ELEMENTOS DA ANÁLISE FUNCIONAL PARA O ESTUDO DA EQUAÇÃO DA CORDA VIBRANTE

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙❊❘●■P❊

P❘❖●❘❆▼❆ ❉❊ ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲

❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲

P❘❖❋▼❆❚

❆➱❉❙❖◆ ◆❆❙❈■▼❊◆❚❖ ●Ó■❙

❊▲❊▼❊◆❚❖❙ ❉❆ ❆◆➪▲■❙❊ ❋❯◆❈■❖◆❆▲ P❆❘❆ ❖ ❊❙❚❯❉❖ ❉❆

❊◗❯❆➬➹❖ ❉❆ ❈❖❘❉❆ ❱■❇❘❆◆❚❊

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙❊❘●■P❊

P❘❖●❘❆▼❆ ❉❊ ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲

❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲

P❘❖❋▼❆❚

❆➱❉❙❖◆ ◆❆❙❈■▼❊◆❚❖ ●Ó■❙

❊▲❊▼❊◆❚❖❙ ❉❆ ❆◆➪▲■❙❊ ❋❯◆❈■❖◆❆▲ P❆❘❆ ❖ ❊❙❚❯❉❖ ❉❆

❊◗❯❆➬➹❖ ❉❆ ❈❖❘❉❆ ❱■❇❘❆◆❚❊

❚r❛❜❛❧❤♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❡r❣✐♣❡ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✭P❘❖❋▼❆❚✮✳

❖❘■❊◆❚❆❉❖❘✿

Pr♦❢✳ ❉r✳ ❆▲❊❏❆◆❉❘❖ ❈❆■❈❊❉❖ ❘❖◗❯❊

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FICHA CATALOGRÁFICA ELABORADA PELA BIBLIOTECA PROFESSOR ALBERTO CARVALHO UNIVERSIDADE FEDERAL DE SERGIPE

G616e

Góis, Aédson Nascimento.

Elementos da análise funcional para o estudo da equação da corda vibrante / Aédson Nascimento Góis; orientador Alejandro Caicedo Roque. – Itabaiana, 2016.

67 f.

Dissertação (Mestrado Profissional em Matemática) – Universidade Federal de Sergipe, 2016.

1. Corda vibrante. 2. Espaços de Banach. 3. Espaços de Hilbert. 4. Ortogonalidade. 5. Séries de Fourier I. Roque, Alejandro Caicedo. II. Título.

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❈♦♥t❡ú❞♦

❘❡s✉♠♦

❆❜str❛❝t

■♥tr♦❞✉çã♦ ✶

✶ Pr❡❧✐♠✐♥❛r❡s ✸

✷ ❊s♣❛ç♦s ◆♦r♠❛❞♦s ✺

✷✳✶ ❊①❡♠♣❧♦s ❞❡ ❊s♣❛ç♦s ◆♦r♠❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷ ❊s♣❛ç♦s ◆♦r♠❛❞♦s ❞❡ ❉✐♠❡♥sã♦ ❋✐♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✸ ❊s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✸ ❊s♣❛ç♦s ❝♦♠ Pr♦❞✉t♦ ■♥t❡r♥♦ ❡ ❊s♣❛ç♦s ❞❡ ❍✐❧❜❡rt ✷✸

✸✳✶ Pr♦❞✉t♦s ■♥t❡r♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✷ ❖rt♦❣♦♥❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✸ ❈♦♠♣❧❡♠❡♥t♦ ❖rt♦❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✹ ❇❛s❡s ❖rt♦♥♦r♥❛✐s ❡♠ ❉✐♠❡♥sã♦ ■♥✜♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✹ ❙ér✐❡s ❞❡ ❋♦✉r✐❡r ✹✼

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆♦ s❡r ❙♦❜❡r❛♥♦ ❙❡♥❤♦r ❞♦ ❯♥✐✈❡rs♦✿ ❏❡♦✈á ❉❡✉s✱ ♣♦r t♦❞❛s ❛s ❜♦❛s ❞á❞✐✈❛s ❝♦♥❝❡❞✐❞❛s✳ ❆ t✐ ❞♦✉ ❣r❛ç❛s✱ ❙❡♥❤♦r✱ ♣♦✐s és ♦ ♠❡✉ ❉❡✉s✱ ♠❡✉ P❛✐✱ ♠❡✉ ❆♠✐❣♦✳ ✧❉✐❣♥♦ és ❞❡ r❡❝❡❜❡r ❛ ❣❧ór✐❛✱ ❛ ❤♦♥r❛ ❡ ♦ ♣♦❞❡r✳✲ ❆♣✳✹✿✶✶ ❊ ❛♦ ❙❡✉ ❋✐❧❤♦ ❆♠❛❞♦✱ ❈r✐st♦ ❏❡s✉s✱ ♣♦r t❡r ❡♥tr❡❣✉❡ s✉❛ ✈✐❞❛ ❝♦♠♦ r❡s❣❛t❡ ❞❛♥❞♦ ❡①❡♠♣❧♦ í♠♣❛r ❞❡ ❛♠♦r ❛❧tr✉íst❛✳ ✲ ▼❛t ✷✵✿✷✽

