q✉✐✈❛❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✼ Ps❡✉❞♦♠étr✐❝❛s ❡ ▼étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✹ ❖ ❚❡♦r❡♠❛ ❞❡ ◆❛s❤✲▼♦s❡r ✹✶

✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✷ ❈♦♥s✐❞❡r❛çõ❡s ■♥✐❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✸ ❯♠ ❚❡♦r❡♠❛ ❞❡ ❋✉♥çã♦ ■♥✈❡rs❛ ♣❛r❛ ❊s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✹ ❯♠❛ ●❡♥❡r❛❧✐③❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ◆❛s❤✲▼♦s❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

❆♣ê♥❞✐❝❡ ✻✻

✹✳✺ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✹✳✻ ❆ ■♥t❡❣r❛❧ ❞❡ ❉❛♥✐❡❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✹✳✼ ❖ ❚❡♦r❡♠❛ ❞❡ ❇❡♣♣♦✲▲❡✈✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✹✳✽ ❖ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ▼♦♥ót♦♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✹✳✾ ❖ ▲❡♠❛ ❞❡ ❋❛t♦✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✹✳✶✵ ❖ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ❉♦♠✐♥❛❞❛ ❞❡ ▲❡❜❡s❣✉❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✹✳✶✶ ❆ ■♥t❡❣r❛❧ ❞❡ ❉❛♥✐❡❧❧ ♣❛r❛ ❙ér✐❡s ❞❡ ◆ú♠❡r♦s ❘❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✹✳✶✷ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s ♣❛r❛ ❙ér✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺

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❘❡s✉♠♦

❙❊❚❚❊✱ ❘♦❜❧ê❞♦ ▼❛❦✬s ▼✐r❛♥❞❛✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❋❡✈❡r❡✐r♦ ❞❡

2013. ❖ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞ ❡ ♦ ❚❡♦r❡♠❛ ❞❡ ◆❛s❤✲▼♦s❡r✳ ❖r✐❡♥t❛❞♦r✿ ❙❛♥❞r♦ ❱✐❡✐r❛ ❘♦♠❡r♦✳

◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s ♦s ❊s♣❛ç♦s ❞❡ ❋ré❝❤❡t ❡ ✉♠ t❡♦r❡♠❛ ❞❡ ❢✉♥çã♦ ✐♠♣❧í❝✐t❛ ♣❛r❛ ❛❧❣✉♥s ❞❡ss❡s ❡s♣❛ç♦s✱ ♦❜t✐❞♦ ♣♦r ❊❦❡❧❛♥❞ ❡♠ ✷✵✶✶✱ ❡ q✉❡ ✐♥❝❧✉✐ ❝♦♠♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ♦ t❡♦r❡♠❛ ❞❡ ◆❛s❤✲▼♦s❡r✳ ◆♦ ✐♥í❝✐♦ ❞❡ss❡ ❡st✉❞♦ ❛♣r❡s❡♥t❛♠♦s ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞✱ q✉❡ s❡rá ✉♠❛ ❢❡rr❛♠❡♥t❛ ❡ss❡♥❝✐❛❧ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛✳ ❊st✉❞❛♠♦s ❝♦♥❝❡✐t♦s ❡❧❡♠❡♥t❛r❡s ❞❡ ❆♥á❧✐s❡ ❈♦♥✈❡①❛ ❡ ❝❛r❛❝t❡r✐③❛♠♦s ♦ s✉❜❞✐❢❡r❡♥❝✐❛❧ ❞❛ ♥♦r♠❛ ❞❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ ❊ss❛ ❝❛r❛❝t❡r✐③❛çã♦ s❡rá ✉s❛❞❛ ✈ár✐❛s ✈❡③❡s ♥❛ ♣r♦✈❛ ❞♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧✳ ❆❧é♠ ❞✐ss♦✱ sã♦ ✉s❛❞♦s três r❡s✉❧t❛❞♦s ❞❛ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛✿ ♦ ▲❡♠❛ ❞❡ ❋❛t♦✉✱ ♦ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ▼♦♥ót♦♥❛ ❡ ♦ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ❉♦♠✐♥❛❞❛ ❞❡ ▲❡❜❡s❣✉❡✳ ❊st❡s três r❡s✉❧t❛❞♦s sã♦ ❡♠♣r❡❣❛❞♦s ❡♠ s✉❛s ✈❡rsõ❡s ♣❛r❛ sér✐❡s ❞❡ ♥ú♠❡r♦s r❡❛✐s✳

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❆❜str❛❝t

❙❊❚❚❊✱ ❘♦❜❧ê❞♦ ▼❛❦✬s ▼✐r❛♥❞❛✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❋❡❜r✉❛r② ♦❢

2013. ❖ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞ ❡ ♦ ❚❡♦r❡♠❛ ❞❡ ◆❛s❤✲▼♦s❡r✳ ❆❞✈✐s♦r✿ ❙❛♥❞r♦ ❱✐❡✐r❛ ❘♦♠❡r♦✳

■♥ t❤✐s t❡①t ✇❡ st✉❞② t❤❡ ❋ré❝❤❡t s♣❛❝❡s ❛♥❞ ❛♥ ✐♠♣❧✐❝✐t ❢✉♥❝t✐♦♥ t❤❡♦r❡♠ ❢♦r s♦♠❡ ♦❢ t❤❡s❡ s♣❛❝❡s✱ ♦❜t❛✐♥❡❞ ❜② ❊❦❡❧❛♥❞ ✐♥ ✷✵✶✶✱ ❛♥❞ ✇❤✐❝❤ ✐♥❝❧✉❞❡s ❛s ❛ s♣❡❝✐❛❧ ❝❛s❡ t❤❡ ◆❛s❤✲ ▼♦s❡r t❤❡♦r❡♠✳ ❆t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤✐s st✉❞② ✇❡ ♣r❡s❡♥t t❤❡ ❊❦❡❧❛♥❞✬s ♣r✐♥❝✐♣❧❡✱ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ❛♥ ❡ss❡♥t✐❛❧ t♦♦❧ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠✳ ❲❡ st✉❞② ❡❧❡♠❡♥t❛r② ❝♦♥❝❡♣ts ♦❢ ❈♦♥✈❡① ❆♥❛❧②s✐s ❛♥❞ ❝❤❛r❛❝t❡r✐③❡ t❤❡ s✉❜❞✐✛❡r❡♥t✐❛❧ ♦❢ t❤❡ ♥♦r♠ ♦❢ ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤✐s ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ✇✐❧❧ ❜❡ ✉s❡❞ r❡♣❡❛t❡❞❧② ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♠❛✐♥ r❡s✉❧t✳ ❆❧s♦✱ t❤r❡❡ r❡s✉❧ts ♦❢ t❤❡ ❚❤❡♦r② ♦❢ ▼❡❛s✉r❡ ❛r❡ ✉s❡❞✿ ❋❛t♦✉✬s ▲❡♠♠❛✱ t❤❡ ▼♦♥♦t♦♥❡ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠ ❛♥❞ t❤❡ ▲❡❜❡s❣✉❡✬s ❉♦♠✐♥❛t❡❞ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠✳ ❚❤❡s❡ t❤r❡❡ r❡s✉❧ts ❛r❡ ❡♠♣❧♦②❡❞ ✐♥ t❤❡✐r ✈❡rs✐♦♥s ❢♦r s❡r✐❡s ♦❢ r❡❛❧ ♥✉♠❜❡rs✳

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■♥tr♦❞✉çã♦

❊①✐st❡♠ ♣r♦❜❧❡♠❛s ♥❛ ❋ís✐❝❛✲▼❛t❡♠át✐❝❛ q✉❡ sã♦ ❡q✉✐✈❛❧❡♥t❡s ❛ ❡①✐stê♥❝✐❛ ❞❡ ❢✉♥çõ❡s ✐♥✈❡rs❛s ❡♠ ❡s♣❛ç♦s ❞❡ ❋ré❝❤❡t✳ Pr♦❜❧❡♠❛s ❝♦♠♦ ❡ss❡s✱ t❛♠❜é♠ ❡①✐st❡♠ ♥❛ ●❡♦♠❡tr✐❛ ❉✐❢❡r❡♥❝✐❛❧✳ P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ♠❡r❣✉❧❤♦s ✐s♦♠étr✐❝♦s ❞❡ ✈❛r✐❡❞❛❞❡s r✐❡♠❛♥♥✐❛♥❛s✱ ◆❛s❤ ❬✶✶❪ ✉s♦✉ ✉♠❛ té❝♥✐❝❛ ✐♥♦✈❛❞♦r❛ q✉❡ ▼♦s❡r ❬✶✵❪ r❡❢♦r♠✉❧♦✉ ❝♦♠♦ ✉♠ t❡♦r❡♠❛ ❛❜str❛t♦ ❞❡ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧ ◆ã♦ ▲✐♥❡❛r✳ ❙❡r❣❡r❛❡rt ❬✶✽❪ ❡s❝❧❛r❡❝❡✉ ♦ ❝♦♥t❡①t♦ ❞♦ t❡♦r❡♠❛ ❛♦ ❢♦r♠✉❧❛r s✉❛s ❤✐♣ót❡s❡s ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ ❞❡ ❛♣❧✐❝❛çõ❡ss ❡♥tr❡ ❡s♣❛ç♦s ❞❡ ❋ré❝❤❡t✳ ❯t✐❧✐③❛♥❞♦ t♦❞♦ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❛❝✉♠✉❧❛❞♦ ❛té ♦ ✐♥í❝✐♦ ❞❛ ❞é❝❛❞❛ ❞❡ ♦✐t❡♥t❛ ❞♦ sé❝✉❧♦ ❳❳✱ ❍❛♠✐❧t♦♥ ❬✹❪ ❡s❝r❡✈❡✉ ✉♠ ❧♦♥❣♦ tr❛t❛❞♦ s♦❜r❡ ♦ ❈á❧❝✉❧♦ ❡♠ ❡s♣❛ç♦s ❞❡ ❋ré❝❤❡t ❡ ♦ ❚❡♦r❡♠❛ ❞❡ ◆❛s❤✲▼♦s❡r q✉❡ é ✉t✐❧✐③❛❞♦ ♥❛ ♣r♦✈❛ ❞♦ t❡♦r❡♠❛ ❑❆▼ ❬✷✶❪✱ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❣r❛✈✐t❛❝✐♦♥❛✐s ❬✻❪✱ ♥♦ ❡st✉❞♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❢r♦♥t❡✐r❛ ❞♦ ❡❧❡tr♦♠❛❣♥❡t✐s♠♦ ❬✶✼❪ ❡ ♥❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞❡ ❡✈♦❧✉çã♦ ❞❛ ❋ís✐❝❛✲▼❛t❡♠át✐❝❛ ❬✶✸❪ ❡ ❬✶✹❪✳