➚ ♠✐♥❤❛ ♠ã❡✱ ❘✐✈❛♥❡✐❞❡ ❙❛♥t♦s ◆❛s❝✐♠❡♥t♦ ●ó✐s q✉❡ ❢♦✐ t❛♠❜é♠ ♣♦r ♠✉✐t♦ t❡♠♣♦ ♠❡✉ ♣❛✐✳ ❆♠✐❣❛✱ ❝♦♠♣❛♥❤❡✐r❛✱ í❝♦♥❡✱ ♠❡♥t♦r❛✱ ♠❡str❛✱ r❡❢❡rê♥❝✐❛✱ ❡①í♠✐♦ ❡①❡♠♣❧♦ ♣r❛ s❡r s❡❣✉✐❞♦ ❞❡ ♣❡rt♦✳ ❊ ❡ss❛ ✈✐tór✐❛ só ♦❝♦rr❡✉ ♣♦r ✈♦❝ê✱ ❡ ♣r❛ ✈♦❝ê✳

➚ ♠✐♥❤❛ ❢❛♠í❧✐❛ ✐♠❡❞✐❛t❛ ❢♦r♠❛❞❛ ♣❡❧♦ ♣❛✐ ▼ár❝✐♦ q✉❡ ♠ã❡ ❡s❝♦❧❤❡✉ ♣r❛ ♠✐♠ ✭❡ ♥ós ❡s❝♦❧❤❡♠♦s ✉♠ ❛♦ ♦✉tr♦ ❝♦♠♦ ♣❛✐ ❡ ✜❧❤♦✱ ❡♠ s❡❣✉✐❞❛✮✱ ❡ ♦ ♠❡✉ ✐r♠ã♦ ●❛❜r✐❡❧ ❡ ♠✐♥❤❛ t✐❛ ✭✐r♠ã ❛❞♦t✐✈❛✮ ❍❡❧❡♥✐❝❡✳ ❆ ✈♦❝ês q✉❡ t✐✈❡r❛♠ ❛ ♣❛❝✐ê♥❝✐❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ❧✐❞❛r ❝♦♠✐❣♦✱ ❡ ♠❡ ❞❡r❛♠ ❢♦rç❛s s✐♠♣❧❡s♠❡♥t❡ ♣♦r ❡①✐st✐r❡♠ ❡♠ ♠✐♥❤❛ ✈✐❞❛✱ ♠❡✉ ♠✉✐t♦ ♦❜r✐❣❛❞♦✳

❆♦s ♠❡✉s ❛✈ós✱ t✐♦s ❡ ♣r✐♠♦s ♣♦r s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈❛r❡♠✳ ▼✉✐t❛s ✈❡③❡s ❛té ♠❡ ❛tr✐✲ ❜✉✐♥❞♦ ❛ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ❞❡ s❡r ♦ ❡①❡♠♣❧♦ ♣❛r❛ ♦s ♠❛✐s ❥♦✈❡♥s ❞❛ ❢❛♠í❧✐❛✳ ❊♥✜♠✱ ❡✉ ❝♦♥s❡❣✉✐✦ ❱♦❝ês t❛♠❜é♠ ❝♦♥s❡❣✉✐rã♦✱ ♠❡✉s ♣r✐♠♦s✳ ❈❛✐q✉❡✱ ❈❧é✈❡rt♦♥✱ ●r❡✐❝❡✱ ●✐❧✈â♥✐❛✱ ▼❛❣♥❛✱ ❘❛q✉❡❧✱ ❘❛②❛♥❡ ❡ ❙té♣❤❛♥✐✱ ♠❡✉s ❝❤❡❣❛❞♦s✱ ❛♠♦ ✈♦❝ês ❆ t✐❛ ▲✉❝✐❡❞❡ q✉❡ ♠❡ ❛❝♦✲ ❧❤❡✉ ❝♦♠♦ ❛ ✉♠ ✜❧❤♦ ♠❡ ❞❛♥❞♦ ♠❛✐s q✉❡ ❣✉❛r✐t❛✱ ✉♠ ❧✉❣❛r ❡♠ s❡✉ ❝♦r❛çã♦✳ ◆ã♦ t❡♥❤♦ ♣❛❧❛✈r❛s ♣r❛ ❡①♣r❡ss❛r ❛ ❣r❛t✐❞ã♦ ❡ ❛❢❡t♦ q✉❡ t❡♥❤♦ ♣❡❧❛ s❡♥❤♦r❛ ❡ ♠❡✉s ✧♣r✐♠♦s✧❏✉❧✐❛♥❛✱ ◆❡t✐♥❤♦ ❡ ❨✉r✐✳