❆ ❞❡♠♦♥str❛çã♦ ✉s✉❛❧ ❞♦ t❡♦r❡♠❛ ❞❡ ◆❛s❤✲▼♦s❡r é ❜❛st❛♥t❡ tr❛❜❛❧❤♦s❛ ❡ ❡♥✈♦❧✈❡ ✉♠ ♣r♦❝❡ss♦ ✐t❡r❛t✐✈♦ ♣❛r❛ s✉❛ ❝♦♥✈❡r❣ê♥❝✐❛✱ ♦ ♠ét♦❞♦ ❞❡ ◆❡✇t♦♥ ♣❛r❛ ❛♣r♦①✐♠❛r r❛í③❡s✳ ❊♠ ❬✼❪✱ ♣♦❞❡♠♦s ✈❡r ♦ ❡♥✉♥❝✐❛❞♦ ❜❡♠ ❝♦♠♦ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❞❡ ◆❛s❤✲▼♦s❡r✳ ❯♠ ❝♦♠♣♦♥❡♥t❡ ❡ss❡♥❝✐❛❧ ❞❛ ❞❡♠♦♥str❛çã♦ é ♦ ✉s♦ ❞❡ ♦♣❡r❛❞♦r❡s ❞❡ s✉❛✈✐③❛çã♦ ❬✾❪✱ q✉❡ sã♦ ♦❜❥❡t♦s q✉❡ ♥❡♠ t♦❞♦s ♦s ❡s♣❛ç♦s ❞❡ ❋ré❝❤❡t ♣♦ss✉❡♠✳ ❊♠ ✷✵✶✶✱ ■✈❛r ❊❦❡❧❛♥❞ ❬✶❪✱ ❞❡♠♦♥str♦✉ ✉♠ t❡♦r❡♠❛ ❞❡ ❢✉♥çã♦ ✐♠♣❧í❝✐t❛ ♣❛r❛ ❡s♣❛ç♦s ❞❡ ❋ré❝❤❡t q✉❡ ✐♥❝❧✉✐ ❝♦♠♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ♦ t❡♦r❡♠❛ ❞❡ ◆❛s❤✲▼♦s❡r✳ ◆❡st❛ ❞❡♠♦♥str❛çã♦✱ ❊❦❡❧❛♥❞ ✉s❛ ✉♠ r❡s✉❧t❛❞♦ ❞❛ ❆♥á❧✐s❡ ❱❛r✐❛❝✐♦♥❛❧ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞✳ ❙ã♦ ✉s❛❞♦s t❛♠❜é♠ três r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s ❞❛ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛✿ ♦ ❧❡♠❛ ❞❡ ❋❛t♦✉✱ ♦ t❡♦r❡♠❛ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♠♦♥ót♦♥❛ ❡ ♦ t❡♦r❡♠❛ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦♠✐♥❛❞❛ ❞❡ ▲❡❜❡s❣✉❡✳ ❈♦♠♣❛r❛❞❛ ❛♦ t❡①t♦ ❝❧áss✐❝♦ ❞❡ ❍❛♠✐❧t♦♥ ❬✺❪✱ ❛ ❛❜♦r❞❛❣❡♠ ❞❡ ❊❦❡❧❛♥❞ é ♠❛✐s s✐♠♣❧❡s✱ ❝♦♠ ❤✐♣ót❡s❡s ♠❛✐s ❢r❛❝❛s ❡ t❡s❡ ♠❛✐s ❢♦rt❡✳ ◆❡ss❛ ♥♦✈❛ ❞❡♠♦♥str❛çã♦ t❛♠❜é♠ sã♦ ✉s❛❞♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s r❡❧❛t✐✈❛♠❡♥t❡ s✐♠♣❧❡s ❞❡ ❛♥á❧✐s❡ ❝♦♥✈❡①❛✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ♦ s✉❜❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥✈❡①❛✱ ❡ ❛❧❣✉♠❛s ♥♦çõ❡s à r❡s♣❡✐t♦ ❞♦s ❡s♣❛ç♦s ❞❡ ❋ré❝❤❡t✳

❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡st❛ ❞✐ss❡rt❛çã♦ é ♦ ❡st✉❞♦ ❞❛ ✈❡rsã♦ ❞❡ ❊❦❡❧❛♥❞ ❞♦ ❚❡♦r❡♠❛ ❞❡ ◆❛s❤✲▼♦s❡r✱ ✉s❛❞♦ ♣❛r❛ ❣❛r❛♥t✐r ❛ ❡①✐stê♥❝✐❛ ❞❡ ❢✉♥çõ❡s ✐♥✈❡rs❛s ❡♥tr❡ ❡s♣❛ç♦s ❞❡ ❋ré❝❤❡t ❡ ❝♦♠ ✈ár✐❛s ❛♣❧✐❝❛çõ❡s ♥❛ ❋ís✐❝❛✲▼❛t❡♠át✐❝❛✳

❙❡❣✉❡ ✉♠❛ ❜r❡✈❡ ❞❡s❝r✐çã♦ ❞❡ ❝❛❞❛ ❝❛♣ít✉❧♦✿

◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ❞❡♠♦♥str❛♠♦s ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞ ❡ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s

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✻ ❙❯▼➪❘■❖

r❡s✉❧t❛❞♦s ❝❧áss✐❝♦s q✉❡ s✉r❣❡♠ ♥❛t✉r❛❧♠❡♥t❡ ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛s ❞❡ s✉❛ ✈❛❧✐❞❡③✳ ▼♦s✲ tr❛♠♦s ❛✐♥❞❛✱ q✉❡ ❡ss❡ ♣r✐♥❝í♣✐♦ é ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ ❛ ❝♦♠♣❧❡t❡③❛ ❞❡ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳

◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ✐♥tr♦❞✉③✐♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❡ ❆♥á❧✐s❡ ❈♦♥✈❡①❛ s❡❣✉✐♥❞♦ ❡s✲ s❡♥❝✐❛❧♠❡♥t❡ ❬✶✷❪✳ ❖ ❝❛♣ít✉❧♦ é ❡ss❡♥❝✐❛❧♠❡♥t❡ ❞❡❞✐❝❛❞♦ ❛ ❞❡♠♦♥str❛r ✉♠ ✐♠♣♦rt❛♥t❡ r❡s✉❧t❛❞♦ s♦❜r❡ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦ s✉❜❞✐❢❡r❡♥❝✐❛❧ ❞❛ ♥♦r♠❛ ❞❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳

❖ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ tr❛t❛ ❞❡ ❡s♣❛ç♦s ♣♦❧✐♥♦r♠❛❞♦s✳ ◆❡ss❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ✈ár✐♦s r❡s✉❧t❛❞♦s ❡ ✐♥tr♦❞✉③✐♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❡s♣❛ç♦s ❞❡ ❋ré❝❤❡t✳ ❊st✉❞❛♠♦s ♦s ❛s♣❡❝t♦s t♦♣♦✲ ❧ó❣✐❝♦s ❞❡ss❛ ❡str✉t✉r❛ ❡ ♠♦str❛♠♦s ❝r✐tér✐♦s q✉❡ ❧✐❣❛♠ ❛ t♦♣♦❧♦❣✐❛ ❞❡ss❡ ❡s♣❛ç♦ à ♠❛♥❡✐r❛ ♠❛✐s ♥❛t✉r❛❧ ❞❡ ❞❡✜♥✐r ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♥❡ss❡s ❡s♣❛ç♦s✳ ❆ ❡①♣♦s✐çã♦ s❡❣✉❡ ❜❛s✐❝❛♠❡♥t❡ ❬✺❪✳ ❆♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ✐♥t❡r❡ss❛♥t❡s ❞❡ss❛s ❡str✉t✉r❛s q✉❡ ❛s ❞✐❢❡r❡♥❝✐❛♠ ❞♦s ❡s✲ ♣❛ç♦s ♥♦r♠❛❞♦s✳ ❈❛r❛❝t❡r✐③❛♠♦s ♦s ❡s♣❛ç♦s ♣♦❧✐♥♦r♠❛❞♦s ♠❡tr✐③á✈❡✐s ❡ ❞❡✜♥✐♠♦s ❛ ❝❧❛ss❡ ❞♦s ❡s♣❛ç♦s ❞❡ ❋ré❝❤❡t✳

◆♦ q✉❛rt♦ ❝❛♣ít✉❧♦✱ ❞❡✜♥✐♠♦s ❛❧❣✉♥s ❡s♣❛ç♦s ♣❛rt✐❝✉❧❛r❡s ❞❡ ❋ré❝❤❡t✳ ❉❡ ♣♦ss❡ ❞❡ss❛ ♥♦✈❛ ❡str✉t✉r❛✱ ❛♣r❡s❡♥t❛♠♦s ♦ t❡♦r❡♠❛ ❞❡ ❢✉♥çã♦ ✐♠♣❧í❝✐t❛ q✉❡ ❢♦✐ ❞❡s❝♦❜❡rt♦ ♣♦r ❊❦❡✲ ❧❛♥❞ ❡♠ ✷✵✶✶✳ ▼♦str❛♠♦s t❛♠❜é♠ ✉♠ t❡♦r❡♠❛ ❞❡ ❢✉♥çã♦ ✐♠♣❧í❝✐t❛ ♣❛r❛ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ ❝♦♠ ❤✐♣ót❡s❡s ♠❛✐s ❢r❛❝❛s ❞♦ q✉❡ ❛s ✈❡rsõ❡s ❝❧áss✐❝❛s ❞❡ss❡ t✐♣♦ ❞❡ t❡♦r❡♠❛✳ ❆ ❞❡♠♦♥str❛çã♦ ❞♦ r❡s✉❧t❛❞♦ ❡♠ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ s❡ ❛ss❡♠❡❧❤❛ à ❞❡♠♦♥str❛çã♦ ♣❛r❛ ❡ss❡ t❡♦r❡♠❛ ❡♠ ❡s♣❛ç♦s ❞❡ ❋ré❝❤❡t✳ ❊♠ ❛♠❜♦s✱ ❤á t❛♥t♦ ♦ ❡♠♣r❡❣♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡✲ ❧❛♥❞ q✉❛♥t♦ ♦ ✉s♦ ❞❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦ s✉❜❞✐❢❡r❡♥❝✐❛❧ ❞❛ ♥♦r♠❛ ❡♠ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✳ ▼♦str❛♠♦s ❛❧❣✉♥s ❝♦r♦❧ár✐♦s ❡ ❛❧❣✉♠❛s s✐♥t❡t✐③❛çõ❡s ❞♦ t❡♦r❡♠❛ ❞❡ ❢✉♥çã♦ ✐♠♣❧í❝✐t❛ ♣❛r❛ ❡s♣❛ç♦s ❞❡ ❋ré❝❤❡t q✉❡ ♦ t♦r♥❛♠ ♠❛✐s ❝♦♠♣r❡❡♥sí✈❡❧ ❡ ♠❛✐s ❛♣❧✐❝á✈❡❧✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ t❡♦r❡♠❛ s❡rá ❞✐✈✐❞❛ ❡♠ três ♣❛ss♦s✳ ▼❛s ❡♠❜♦r❛ s❡❥❛ ✉♠ ♣♦✉❝♦ ❧♦♥❣❛✱ é ♠✉✐t♦ ♠❛✐s ❝♦♠♣r❡❡♥sí✈❡❧ q✉❡ ❛s ✈❡rsõ❡s ❝❧áss✐❝❛s ❞♦ t❡♦r❡♠❛ ❞❡ ◆❛s❤✲▼♦s❡r✳

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❈❛♣ít✉❧♦ ✶

❖ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞

✶✳✶ ■♥tr♦❞✉çã♦

◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞✱ ✉♠ r❡s✉❧t❛❞♦ ♠✉✐t♦ ✉s❛❞♦ ❡♠ ❛♥á❧✐s❡ ❝♦♥✈❡①❛✳ ▼♦str❛r❡♠♦s q✉❡ ❞❡ ♣♦ss❡ ❞❡ss❡ t❡♦r❡♠❛✱ ♣♦❞❡♠♦s ♣r♦✈❛r ♠✉✐t♦s ♦✉tr♦s t❡♦r❡♠❛s ❝♦♥❤❡❝✐❞♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✳ ▼♦str❛r❡♠♦s ❛✐♥❞❛ q✉❡ ❛ ✈❛❧✐❞❡③ ❞❡ss❡ ♣r✐♥❝í♣✐♦ ❡♠ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ é ❡q✉✐✈❛❧❡♥t❡ à s✉❛ ❝♦♠♣❧❡t❡③❛ ❬✶✾❪✳ ❆ ❡ssê♥❝✐❛ ❞❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ♣r✐♥❝í♣✐♦ ❡stá ♥♦ ❚❡♦r❡♠❛ ❞❛ ■♥t❡rs❡❝çã♦ ❞❡ ❈❛♥t♦r✳ ❆❧❣✉♠❛s ✈❡③❡s ✉s❛r❡♠♦s ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❛❜✉s✐✈❛ q✉❡ ♥ã♦ ❞✐❢❡r❡♥❝✐❛ ❢✉♥çõ❡s ❞❡ ❢✉♥❝✐♦♥❛✐s✳ ❖s ❝❛s♦s ♦♥❞❡ ❞❡ ❢❛t♦ ❡ss❡s ❝♦♥❝❡✐t♦s ♥ã♦ ❝♦✐♥❝✐❞❡♠ s❡rã♦ ❞❡st❛❝❛❞♦s ♣❡❧♦ ❝♦♥t❡①t♦✳ ❈♦♠❡❝❡♠♦s ❡♥tã♦ ♣♦r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡ ❞❡✜♥✐çõ❡s q✉❡ ♥♦s s❡rã♦ út❡✐s✿

❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ q✉❡ é ❍❛✉s❞♦r✛✳ ❯♠ ❢✉♥❝✐♦♥❛❧ Φ : X →

R_{∪ {∞}} é ❞✐t♦ s❡r s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡✱ ♦✉ ❛❜r❡✈✐❛❞❛♠❡♥t❡ s❝✐ s❡ ♣❛r❛ t♦❞♦

a∈R ♦ ❝♦♥❥✉♥t♦

{x∈X|Φ(x)> a}

é ❛❜❡rt♦✳

❚❡♦r❡♠❛ ✶✳✶✳ ❙❡❥❛♠X ✉♠ ❡s♣❛ç♦ ❍❛✉s❞♦r✛ ❝♦♠♣❛❝t♦ ❡Φ :X→R_∪{∞}✉♠ ❢✉♥❝✐♦♥❛❧ s❝✐✳ ❊♥tã♦✿

✐✮ Φ é ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡❀

✐✐✮ ❖ í♥✜♠♦ ❞❡ Φ(X) é ❛❧❝❛♥ç❛❞♦ ❡♠ ❛❧❣✉♠ ♣♦♥t♦ x0 ∈X✳

❉❡♠✳✿

(13)

✽ ✶✳✶✳ ■◆❚❘❖❉❯➬➹❖

✐✮ ◆♦t❡ q✉❡ ♣❛r❛ ❝❛❞❛ n∈N ♦ ❝♦♥❥✉♥t♦

An ={x∈X|Φ(x)>−n}

é ❛❜❡rt♦✳ ◆♦t❡ ❛✐♥❞❛ q✉❡

X = [

n∈N

An.

P❡❧❛ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡X✱ s❡❣✉❡ q✉❡ ❡①✐t❡ n0 ∈Nt❛❧ q✉❡

n0

[

i=1

Ai =X.

▲♦❣♦✱ Φ(x)>−n0✱ ∀x∈X✳

✐✐✮ ❙❡❥❛l = infXΦ✳ ❊♥tã♦✱ t❡♠♦s l >−∞✳ ❙❡ l ♥ã♦ ❢♦ss❡ ❛t✐♥❣✐❞♦✱ t❡rí❛♠♦s [

n∈N

{x∈X|Φ(x)> l+ 1

n}=X.

P❡❧❛ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡X✱ s❡❣✉❡ q✉❡ ❡①✐st❡ n1 ∈Nt❛❧ q✉❡

X =

n1

[

i=1

{x∈X|Φ(x)> l+ 1

n}.

❖✉ s❡❥❛✱ ♣❛r❛ t♦❞♦ x∈ X✱ t❡♠♦sΦ(x)> l+ 1

n1✳ ▼❛s ✐st♦ ❝♦♥tr❛r✐❛ ♦ ❢❛t♦ ❞❡

l s❡r ❛ ♠❛✐♦r ❞❛s ❝♦t❛s ✐♥❢❡r✐♦r❡s ❞❡ Φ(X)✳

❉❡✜♥✐çã♦ ✶✳✷✳ ❯♠ ❢✉♥❝✐♦♥❛❧ Φ : X → R_{∪ {∞}}✱ é ❞✐t♦ s❡q✉❡♥❝✐❛❧♠❡♥t❡ s❡♠✐❝♦♥✲ tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ s❡ ♣❛r❛ t♦❞❛ s❡q✉ê♥❝✐❛ (xn)n∈N ❝♦♠ limxn = x0 t❡♠♦s Φ(x0) ≤

lim inf Φ(xn)✳

❚❡♦r❡♠❛ ✶✳✷✳

✐✮ ❚♦❞♦ ❢✉♥❝✐♦♥❛❧ s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ Φ : X → R_{∪ {∞}} é s❡q✉❡♥❝✐❛❧♠❡♥t❡ s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡❀

✐✐✮ ❙❡X s❛t✐s❢❛③ ♦ Pr✐♠❡✐r♦ ❆①✐♦♠❛ ❞❛ ❊♥✉♠❡r❛❜✐❧✐❞❛❞❡✱ ❡♥tã♦ t♦❞♦ ❢✉♥❝✐♦♥❛❧ s❡q✉❡♥✲ ❝✐❛❧♠❡♥t❡ s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡✳

❉❡♠✳✿

✐✮ ❙❡❥❛♠ x0 ∈X ❡ (xn)n∈N⊂X t❛✐s q✉❡ xn→x0✳

Pr✐♠❡✐r♦✱ s✉♣♦♥❤❛♠♦s q✉❡ Φ(x0) < ∞✳ P❛r❛ ❝❛❞❛ ε > 0✱ ❝♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦

(14)

✾ ✶✳✷✳ ❋❖❘▼❆❙ ❋❘❆❈❆ ❊ ❋❖❘❚❊ ❉❖ P❘■◆❈❮P■❖ ❉❊ ❊❑❊▲❆◆❉

❡①✐st❡ n0 ∈ N t❛❧ q✉❡ xn ∈ Aε ♣❛r❛ t♦❞♦ n ≥ n0✳ P❛r❛ ❡ss❡s ✈❛❧♦r❡s ❞❡ n ≥ n0✱

t❡r❡♠♦sΦ(xn)>Φ(x0)−ε✱ ♦✉ s❡❥❛

Φ(xn)−Φ(x0)>−ε.

P♦rt❛♥t♦✱

lim inf Φ(xn)≥Φ(x0)−ε.

P❡❧❛ ❛r❜✐tr❛r✐❡❞❛❞❡ ❞❡ ε✱ s❡❣✉❡ q✉❡ Φ(x0)≤lim inf Φ(xn)✳ ❚r❛t❡♠♦s ❛❣♦r❛ ❞♦ ❝❛s♦

Φ(x0) = ∞✳ ❙❡❥❛ A = {x ∈ X|Φ(x) > M}✱ ♣❛r❛ ❛❧❣✉♠ M > 0 ❛r❜✐trár✐♦✳

Pr♦❝❡❞❡♥❞♦ ❞❡ ❢♦r♠❛ s✐♠✐❧❛r ❛❧❝❛♥ç❛♠♦s ❛ ❝♦♥❝❧✉sã♦ ❞❡s❡❥❛❞❛✳ ✐✐✮ ◗✉❡r❡♠♦s ♣r♦✈❛r q✉❡ ♦ ❝♦♥❥✉♥t♦

F ={x∈X|Φ(x)≤a}

é ❢❡❝❤❛❞♦✳ ❙✉♣♦♥❤❛ q✉❡ ✐st♦ ♥ã♦ ♦❝♦rr❡ss❡✳ ❊♥tã♦ ❡①✐st❡ x0 ∈ F −F✱ ♦✉ s❡❥❛✱

Φ(x0)> a✳ ❙❡❥❛On ✉♠❛ ❜❛s❡ ❡♥✉♠❡rá✈❡❧ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❛❜❡rt❛s ❞❡x0✳ P❛r❛ ❝❛❞❛

n∈N ❡①✐st❡ _xn_∈_F _{∩ O}_n ❡ ❧♦❣♦ _xn_→_x₀✳ ❈♦♠♦ _Φé s❡q✉❡♥❝✐❛❧♠❡♥t❡ s❡♠✐❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡

Φ(xn)≤a.

❈♦♥❝❧✉í♠♦s q✉❡Φ(x0)≤a✳

❈♦r♦❧ár✐♦ ✶✳✶✳ ❙❡Xé ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✱ ❡♥tã♦ ❛s ❞✉❛s ❞❡✜♥✐çõ❡s ❞❛❞❛s ♣❛r❛ ✉♠ ❢✉♥❝✐♦✲ ♥❛❧ s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ❡ s❡q✉❡♥❝✐❛❧♠❡♥t❡ s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ❝♦✐♥❝✐❞❡♠✳ ❉❡✜♥✐çã♦ ✶✳✸✳ ❙❡❥❛♠ Φ : X → R_{∪ {∞}} ❡ _x₀ _∈ _X✳ ❉✐③❡♠♦s q✉❡ _Φ é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ❡♠ x0 s❡ ♣❛r❛ t♦❞♦ a∈ R ❝♦♠ a < Φ(x0) ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ V

❞❡x0 t❛❧ q✉❡ a <Φ(x)♣❛r❛ t♦❞♦ x∈V✳

➱ ❢á❝✐❧ ✈❡r q✉❡ ✉♠ ❢✉♥❝✐♦♥❛❧ s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ✉♠ ❢✉♥❝✐♦♥❛❧ q✉❡ é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡✳ ❉❡✜♥✐çõ❡s ❡ ❛✜r♠❛çõ❡s s❡♠❡❧❤❛♥t❡s ♣♦❞❡♠ s❡r ❢❡✐t❛s ♣❛r❛ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ✐♥❢❡r✐♦r s❡q✉❡♥❝✐❛❧✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ s❡ ❞❡✜♥❡ s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r ✭s❝s✮ ❡ ✈❛❧❡♠ ♣r♦♣r✐❡❞❛❞❡s ❛♥á❧♦❣❛s✳

❖❜s❡r✈❛çã♦✿ ◆♦t❡ q✉❡ s❡ V+ _{❞❡♥♦t❛ t♦❞♦s ♦s ❢✉♥❝✐♦♥❛✐s} _scs _❡ _V− _{❞❡♥♦t❛ t♦❞♦s ♦s}

❢✉♥❝✐♦♥❛✐ssci ❛♠❜♦s s♦❜r❡ ♦ ♠❡s♠♦ ❝♦♥❥✉♥t♦ X✱ ❡♥tã♦ t❛♥t♦V+ q✉❛♥t♦V− sã♦ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❝♦♠ r❡❧❛çã♦ ❛s ♦♣❡r❛çõ❡s ❝❛♥ô♥✐❝❛s ❡ ❛✐♥❞❛ ♦ ❝♦♥❥✉♥t♦ V = C(X) ❞❛s ❢✉♥çõ❡s

❝♦♥tí♥✉❛s ❞❡ X ❡♠ R_{∪ {}₊_∞} é t❛❧ q✉❡ _V ₌_V+_∩_V−_✳

✶✳✷ ❋♦r♠❛s ❋r❛❝❛ ❡ ❋♦rt❡ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞

❚❡♦r❡♠❛ ✶✳✸ ✭Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞ ✲ ❋♦r♠❛ ❋r❛❝❛✮✳ ❙❡❥❛♠ (X, d) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦

(15)

✶✵ ✶✳✷✳ ❋❖❘▼❆❙ ❋❘❆❈❆ ❊ ❋❖❘❚❊ ❉❖ P❘■◆❈❮P■❖ ❉❊ ❊❑❊▲❆◆❉

❖❜s❡r✈❛çã♦✳✿ ❆ ♣❛rt✐r ❞❛ ❢♦r♠❛ ❢r❛❝❛✱ ♣♦❞❡♠♦s ❞❡❞✉③✐r ❞✐✈❡rs❛s ✈❛r✐❛♥t❡s ❞♦ Pr✐♥❝í✲ ♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞✳ ❆ ♣r✐♠❡✐r❛ ❡ t❛❧✈❡③ ✉♠❛ ❞❛s ♠❛✐s ✉s❛❞❛s é ❛ s❡❣✉✐♥t❡✿ ❆❞✐❝✐♦♥❛✲s❡ à ❢♦r♠❛ ❢r❛❝❛ ❞❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡ ❛ s❡❣✉✐♥t❡ ❤✐♣ót❡s❡✿ ∀ε > 0✱ s❡❥❛ x0 ∈ X

t❛❧ q✉❡ Φ(x0) <infX Φ +ε. ◆❡st❡ ❝❛s♦✱ é ♣♦ssí✈❡❧ s❡ ♦❜t❡r ✉♠ ❡❧❡♠❡♥t♦ x ∈ X t❛❧ q✉❡✿

Φ(x)≤Φ(x0) ❡ ❛✐♥❞❛Φ(x)<Φ(x) +εd(x, x)✱ ♣❛r❛ q✉❛❧q✉❡r x∈X✳

❆ ❢♦r♠❛ ❢r❛❝❛ ❞❡st❡ t❡♦r❡♠❛ é s✉✜❝✐❡♥t❡ ♣❛r❛ ❞❡♠♦♥str❛r ❛ ❢♦r♠❛ ❢♦rt❡✳

❚❡♦r❡♠❛ ✶✳✹ ✭Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞ ✲ ❋♦r♠❛ ❋♦rt❡✮✳ ❙❡❥❛♠ (X, d) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦

❝♦♠♣❧❡t♦ ❡ Φ : X → R_{∪ {}₊_∞} ✉♠ ❢✉♥❝✐♦♥❛❧ sci✱ ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡ ❡ ♥ã♦ ✐❞❡♥t✐✲ ❝❛♠❡♥t❡ ✐♥✜♥✐t♦✳ ❉❛❞♦ ε > 0✱ s❡❥❛ x ∈ X t❛❧ q✉❡ Φ(x) ≤ infX Φ +ε. ❊♥tã♦✱ ♣❛r❛ ❝❛❞❛ λ >0✱ ❡①✐st❡ xλ ∈X t❛❧ q✉❡ ✿

✐✮ Φ(xλ)≤Φ(x);

✐✐✮ d(xλ, x)≤λ;

✐✐✐✮ Φ(xλ)<Φ(x) + ε

λd(x, xλ) ♣❛r❛ t♦❞♦ x6=xλ ❉❡♠✳✿

❖ ❝♦♥❥✉♥t♦ Y = {y ∈ X|Φ(y) +d(x0, y) ≤ Φ(x0)} é ❢❡❝❤❛❞♦ ❡♠ X✱ ♣♦✐s d(x0,·) é

❝♦♥tí♥✉♦ ❡ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱ X −Y = {y ∈X|Φ(y) +d(x0, y)> Φ(x0)} é ❛❜❡rt♦ ❡♠ X✳

▲♦❣♦✱ ❝♦♠ ❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛ ❞❡X✱Y t❛♠❜é♠ é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✳ ❆♣❧✐❝❛♥❞♦ ❛ ❢♦r♠❛ ❢r❛❝❛ ♣❛r❛ ❡st❡ ❡s♣❛ç♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❡①✐st❡x∈Y t❛❧ q✉❡ Φ(x)<Φ(y) +d(y, x)

q✉❛❧q✉❡r q✉❡ s❡❥❛ y ∈ Y − {x0} ❡ ❛✐♥❞❛ Φ(x) +d(x, x0) ≤ Φ(x0)✳ ❊♠ ♣❛rt✐❝✉❧❛r t❡♠♦s✿

|Φ(x)−Φ(x0)| ≤ ε✱ ♣♦✐s s❡ ♥ã♦ ❢♦ss❡ ❛ss✐♠✱ t❡rí❛♠♦s infXΦ +ε > Φ(x0) > ε+ Φ(x) ❡

♣♦r ❝♦♥s❡❣✉✐♥t❡ Φ(x) < infXΦ✳ ❊st❛ ❝♦♥tr❛❞✐çã♦ ♥♦s ❝♦♥❞✉③ à ♥♦ss❛ t❡s❡✳ ◆♦t❡ ❛❣♦r❛ q✉❡|Φ(x0)−Φ(x)|=d(x0, x)≤ε✳ ❆♣❧✐❝❛♥❞♦ ♦ ú❧t✐♠♦ r❡s✉❧t❛❞♦ ♣❛r❛ ❛ ♠étr✐❝❛ γ.d✱ ❝♦♠

γ >0✱ ♦❜t❡♠♦s ❛ ❢♦r♠❛ ❢♦rt❡ ❝♦♠x0 t♦♠❛❞♦ ❝♦♠♦ x ❡ xt♦♠❛❞♦ ❝♦♠♦ xλ.

Pr♦✈❡♠♦s ❛❣♦r❛ ❛ ❢♦r♠❛ ❢r❛❝❛ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞

❈♦♠♦ infXΦ> −∞✱ ❡①✐st❡ x0 ∈ X t❛❧ q✉❡ Φ(x0)< infXΦ + 1✳ ❈♦♥str✉✐r❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ (xi)i∈N ⊂X ❡ ✉♠❛ ❝❛❞❡✐❛ ❞❡❝r❡s❝❡♥t❡ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❢❡❝❤❛❞♦s {Si}i∈N ❞❡X

❛tr❛✈❡s ❞♦ s❡❣✉✐♥t❡ ♣r♦❝❡❞✐♠❡♥t♦ r❡❝✉rs✐✈♦✿ s✉♣♦♥❤❛ q✉❡xj s❡❥❛ ❝♦♥❤❡❝✐❞♦✳ ❉❡✜♥❛ ❡♥tã♦ Sj :={x∈ X|Φ(xj)≥Φ(x) +d(x, xj)} ❡ t♦♠❡ ❡♥tã♦ xj+1 ❝♦♠♦ s❡♥❞♦ ♦ ❡❧❡♠❡♥t♦ ❞❡ Sj

q✉❡ s❛t✐s❢❛③

Φ(xj+1)<inf

Sj

Φ + 1

j+ 1.

❖❜✈✐❛♠❡♥t❡✱ ♣❛r❛ ❝❛❞❛ j ∈ N✱ t❡♠♦s _Sj ₆₌ _∅✱ ♣♦✐s _xj _∈ Sj✳ ❈♦♠♦ _Φ(_·_{) +}_d₍_·_{, xj}₎ é ✉♠❛ ❛♣❧✐❝❛çã♦sci✱ ❡♥tã♦ ❝❛❞❛Sj é ❢❡❝❤❛❞♦✳ ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ♣❛r❛ ❝❛❞❛j = 0,1,2, ... t❡♠♦sSj+1 ⊂Sj✳ ❉❡ ❢❛t♦✱ ✜①❛❞♦j ∈N✱ s❡❥❛x∈Sj+1✳ ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱

x∈Sj+1 ❡ q✉❡ xj+1∈Sj✱ ♦❜t❡♠♦s

(16)

✶✶ ✶✳✷✳ ❋❖❘▼❆❙ ❋❘❆❈❆ ❊ ❋❖❘❚❊ ❉❖ P❘■◆❈❮P■❖ ❉❊ ❊❑❊▲❆◆❉

■st♦ ♠♦str❛ q✉❡ x∈Sj✳ P♦rt❛♥t♦✱ Sj+1 ⊂Sj ❝♦♠♦ ❞❡s❡❥á✈❛♠♦s ♠♦str❛r✳ ❆❧é♠ ❞✐ss♦✱

diam(Sj) ≤ 2

j✱ ♣❛r❛ j = 1,2,3, ...✳ ❈♦♠ ❡❢❡✐t♦✱ ❡s❝♦❧❤✐❞♦ j ∈ {1,2,3,4, ...}✱ s❡❥❛ x ∈ Sj✳ ❙❛❜❡♠♦s q✉❡x∈Sj−1✳ ❈♦♠♦Φ(xj)<infSjΦ +

1

j✱ ❡♥tã♦ d(x, xj)≤Φ(xj)−Φ(x)≤Φ(xj)− inf

Sj−1

Φ< 1 j.

❈♦♠♦ d(x, y) ≤ d(x, xj) +d(y, xj) < 1 j +

1

j =

2

j✱ ❡♥tã♦ ❞❡ ❢❛t♦ t❡♠♦s diam(Sj) ≤

2

j✳ ❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ■♥t❡rs❡❝çã♦ ❞❡ ❈❛♥t♦r✱ ❝♦♥❝❧✉í♠♦s q✉❡

\

i≥1

Si ={x}

❝♦♠ x∈X✳ ❱❛♠♦s ❛❣♦r❛ ♣r♦✈❛r q✉❡

Φ(x)<Φ(x) +d(x, x)

♣❛r❛ t♦❞♦x6=x✳

❉❡ ❢❛t♦✱ s❡ ✐st♦ ♥ã♦ ♦❝♦rr❡ss❡ t❡rí❛♠♦s✱ ♣❛r❛ ❛❧❣✉♠x∈X−{x}✱Φ(x)−d(x, x)≥Φ(x)✳

❆ss✐♠✱ ♣❛r❛ ❝❛❞❛ j ∈N✱ t❡♠♦s✿

Φ(x) +d(x, x)≤Φ(x)≤Φ(xj)−d(x, xj)

❡ ♣♦rt❛♥t♦✱

Φ(x)≤Φ(xj)−d(x, x)−d(x, xj)≤Φ(xj)−d(x, xj)

♣❛r❛ t♦❞♦j ∈N✳ ■st♦ é✱

x∈ [

i≥0

Si ={x}.

◆♦ ❡♥t❛♥t♦✱ x6=x✳ ❊st❛ ❝♦♥tr❛❞✐çã♦ ❝♦♥❝❧✉✐ ♥♦ss❛ ❛✜r♠❛çã♦✳

(17)

✶✷ ✶✳✸✳ ❈❖◆❙❊◗❯✃◆❈■❆❙ ■▼❊❉■❆❚❆❙ ❉❖ P❘■◆❈❮P■❖ ❉❊ ❊❑❊▲❆◆❉

❖ ❝♦♥❡ s✉♣❡r✐♦r q✉❡ t❡♠ ❣❡r❛tr✐③ ❞❡ ✐♥❝❧✐♥❛çã♦ ε

λ ♣♦❞❡ s❡r ♠♦✈✐❞♦ ♣❛r❛ ✉♠❛ ♣♦s✐çã♦ t❛❧ q✉❡ s❡✉ ✈ért✐❝❡ s❡❥❛ ♦ ú♥✐❝♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❝♦♠ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦✳ ❊st❡ ♠♦✈✐♠❡♥t♦ é ❢❡✐t♦ ❢❛③❡♥❞♦✲s❡ ✉♠❛ tr❛♥s❧❛çã♦ ❤♦r✐③♦♥t❛❧ s❡❣✉✐❞❛ ❞❡ ✉♠❛ tr❛♥s❧❛çã♦ ✈❡rt✐❝❛❧ ❛té ❢❛③❡r ❝♦♠ q✉❡ ♦ ✈ért✐❝❡ ❝♦✐♥❝✐❞❛ ❝♦♠ ♦ ♣♦♥t♦ ❢♦r♠❛❞♦ ♣♦r xλ ❡ s✉❛ ✐♠❛❣❡♠ ♣❡❧❛ ❢✉♥çã♦✳ ■st♦ ♥♦s ❞✐③ q✉❡ ❛ r❡❣✐ã♦ ✐♥t❡r♥❛ ❛♦ ❝♦♥❡ é ✉♠❛ r❡❣✐ã♦ q✉❡ ✜❝❛ ✏✐s♦❧❛❞❛✑ ❞♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦✳

✶✳✸ ❈♦♥s❡q✉ê♥❝✐❛s ■♠❡❞✐❛t❛s ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞

❚❡♦r❡♠❛ ✶✳✺ ✭❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❈❛r✐st✐✮✳ ❙❡❥❛ ψ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦ X ❡♠ s✐ ♠❡s♠♦✳ ❙❡ ♣❛r❛ t♦❞♦ x ∈ X t❡♠✲s❡ d(x, ψ(x)) ≤ Φ(x)−

Φ(ψ(x))✱ ♣❛r❛ ❛❧❣✉♠ ❢✉♥❝✐♦♥❛❧ Φ sci ❡ ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡✱ ❡♥tã♦ ψ t❡♠ ✉♠ ♣♦♥t♦ ✜①♦✳

❉❡♠✳✿

P❡❧❛ ❢♦r♠❛ ❢r❛❝❛ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞✱ ❡①✐st❡x∈X t❛❧ q✉❡ Φ(x)<Φ(x) +d(x, x)✱

♣❛r❛ t♦❞♦ x ∈ X − {x}✳ ▼❛s ♣❡❧❛ ❤✐♣ót❡s❡ ❞♦ t❡♦r❡♠❛✱ Φ(x) ≥ Φ(ψ(x)) +d(x, ψ(x))✳

P♦rt❛♥t♦✱ ψ(x) ♥ã♦ ♣❡rt❡♥❝❡ ❛X− {x}✱ ♦✉ s❡❥❛✱ ψ(x) =x✳

❚❡♦r❡♠❛ ✶✳✻ ✭❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✮✳ ❙❡❥❛ (X, d) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠✲

♣❧❡t♦✱ ❡ s❡❥❛ ψ : X → X ✉♠❛ ❝♦♥tr❛çã♦✱ ✐st♦ é✱ d(ψ(x), ψ(y)) ≤ kd(x, y) ♣❛r❛ t♦❞♦s

x, y ∈X ❡ 0≤k < 1✳ ❊♥tã♦ ψ t❡♠ ✉♠ ♣♦♥t♦ ✜①♦✳ ❉❡♠✳✿

❉❡ ❢❛t♦✱ ❞❡✜♥✐♥❞♦ Φ(x) = 1

1−kd(x, ψ(x))✱ ❛s ❝♦♥❞✐çõ❡s ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r sã♦ s❛t✐s❢❡✐t❛s ❡ ♣♦rt❛♥t♦✱ψ t❡♠ ✉♠ ♣♦♥t♦ ✜①♦✳

❚❡♦r❡♠❛ ✶✳✼ ✭❚❡♦r❡♠❛ ❞❛ ❊①✐stê♥❝✐❛ ❞❡ ❚❛❦❛❤❛s❤✐✮✳ ❙❡❥❛♠ (X, d) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦

❝♦♠♣❧❡t♦ ❡ Φ :X →R ✉♠ ❢✉♥❝✐♦♥❛❧ _sci✱ ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡✳ ❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ t♦❞♦ x ∈ X s❛t✐s❢❛③❡♥❞♦ Φ(x) > infXΦ✱ ❡①✐st❛ y ∈ X − {x} t❛❧ q✉❡ Φ(y) < Φ(x)−d(x, y)✳ ❊♥tã♦ Φ ♣♦ss✉✐ ✉♠ ♠í♥✐♠♦✳

❉❡♠✳✿

❱❛♠♦s s✉♣♦r q✉❡ ❛ ❛✜r♠❛çã♦ s❡❥❛ ❢❛❧s❛✱ ♦✉ s❡❥❛✱ q✉❡ ♣❛r❛ t♦❞♦ x ∈ X✱ t✐✈éss❡♠♦s

Φ(x) > infXΦ✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ x ∈ X✱ ❡①✐st❡ yx ∈ X − {x} t❛❧ q✉❡ Φ(yx) < Φ(x)− d(x, yx)✳ P❡❧♦ Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞✱ ❡①✐st❡ x ∈ X t❛❧ q✉❡ Φ(x) < Φ(z) +

d(z, x)✱ ♣❛r❛ t♦❞♦ z ∈ X − {x}✳ ❈♦♠♦ x 6= yx✱ ❡♥tã♦ ❢❛③❡♥❞♦ yx = z✱ t❡♠♦s Φ(x) <

(18)

✶✸ ✶✳✹✳ ❈❆❘❆❈❚❊❘■❩❆➬➹❖ ❉❊ ❯▼ ❊❙P❆➬❖ ▼➱❚❘■❈❖ ❈❖▼P▲❊❚❖

✶✳✹ ❈❛r❛❝t❡r✐③❛çã♦ ❞❡ ✉♠ ❊s♣❛ç♦ ▼étr✐❝♦ ❈♦♠♣❧❡t♦

▼♦str❛r❡♠♦s ❛❣♦r❛✱ q✉❡ ❛ ❝♦♠♣❧❡t❡③❛ ❞♦ ❡s♣❛ç♦ ♠étr✐❝♦(X, d)é ❡q✉✐✈❛❧❡♥t❡ à ✈❛❧✐❞❡③

❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞ ♥❡st❡ ❡s♣❛ç♦ ❝♦♠♦ ♠♦str❛❞♦ ♣♦r ❙✉❧❧✐✈❛♥ ❬✶✾❪✳

❚❡♦r❡♠❛ ✶✳✽✳ ❙❡❥❛ (X, d) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳ ❊♥tã♦ X é ❝♦♠♣❧❡t♦ s❡✱ ❡ s♦♠❡♥t❡ s❡ t♦❞♦ ❢✉♥❝✐♦♥❛❧ ❝♦♥tí♥✉♦ F :X →R₊✱ ♥ã♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ✐♥✜♥✐t♦✱ ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡ ❡ ♣❛r❛ t♦❞♦ ε >0✱ ❡①✐st❡ ✉♠ ♣♦♥t♦ v ∈X s❛t✐s❢❛③❡♥❞♦✿

✐✮ F(v)≤infMF +ε❀

✐✐✮ F(w) +εd(v, w)> F(v), ∀w6=v. ❉❡♠✳✿

❆ ❞✐r❡çã♦ ✏s♦♠❡♥t❡ s❡✑ s❡❣✉❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❊❦❡❧❛♥❞✳ P❛r❛ ❛ r❡❝í♣r♦❝❛✱ ❛ss✉♠❛ q✉❡ (X, d) é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❛r❜✐trár✐♦ ❡♠ q✉❡ ✈❛❧❡♠ ❛s ❤✐♣ót❡s❡s i) ❡ ii) ❞♦

t❡♦r❡♠❛✳ ❙❡❥❛(yn)⊂X ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ (yn)t❡♠ ❧✐♠✐t❡

❡♠ X✳ ❈♦♥s✐❞❡r❡ ♦ ❢✉♥❝✐♦♥❛❧ F :X →R✱ ❞❛❞♦ ♣♦r

F(x) = lim

n→∞d(yn, x).

▼♦str❡♠♦s q✉❡ F é ❝♦♥tí♥✉♦✳ ❉❡ ❢❛t♦✱ s❡❥❛ (xm) ⊂ X t❛❧ q✉❡ xm → x ∈ X✳ ➱ s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ F(xm)→F(x)✳ P❛r❛ ❝❛❞❛ ♣❛r✱m, n∈N✱ ✈❛❧❡♠✿

0≤d(xm, yn)≤d(xm, x) +d(yn, x) ✭✶✳✶✮

❡

0≤d(yn, x)≤d(yn, xm) +d(xm, x) ✭✶✳✷✮

❋✐①❛♥❞♦ m ❡ t♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ ❝♦♠ n → ∞❡♠ (1.1) ❡(1.2)✱ t❡♠♦s✿

|F(xm)−F(x)| ≤d(xm, x).

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ F(xm)→F(x)✱ ❡ ♣♦rt❛♥t♦✱F é ❝♦♥tí♥✉♦✳

◆♦t❡ q✉❡F é ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r0✳ ▼♦str❡♠♦s q✉❡✱ ❝♦♠ ♠❛✐s ❡❢❡✐t♦✱inf

X F = 0. ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛ q✉❡ infXF =c >0✳ ❈♦♠♦ (yn) é ❞❡ ❈❛✉❝❤②✱ ❡①✐st❡ N0 ∈N t❛❧ q✉❡

s❡ m, n≥ N0✱ t❡♠✲s❡ d(ym, yn) <

c

2✳ ❉❛í✱ F(ym)<

c

2. ▼❛s ✐st♦ ❝♦♥tr❛r✐❛ ♦ ❢❛t♦ ❞❡ c s❡r

(19)

✶✹ ✶✳✹✳ ❈❆❘❆❈❚❊❘■❩❆➬➹❖ ❉❊ ❯▼ ❊❙P❆➬❖ ▼➱❚❘■❈❖ ❈❖▼P▲❊❚❖

P❛r❛ ♠♦str❛r q✉❡ X é ❝♦♠♣❧❡t♦✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ F t❡♠ ✉♠ ♠í♥✐♠♦ ❡♠ ❛❧❣✉♠ v ∈X✱ ♦✉ s❡❥❛ F(v) = 0✳

❋✐①❡♠♦s ❡♥tã♦ ✉♠ ♥ú♠❡r♦ 0< ε < 1✳

❉❡ i) ❡ ii)✱ ❡①✐st❡ v ∈X ❝♦♠ F(v)≤ε ❡ F(w) +εd(v, w)> F(v), ∀w6=v. ❋✐①❛♥❞♦ n∈N✱ t❡♠♦s✿

d(v, w)≤d(yn, v) +d(yn, w). ✭✶✳✸✮ ❚♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ ❝♦♠ n → ∞❡♠ (1.3)✱ t❡♠♦s✿

d(v, w)≤F(v) +F(w).

❉❛❞♦ η >0q✉❛❧q✉❡r✱ é ♣♦ssí✈❡❧ s❡ ❡s❝♦❧❤❡r w∈X t❛❧ q✉❡ F(w)≤η✳

❉❡ ❢❛t♦✱ ❝♦♠♦ (yn) é ❞❡ ❈❛✉❝❤②✱ ❡①✐st❡ N ∈ N ❞❡ ♠♦❞♦ q✉❡ s❡ _{m, n} _≥ _N✱ ❡♥tã♦ d(yn, ym)< η✳

❙❡❥❛ ❡♥tã♦✱ w=yN+1✳ ❊♥tã♦

F(w) =F(yN+1)< η.

P♦rt❛♥t♦✱ t♦♠❛♥❞♦✲s❡ ♦ ❧✐♠✐t❡ ❝♦♠ n→ ∞ ❡♠ (1.3)t❡♠♦s✿

d(v, w)≤F(v) +F(w)≤ε+η.

❉❛ ❝♦♥❞✐çã♦ ii)✱ ♣❛r❛ w=yN+1✱ t❡♠♦s✿

F(v)< F(w) +εd(w, v)≤η+ε(ε+η) =η+ε2+εη. P❡❧❛ ❛r❜✐tr❛r✐❡❞❛❞❡ ❞❡ η >0✱ s❡❣✉❡ q✉❡ F(v)≤ε2✳

❘❡♣❡t✐♥❞♦✲s❡ ❡ss❡ ♠❡s♠♦ ❛r❣✉♠❡♥t♦ ❝♦♠ ♦✉tr♦ ❡❧❡♠❡♥t♦ w′ _{❝♦♥✈❡♥✐❡♥t❡✱ ❝♦♥❝❧✉✐✲s❡}

q✉❡ F(v)≤ε3✳

(20)

❈❛♣ít✉❧♦ ✷

❊❧❡♠❡♥t♦s ❞❡ ❆♥á❧✐s❡ ❈♦♥✈❡①❛

✷✳✶ ■♥tr♦❞✉çã♦

◆❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉③✐r❡♠♦s ❝♦♥❝❡✐t♦s ❡ ❛♣r❡s❡♥t❛r❡♠♦s r❡s✉❧t❛❞♦s q✉❡ ♥♦s s❡rã♦ ♠✉✐t♦ út❡✐s✳ ❋❛r❡♠♦s ✉♠ tr❛t❛♠❡♥t♦ s♦❜r❡ s✉❜❞✐❢❡r❡♥❝✐❛✐s ❞❡ ❢✉♥çõ❡s ❝♦♥✈❡①❛s ❡ ❡st❛r❡✲ ♠♦s ❡♠ ❝♦♥❞✐çõ❡s ❞❡ ❞❡s❝r❡✈❡r ♦s s✉❜❣r❛❞✐❡♥t❡s ❞❡ ❛❧❣✉♠❛s ❞❡ss❛s ❢✉♥çõ❡s✳ ❊st❡ ❝❛♣ít✉❧♦ s❡❣✉❡ ❜❛s✐❝❛♠❡♥t❡ ❬✶✷❪✳

✷✳✷ ❆ ❉❡r✐✈❛❞❛ ❞❡ ●ât❡❛✉①

❉❡✜♥✐çã♦ ✷✳✶✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦✱X∗ _{♦ s❡✉ ❡s♣❛ç♦ ❞✉❛❧ t♦♣♦❧ó❣✐❝♦✱}

Φ :X →R ❡ _p_∈_X∗✳ ❉✐③❡♠♦s q✉❡_p é ❛ ❞❡r✐✈❛❞❛ ❞❡ ●ât❡❛✉① ❞❡_Φ ❡♠ _x s❡✿

lim

h→0

Φ(x+hv)−Φ(x)− hp, hvi

h = 0,

q✉❛❧q✉❡r q✉❡ s❡❥❛ v ∈X✳ ❖♥❞❡h·,·i:X×X∗ _→_R _{❞❡✜♥✐❞❛ ♣♦r} _h_{v, p}_i₌_p₍_v₎_✳

❆ ❞❡r✐✈❛❞❛ p t❛♠❜é♠ é ❝❤❛♠❛❞❛ ❞❡ ❣r❛❞✐❡♥t❡ ❞❡ Φ ❡ é ✐♥❞✐❝❛❞❛ ♣♦r Φ′ _♦✉ _∇_Φ_{✳ ❙❡}

Φ ❢♦r ❞✐❢❡r❡♥❝✐á✈❡❧ s❡❣✉♥❞♦ ❋ré❝❤❡t ❡❧❛ t❛♠❜é♠ s❡rá ❞✐❢❡r❡♥❝✐á✈❡❧ s❡❣✉♥❞♦ ●ât❡❛✉① ❡ ❛s

❞❡r✐✈❛❞❛s ❞❛s ❞✉❛s ❞❡✜♥✐çõ❡s ❝♦✐♥❝✐❞✐rã♦✳ ❈♦♥t✉❞♦✱ ❛ ❡①✐stê♥❝✐❛ ❞❛ ❞❡r✐✈❛❞❛ ❞❡ ●ât❡❛✉① ♥ã♦ ❣❛r❛♥t❡ s❡q✉❡r ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ Φ✳

(21)

✶✻ ✷✳✸✳ ❈❖◆❏❯◆❚❖❙ ❈❖◆❱❊❳❖❙

✷✳✸ ❈♦♥❥✉♥t♦s ❈♦♥✈❡①♦s

❉❡✜♥✐çã♦ ✷✳✷✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ C ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ é ❞✐t♦ ❝♦♥✈❡①♦ s❡✱ ♣❛r❛ q✉❛✐s✲ q✉❡r x, y ∈X ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛

[x, y] ={(1−t)x+ty|t∈[0,1]}

❡stá ❝♦♥t✐❞♦ ❡♠C✳

❊①❡♠♣❧♦ ❞❡ ❈♦♥❥✉♥t♦ ❈♦♥✈❡①♦✿

x

y

❊①❡♠♣❧♦ ❞❡ ❈♦♥❥✉♥t♦ ◆ã♦ ❈♦♥✈❡①♦✿

x

y

✷✳✹ ❋✉♥❝✐♦♥❛✐s ❈♦♥✈❡①♦s

❉❡✜♥✐çã♦ ✷✳✸✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❯♠ ❢✉♥❝✐♦♥❛❧ Φ : X → R_{∪ {}₊_∞} é ❞✐t♦ ❝♦♥✈❡①♦ s❡

Φ((1−t)x+ty)≤(1−t)Φ(x) +tΦ(y)

♣❛r❛ ❝❛❞❛ t ∈[0,1]✳

❖ ❝♦♥❥✉♥t♦ Dom(Φ) = {x ∈ X|Φ(x) ∈ R_} é ❝❤❛♠❛❞♦ ❞❡ ❉♦♠í♥✐♦ ❊❢❡t✐✈♦ ❞❡

(22)

✶✼ ✷✳✹✳ ❋❯◆❈■❖◆❆■❙ ❈❖◆❱❊❳❖❙

x, y ∈Dom(Φ)✱ x6=y ❡ ❛✐♥❞❛ t∈(0,1)✳ ➱ ó❜✈✐♦ q✉❡ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❛❝✐♠❛ só ♣r❡❝✐s❛♠

s❡r ✈❡r✐✜❝❛❞❛s q✉❛♥❞♦ Φ(x) ❡ Φ(y) sã♦ ✜♥✐t♦s✳ ❚❛♠❜é♠ é ó❜✈✐♦ q✉❡ Dom(Φ) é ✉♠

s✉❜❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦ ❞❡ X s❡♠♣r❡ q✉❡ Φ❢♦r ❝♦♥✈❡①♦✳

❆ss✐♠✱ ❞✐③❡♠♦s q✉❡ Φ : Dom(Φ) → R é ❝♦♥✈❡①♦ ✭♦✉ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦✮ s❡ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s r❡❧❡✈❛♥t❡s sã♦ s❛t✐s❢❡✐t❛s ♣❛r❛ x, y ∈Dom(Φ)✳

❖ ❊♣í❣r❛❢♦ ❞♦ ❢✉♥❝✐♦♥❛❧ Φ é ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡ X×R ❞❡✜♥✐❞♦ ♣♦r

Epi(Φ) ={(x, a)∈X×R_|_Φ(_x₎_≤_a_}_.

❚❡♦r❡♠❛ ✷✳✶✳ ❙❡❥❛✱ X ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ Φ : X → R _{∪ {}₊_∞}✳ ❖ ❢✉♥❝✐♦♥❛❧ _Φ é ❝♦♥✈❡①♦ s❡ ❡ s♦♠❡♥t❡ s❡✱ Epi(Φ) é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦✳

❉❡♠✳✿

❙❡❥❛ Φ : X → R_{∪ {}₊_∞} ✉♠ ❢✉♥❝✐♦♥❛❧ ❝♦♥✈❡①♦✳ ❉❛❞♦s ₍_x₁_{, a}₁₎_,₍_x₂_{, a}₂₎ _∈ _Epi_(Φ) ❡ t∈[0,1]✱ t❡♠♦s✿

Φ(x1)≤a1 ❡ Φ(x2)≤a2✳ ❈♦♠♦ Φé ❝♦♥✈❡①♦✱

Φ((1−t)x1+tx2)≤(1−t)Φ(x1) +tΦ(x2)≤(1−t)a1+ta2.

▲♦❣♦✱

((1−t)x1+tx2,(1−t)a1+ta2) = (1−t)(x1, a1) +t(x2, a2)∈Epi(Φ).

P♦rt❛♥t♦✱ Epi(Φ) é ❝♦♥✈❡①♦✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ Epi(Φ) ✉♠ ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦✳ ❉❛❞♦s x1, x2 ∈ X ❡ 0 ≤ t ≤ 1✱

t❡♠♦s✿ (x1,Φ(x1)),(x2,Φ(x2))∈Epi(Φ)✳ ❉❡ss❡ ♠♦❞♦✱

(1−t)(x1,Φ(x1)) +t(x2,Φ(x2)) = ((1−t)x1+tx2,(1−t)Φ(x1) +tΦ(x2))∈Epi(Φ),

♦✉ s❡❥❛✱

Φ((1−t)x1 +tx2)≤(1−t)Φ(x1+tΦ(x2)

❡ ♣♦rt❛♥t♦✱ Φé ❝♦♥✈❡①♦✳

(23)

✶✽ ✷✳✹✳ ❋❯◆❈■❖◆❆■❙ ❈❖◆❱❊❳❖❙

Pr♦♣♦s✐çã♦ ✷✳✶✳ ❙❡❥❛ Φ :X →R_{∪ {}₊_∞} ❝♦♥✈❡①♦✳ ❙❡ _Φ é ❝♦♥tí♥✉♦ ❡♠ _x₀ _∈_Dom_(Φ) ❡♥tã♦ ❡①✐st❡♠ M > 0 ❡ δ >0 t❛✐s q✉❡

|Φ(x)−Φ(y)| ≤M.kx−y k para todo x, y ∈B(x0;δ).

❉❡♠✳✿

❈♦♠♦Φé ❝♦♥tí♥✉♦ ❡♠x0✱ ❡♥tã♦ ❡❧❡ é ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞♦ ♥❡st❡ ♣♦♥t♦❀ ♦✉ s❡❥❛✱ ❡①✐st❡♠

M1 >0❡δ >0t❛✐s q✉❡|Φ| ≤M1 ❡♠B(x0,2δ).❉❛❞♦sx, y ∈B(x0, δ)✱ s❡❥❛♠λ=kx−yk

❡ z =y+ _λδ(y−x). ❈♦♠♦

kz−x0 k=ky−x0 +

δ

λ(y−x)k≤ky−x0 k+ δ

λ ky−xk≤δ+δ = 2δ. ❊♥tã♦z ∈B(x0; 2δ). ❏á q✉❡

y= λ

(24)

✶✾ ✷✳✺✳ ❖ ❙❯❇❉■❋❊❘❊◆❈■❆▲ ❉❊ ❯▼ ❋❯◆❈■❖◆❆▲ ❈❖◆❱❊❳❖

é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❝♦♥✈❡①❛ ❞❡ z ❡ x ✭❝♦♥t✐❞❛ ❡♠ B(x0; 2δ)✮

❡♥tã♦✱ ♣❡❧❛ ❝♦♥✈❡①✐❞❛❞❡ ❞❡ Φ✱

Φ(y)≤ λ

λ+δΦ(z) + δ

λ+δΦ(x) ❡ ♣♦rt❛♥t♦

Φ(y)−Φ(x)≤ λ

λ+δΦ(z)+

δ λ+δ −1

Φ(x)

= λ

λ+δ(Φ(z)−Φ(x))≤ λ

δ.2M1 =

kx−yk

δ .2M1 ❚r♦❝❛♥❞♦ x ❡ y ♦❜t❡♠♦s ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✱ ❝♦♠M = 2M1

δ .

✷✳✺ ❖ ❙✉❜❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠ ❋✉♥❝✐♦♥❛❧ ❈♦♥✈❡①♦

❉❡✜♥✐çã♦ ✷✳✹✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡ Φ : X → R_{∪ {}₊_∞} ✉♠ ❢✉♥❝✐♦♥❛❧ ❝♦♥✈❡①♦✳ ❖ s✉❜❞✐❢❡r❡♥❝✐❛❧ ❞❡Φ é ❛ ❛♣❧✐❝❛çã♦

∂Φ :X →2X∗

dada por

∂Φ(a) = {ℓ ∈X∗|Φ(y)≥Φ(a) +hℓ, y−ai, para todo y ∈X}.

❖s ❡❧❡♠❡♥t♦s ❞❡∂Φ(a)sã♦ ❝❤❛♠❛❞♦s ❞❡ s✉❜❣r❛❞✐❡♥t❡s ❞❡Φ❡♠X✳ ◆❛❞❛ ✐♠♣❡❞❡ q✉❡✱ ♣❛r❛ ❛❧❣✉♠ a∈X✱ ∂Φ(a) s❡❥❛ ✈❛③✐♦✳ ❊st❡ ❝❛s♦ s❡ a6∈Dom(Φ) ♦✉ Φ♥ã♦ é ♣ró♣r✐❛✳

Φ(a) +hℓ1, y−ai

Φ(a) +hℓ2, y−ai

ℓ1❡ℓ2sã♦ s✉❜❣r❛❞✐❡♥t❡s ❞❡Φ❡♠a✳ ●❡♦♠❡tr✐❝❛♠❡♥t❡✱ ♦ s✉❜❣r❛❞✐❡♥t❡ sã♦ ♦s ❢✉♥❝✐♦♥❛✐s

(25)

✷✵ ✷✳✻✳ ❆ ❉❊❘■❱❆❉❆ ❉■❘❊❈■❖◆❆▲ ▲❆❚❊❘❆▲ ❉❊ ❯▼ ❋❯◆❈■❖◆❆▲ ❈❖◆❱❊❳❖

✷✳✻ ❆ ❉❡r✐✈❛❞❛ ❉✐r❡❝✐♦♥❛❧ ▲❛t❡r❛❧ ❞❡ ✉♠ ❋✉♥❝✐♦♥❛❧ ❈♦♥✲

✈❡①♦

❙❡❥❛Φ ✉♠ ❢✉♥❝✐♦♥❛❧ ❝♦♥✈❡①♦ ❡x0 ∈Dom(Φ)✳ P❛r❛ ❝❛❞❛y∈X✱ ❛ ❞❡r✐✈❛❞❛ ❞✐r❡❝✐♦♥❛❧

à ❞✐r❡✐t❛ ❞❡ Φ❡♠ x0 é✿

Φ′₊(x0;y) = lim

t→0+

Φ(x0+ty)−Φ(x0)

t .

❙❡❥❛ ϕ(t) := Φ(x0+ty)✳ ➱ ó❜✈✐♦ q✉❡

ϕ′₊(0) = Φ′₊(x0;y).

▲♦❣♦ ♣❛r❛ ♣r♦✈❛r q✉❡ ❡①✐st❡ ❛ ❞❡r✐✈❛❞❛ ❞✐r❡❝✐♦♥❛❧ à ❞✐r❡✐t❛ ❞❡ Φ❡♠ x0 ❜❛st❛ ♠♦str❛r q✉❡

❡①✐st❡

ϕ′₊(0).

➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ϕ é ❝♦♥✈❡①❛✱ ❝♦♠ 0 ∈ Dom(ϕ). ❙❡❥❛♠ t1 ≥ t2 > 0✳ ▲♦❣♦✱ ❡①✐st❡

λ∈(0,1]✱t❛❧ q✉❡ t2 =λt1 = (1−λ)0 +λt1. P❡❧❛ ❝♦♥✈❡①✐❞❛❞❡ ❞❡ ϕ✱

ϕ(t2)≤(1−λ)ϕ(0) +λϕ(t1)

❡ ♣♦rt❛♥t♦✱

ϕ(t2)−ϕ(0)

t2

≤ (1−λ)ϕ(0) +λϕ(t1)−ϕ(0)

t2

= λ(ϕ(t1)−ϕ(0))

t2

= λ(ϕ(t1)−ϕ(0))

λt1

= ϕ(t1)−ϕ(0)

t1

.

❆ss✐♠ ❛ ❢✉♥çã♦ ϕ(t)−ϕ(0)

t é ♠♦♥ót♦♥❛ q✉❛♥❞♦ t →0

+_. _{▲♦❣♦✱ s❡ ❡❧❛ é ❧✐♠✐t❛❞❛ ♦ r❡s✉❧t❛❞♦}

(26)

✷✶ ✷✳✻✳ ❆ ❉❊❘■❱❆❉❆ ❉■❘❊❈■❖◆❆▲ ▲❆❚❊❘❆▲ ❉❊ ❯▼ ❋❯◆❈■❖◆❆▲ ❈❖◆❱❊❳❖

❖❜s❡r✈❛çã♦✳✿ ❖ ❛r❣✉♠❡♥t♦ ❛❝✐♠❛ t❛♠❜é♠ ♠♦str❛ q✉❡

Φ′₊(x0;y) = inf

t>0

Φ(x0+ty)−Φ(x0)

t

Pr♦♣♦s✐çã♦ ✷✳✷✳ ❙❡ x0 ∈ Int(DomΦ) ❡♥tã♦ ♣❛r❛ ❝❛❞❛ y ∈ X ❛ ❞❡r✐✈❛❞❛ ❞✐r❡❝✐♦♥❛❧ à

❞✐r❡✐t❛ Φ′

+(x0;y) é ✉♠ ♥ú♠❡r♦ r❡❛❧ ❡

Φ′₊(x0)(y) = Φ′+(x0;y)

❞❡✜♥❡ ✉♠ ❢✉♥❝✐♦♥❛❧ s✉❜❧✐♥❡❛r ❞❡ ❳✳

❉❡♠✳✿

❙❡❥❛ ψ(t) = Φ(x0−ty). ❙❡ t1 ≥t2 >0 ❡♥tã♦

ψ(t2)−ψ(0)

t2

≤ ψ(t1)−ψ(0)

t1

,

♦✉ s❡❥❛✱

−ψ(t2)−ψ(0)

t2

≥ −ψ(t1)−ψ(0)

t1

.

▲♦❣♦✱

−

ψ(t)−ψ(0)

t

é ♥ã♦ ❝r❡s❝❡♥t❡ ❝♦♥❢♦r♠❡ t→0+_{✱ ❡♥q✉❛♥t♦ q✉❡}

ϕ(t)−ϕ(0)

t é ♥ã♦ ❞❡❝r❡s❝❡♥t❡✳

❉❡✈✐❞♦ à ❝♦♥✈❡①✐❞❛❞❡ ❞❡ Φ✱ t❡♠♦s✿

x0 =

1

2(x0−2ty) + 1

2(x0+ 2ty) Φ(x0)≤

1

2Φ(x0−2ty) + 1

2Φ(x0+ 2ty).

❉❛í✱

2Φ(x0)≤Φ(x0−2ty) + Φ(x0+ 2ty)−Φ(x0−2ty) + Φ(x0)

≤Φ(x0+ 2ty)−Φ(x0)−

[ψ(2t)−ψ(0)] 2t ≤

[ϕ(2t)−ϕ(0)] 2t .

(27)

✷✷ ✷✳✻✳ ❆ ❉❊❘■❱❆❉❆ ❉■❘❊❈■❖◆❆▲ ▲❆❚❊❘❆▲ ❉❊ ❯▼ ❋❯◆❈■❖◆❆▲ ❈❖◆❱❊❳❖

❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ sã♦ ♥ú♠❡r♦s r❡❛✐s✳ ❆ss✐♠✱ s❡ t1 ≥t2 >0✱ ❡♥tã♦

−[ψ(2t1)−ψ(0)]

2t1

≤ −[ψ(2t2)−ψ(0)]

2t2

≤ ψ(2t2)−ψ(0)

2t2

≤ ψ(2t1)−ψ(0)

2t1

▲♦❣♦ ♦ ❧✐♠✐t❡ ❞❡

−[ψ(t)−ψ(0)]

t ,

♣❛r❛ t→0+_{✱ é ✉♠ ♥ú♠❡r♦ r❡❛❧✱ ♦ ♠❡s♠♦ ♦❝♦rr❡♥❞♦ ❝♦♠ ♦ ❧✐♠✐t❡ ❞❡}

ψ(t)−ψ(0)

t

♣❛r❛ t→0+_✳

❈♦♠♦ ♦ ♣r✐♠❡✐r♦ ❧✐♠✐t❡ é −Φ′

+(x0;−y) ❡♥tã♦✿

−Φ′₊(x0;−y)≤Φ′+(x0;y).

Φ′

+(x0)(y) é ♣♦s✐t✐✈❛♠❡♥t❡ ❤♦♠♦❣ê♥❡♦ ♣♦✐s❀ ♣❛r❛ s >0✿

Φ′₊(x0)(sy) = lim

t→0+

Φ(x0+t(sy))−Φ(x0)

t =stlim→0+

Φ(x0+ (ts)y)−Φ(x0)

st

=s lim

r→0+

Φ(x0+ry)−Φ(x0)

r =s.Φ

′

+(x0)(y),

♦♥❞❡ r=st✳

P❛r❛ ♠♦str❛r ❛ s✉❜❛❞✐t✐✈✐❞❛❞❡ ✉s❛♠♦s ♥♦✈❛♠❡♥t❡ q✉❡ Φ é ❝♦♥✈❡①♦ ❝♦♠♦ s❡❣✉❡✿

Φ(x0+t(y+z)) = Φ

1

2(x0+ 2ty) + 1

2(x0+ 2tz)

≤ 1

2Φ(x0+ 2ty) + 1

2Φ(x0+ 2tz),

♦✉ s❡❥❛✱

Φ(x0+t(y+z))−Φ(x0)≤

1 2

Φ(x0+ 2ty)−Φ(x0)

+ 1 2

Φ(x0+ 2tz)−Φ(x0)

.

❙❡ t >0✿

Φ(x0+t(y+z))−Φ(x0)

t ≤

Φ(x0+ 2ty)−Φ(x0)

2t

+

Φ(x0+ 2tz)−Φ(x0)

2t

.

❚♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ ♣❛r❛ t→0+_{✱ t❡♠♦s✿}

(28)

✷✸ ✷✳✻✳ ❆ ❉❊❘■❱❆❉❆ ❉■❘❊❈■❖◆❆▲ ▲❆❚❊❘❆▲ ❉❊ ❯▼ ❋❯◆❈■❖◆❆▲ ❈❖◆❱❊❳❖

➱ ó❜✈✐♦ q✉❡ Φé ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ s❡♥t✐❞♦ ❞❡ ●ât❡❛✉① ❡♠ x0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱

Φ′₋(x0;y) = −Φ′+(x0;−y) = Φ′+(x0;y)

♣❛r❛ ❝❛❞❛ y∈X✳ ❈♦♠♦ ✉♠ ❢✉♥❝✐♦♥❛❧ s✉❜❧✐♥❡❛rp é ❧✐♥❡❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ p(−x) =−p(x)

♣❛r❛ t♦❞♦x∈X✱ ❡♥tã♦ Φ é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ s❡♥t✐❞♦ ❞❡ ●ât❡❛✉① ❡♠ x0✱ s❡✱ ❡ s♦♠❡♥t❡ s❡

y→Φ′₊(x0)(y)

é ❧✐♥❡❛r ❡♠ y✳

Pr♦♣♦s✐çã♦ ✷✳✸✳ ❙❡ Φ é ✉♠ ❢✉♥❝✐♦♥❛❧ ❝♦♥✈❡①♦ ❡ ❝♦♥tí♥✉♦ ❡♠ x0 ∈ Dom(Φ)✱ ❡♥tã♦

∂Φ(x0) é ✉♠ ❢r❛❝♦✲∗ ❝♦♠♣❛❝t♦✳

❉❡♠✳✿

❙❛❜❡♠♦s q✉❡∂Φ(x0)é ❢r❛❝♦✲∗❢❡❝❤❛❞♦✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❇❛♥❛❝❤✲❆❧❛♦❣❧✉✱ ❜❛st❛ ♣r♦✈❛r

q✉❡ ❡❧❡ é ❧✐♠✐t❛❞♦ ♣❛r❛ ♦❜t❡r ❛ ❝♦♥❝❧✉sã♦ ❞❡s❡❥❛❞❛✳ ❈♦♠♦ Φ é ❝♦♥tí♥✉❛ ❡♠ x0✱ ❡❧❛ é

❧♦❝❛❧♠❡♥t❡ ❧✐♣s❝❤✐t③✐❛♥❛ ❡♠ x0✱ ♦✉ s❡❥❛✱ ❡①✐st❡♠ M > 0 ❡ U ✈✐③✐♥❤❛♥ç❛ ❞❡ x0✱ ❞❡ ♠♦❞♦

q✉❡

|Φ(x)−Φ(y)| ≤M kx−yk;∀x, y ∈U. ❙❡x∈U ❡ ℓ∈∂Φ(x)✱ ❡♥tã♦ ♣❛r❛ t♦❞♦ y∈U✱ t❡♠♦s✿

hℓ, y−xi ≤Φ(y)−Φ(x)≤M ky−xk,

♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡

ℓ, y−x

ky−xk

≤M

❡ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱kℓ k≤M.

Pr♦♣♦s✐çã♦ ✷✳✹✳ ❙❡❥❛ Φ : X → R _{∪ {}₊_∞} ✉♠ ❢✉♥❝✐♦♥❛❧ ❝♦♥✈❡①♦✳ ❙❡ _Φ é ❝♦♥tí♥✉♦ ❡♠ x0 ∈Int(Dom(Φ) ❡♥tã♦ Φ′+(x0) é ✉♠ ❢✉♥❝✐♦♥❛❧ s✉❜❧✐♥❡❛r ❝♦♥tí♥✉♦ ❡♠ X✱ ❡ ♣♦rt❛♥t♦

Φ′₍_x

0) ✭q✉❛♥❞♦ ❡①✐st❡✮ é ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❝♦♥tí♥✉♦✳

❉❡♠✳✿

❉❛❞♦ x0 ∈Int(DomΦ)✱ ❝♦♠♦Φé ❝♦♥tí♥✉♦ ❡♠ x0✱ ❡①✐st❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛B ❞❡x0 ❡

M > 0❞❡ ♠♦❞♦ q✉❡

Φ(x0+tx)−Φ(x0)≤M.tkxk,

❞❡s❞❡ q✉❡ t >0s❡❥❛ ♣❡q✉❡♥♦ ♦ ❜❛st❛♥t❡ ♣❛r❛ q✉❡ x0+tx∈B✳ ▲♦❣♦✱ ♣❛r❛ t♦❞♦ x∈X✱

(29)

✷✹ ✷✳✻✳ ❆ ❉❊❘■❱❆❉❆ ❉■❘❊❈■❖◆❆▲ ▲❆❚❊❘❆▲ ❉❊ ❯▼ ❋❯◆❈■❖◆❆▲ ❈❖◆❱❊❳❖

♦ q✉❡ ♠♦str❛ q✉❡

Φ′₊(x0)

é ❝♦♥tí♥✉♦✳ ◆♦t❡ q✉❡ ❞❛❞♦s x∗ ∈ X∗ ❡ Φ : X → R_{∪ {}₊_∞} ❝♦♥✈❡①♦ ❡ _x₀ _∈ _Dom_Φ✱ ❛ ❝♦♥❞✐çã♦✿

✭✐✮ hx∗, x−x0i ≤Φ(x)−Φ(x0), x∈X é ❡q✉✐✈❛❧❡♥t❡ ❛

✭✐✐✮ hx∗_{, y}_{i ≤}_Φ′

+(x0;y), y ∈X✱ ♣♦✐s ❞❛❞♦(i) t❡♠♦s✿

hx∗,(x0+ty)−x0i=thx∗, yi ≤Φ(x0+ty)−Φ(x0),

♣❛r❛ t >0✱ ♦ q✉❡ ✐♠♣❧✐❝❛(ii)✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ss✉♠✐♥❞♦ q✉❡x∗ s❛t✐s❢❛ç❛ (ii)✱ ❞❡✜♥✐♠♦s y=x−x0 ❞❡ ♠♦❞♦ q✉❡

(1) hx∗, x−x0i ≤Φ′+(x0;x−x0)

≤ Φ(x0+t(x−x0))−Φ(x0)

t , para todo t >0. P❛r❛ t= 1✱ t❡♠♦s (i)✳

Pr♦♣♦s✐çã♦ ✷✳✺✳ ❙❡❥❛ Φ : X −→ R_{∪ {∞}} ❝♦♥tí♥✉♦ ❡♠ _x₀ _∈ _Int₍_Dom_Φ)✳ ❊♥tã♦ _Φ é ❞❡r✐✈á✈❡❧ ♥♦ s❡♥t✐❞♦ ❞❡ ●ât❡❛✉① s❡ ❡ s♦♠❡♥t❡ s❡ ∂Φ(x0) é ✉♠ ❝♦♥❥✉♥t♦ ✉♥✐tár✐♦✳

❉❡♠✳✿

✭⇒✮ ❙❡❥❛ Φ ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ s❡♥t✐❞♦ ❞❡ ●ât❡❛✉①✳ ❈♦♠♦ Φ′₍_x

0;y) = Φ′+(x0;y)✱❡♥tã♦ ♣❡❧❛

✐♥❡q✉❛çã♦(1) ❞❛ ♦❜s❡r✈❛çã♦ ❛♥t❡r✐♦r✱ t❡♠♦s ♣❛r❛t= 1✳ Φ′(x0)(x−x0)≤Φ(x)−Φ(x0),

♦✉ s❡❥❛✱

Φ′(x0)∈∂Φ(x0).

❆❧é♠ ❞✐ss♦✱ s❡

hx∗, x−x0i ≤Φ(x)−Φ(x0), x∈X,

❡♥tã♦

hx∗, yi ≤Φ′₊(x0)(y)

♠❛s

Φ′₊(x0)(y) = Φ′(x0)(y),∀y∈X.

▲♦❣♦✿

hx∗ −Φ′(x0), yi ≤0;y∈X.

▼❛s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡str✐t❛ ❛❝✐♠❛ é ✐♠♣♦ssí✈❡❧ ♣♦✐s t❡rí❛♠♦s ♣❛r❛ ❛❧❣✉♠ y✿

(30)

✷✺ ✷✳✻✳ ❆ ❉❊❘■❱❆❉❆ ❉■❘❊❈■❖◆❆▲ ▲❆❚❊❘❆▲ ❉❊ ❯▼ ❋❯◆❈■❖◆❆▲ ❈❖◆❱❊❳❖

❡ ♣♦rt❛♥t♦✿

−hx∗−Φ′(x0), yi>0

♦✉ ❛✐♥❞❛

hx∗−Φ′(x0),−yi>0.

▲♦❣♦✱♣❡❧❛ ❝♦♥❞✐çã♦ ❞❡ ❞✉❛❧✐❞❛❞❡✱

x∗ = Φ′(x0).

✭⇐✮ ❙❡❥❛ ❛❣♦r❛x∗ ♦ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞❡ ∂Φ(x0)✱ ♦✉ s❡❥❛✱ x∗ é ♦ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞❡X∗ q✉❡

s❛t✐s❢❛③

hx∗, x−x0i ≤Φ(x)−Φ(x0)

♦ q✉❡ ❡q✉✐✈❛❧❡ ❛ ❞✐③❡r q✉❡ x∗ _{é ♦ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞❡} _X∗ _{t❛❧ q✉❡} hx∗, yi ≤Φ′₊(x0)(y);∀y∈X.

❇❛st❛ ✉s❛r ♦ s❡❣✉✐♥t❡ ❧❡♠❛✿

▲❡♠❛ ✷✳✶✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ ❡ Φ :X →R ✉♠ ❢✉♥❝✐♦♥❛❧ s✉❜❧✐♥❡❛r ❝♦♥tí♥✉♦ ❞❡ ♠♦❞♦ q✉❡ Φ❞♦♠✐♥❛ ✉♠ ú♥✐❝♦ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛rf✳ ◆❡ss❛s ❝♦♥❞✐çõ❡s✱Φé ❧✐♥❡❛r ❡ ❛✐♥❞❛✱ Φ =f✳

❉❡♠✳✿ ❙❡❥❛♠Φ❡f ♥❛s ❝♦♥❞✐çõ❡s ❞♦ ❡♥✉♥❝✐❛❞♦✳ ❙✉♣♦♥❤❛ q✉❡ f ♥ã♦ ❝♦✐♥❝✐❞❛ ❝♦♠Φ✱ ♦✉

s❡❥❛✱ ❡①✐st❡a∈X t❛❧ q✉❡f(a)<Φ(a)✳ ◆❡st❛s ❝♦♥❞✐çõ❡s✱a6= 0 ♣♦✐s ❡♠0f ❝♦✐♥❝✐❞❡ ❝♦♠

Φ✳ ❆ss✐♠✱ s❡❥❛M =hai✳ ❙❡❥❛ ❛✐♥❞❛ b= Φ(a) +f(a)

2 ✳ ◆♦t❡ q✉❡ f(a)< b <Φ(a)✳

❉❡✜♥❛ ❛❣♦r❛ ♦ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛rg s♦❜r❡M ❞❛❞♦ ♣♦rg(λa) =λb✳ ❈♦♠♦ Φ(a)> g(a)> f(a)✱ s❡❣✉❡ q✉❡ g 6=f|M✳

❖❜s❡r✈❡ ❛❣♦r❛ q✉❡ s❡ λ ≥ 0 ❡♥tã♦ Φ(λa) > g(λa) > f(λa)✳ ❖✉ s❡❥❛✱ s❡ λ ≥ 0 ❡♥tã♦

g é ❞♦♠✐♥❛❞♦ ♣♦r Φ✳ ❙❡ λ < 0✱ ❝♦♠♦ g(a) > f(a)✱ t❡♠♦s λg(a) < λf(a) ♦ q✉❡ ❡q✉✐✈❛❧❡

❛ g(λa) < f(λa)✳ ◆♦ ❡♥t❛♥t♦✱ f é ❞♦♠✐♥❛❞❛ ♣♦r Φ ❡♠ X✳ ❊♥tã♦✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r g(λa) <Φ(λa)✳ ■st♦ ♠♦str❛ q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r g s♦❜r❡ M é ❞♦♠✐♥❛❞♦ ♣❡❧❛ r❡str✐ç❛♦ ❞❡Φ❛♦ s✉❜❡s♣❛ç♦M✳ P❡❧♦ t❡♦r❡♠❛ ❞❡ ❍❛❤♥✲❇❛♥❛❝❤✱ ♣♦❞❡♠♦s ❡st❡♥❞❡rg ❛ ✉♠ ❢✉♥❝✐♦♥❛❧ G✱ ❞❡ ♠♦❞♦ q✉❡ kG k=k g k ❡ q✉❡ ❛✐♥❞❛ é ❞♦♠✐♥❛❞♦ ♣♦r Φ ❡♠ X✳ ❏á q✉❡ g(a)> f(a)✱