➚ ✈♦❝ê✱ ❏✉❧✐ ❑❡❧❧❡ ●ó✐s ❈♦st❛✱ ♠✐♥❤❛ ▼❊▼❆✱ ♠✉✐t♦ ♦❜r✐❣❛❞♦✳ ❋♦r❛♠ t❛♥t❛s ❛s ♥♦✐t❡s ❞❡ ✐♥❝❡♥t✐✈♦ ❡ t❛♥t♦s ♦s ❞✐❛s ❞❡ ❝✉♠♣❧✐❝✐❞❛❞❡✳ ◆ã♦ ❡s❝♦❧❤❡♠♦s ❡st❛r✱ ❡♠ ✷✵✵✵✱ ♠❛tr✐❝✉❧❛❞♦s ♥❛ ♠❡s♠❛ ❝❧❛ss❡ ❞❡ ✺❛ _{sér✐❡ ❇ ✭♥♦ ❈♦❧é❣✐♦ ▼✉❧✳ ❏♦s✉é P❛ss♦s✱ ❡♠ ❘✐❜❡✐ró♣♦❧✐s✲❙❊✮✱ ♠❛s}

♦♣t❛♠♦s s✐♠ ♣♦r ❡st❛❜❡❧❡❝❡r ♣❛r❝❡r✐❛ P❖❘ ❚❖❉❆ ❆ ❱■❉❆✳ ❱ár✐❛s ❝♦♥✈❡rs❛s✱ ❞❡s❛❜❛❢♦s✱ ❧❛♠úr✐❛s✱ ①✐♥❣❛s✱ ❧á❣r✐♠❛s ❞❡rr❛♠❛❞❛s ❡ s♦rr✐s♦s ❝♦♠♣❛rt✐❧❤❛❞♦s✳ ❱♦❝ê ♠❡❧❤♦r ❞♦ q✉❡ ♥✐♥❣✉é♠ s❛❜❡ t♦❞♦s ♦s ♦❜stá❝✉❧♦s ❡ ❤✐stór✐❛s q✉❡ ♣r❡❝❡❞❡r❛♠ ❛té ♦ ♠♦♠❡♥t♦✳ ❚❡ ❛♠♦ ♠✉✐t♦✦

➚s ♠✐♥❤❛s ❡①✲♣r♦❢❡ss♦r❛s✱ t✐❛ ❊❞♥❛✱ ❊▲✐❛♥❛✱ ▼❛r✐♥❡✉s❛✱ ▼❛r❧❡♥❡✱ ❞❡♥tr❡ ♦✉tr❛s✱ ❡❞✉✲ ❝❛❞♦r❛s ❡♠ ♠✐♥❤❛ t❡♥r❛ ✐♥❢â♥❝✐❛✳ ❈❡❞♦ ♣❡r❝❡❜❡r❛♠ ♣♦t❡♥❝✐❛❧ ❡ ♠❡ ✐♥❝❡♥t✐✈❛r❛♠ ❛ s❡❣✉✐r ❡♠ ❢r❡♥t❡✳ ❆♦ ♣r♦❢❡ss♦r ❈❧❡✐❞✐♥❛❧❞♦ q✉❡ ♠❡ ♦♣♦rt✉♥✐③♦✉ ❛♣r✐♠♦r❛r ♠❡✉s ❡st✉❞♦s ❝♦♥✲ s❡❣✉✐♥❞♦ ✉♠❛ ❜♦❧s❛ ♥✉♠ ❝♦❧é❣✐♦ ♣❛rt✐❝✉❧❛r ♥✉♠❛ ❝✐❞❛❞❡ ✈✐③✐♥❤❛ ♠❛✐s ❞❡s❡♥✈♦❧✈✐❞❛✳ ➚ ❙✐❧✈â♥✐❛ ●♦♠❡s ▲✐s❜♦❛✱ ✧♠ã❡ ❙✐❧✧✱ ✧♠ã❡ ♣r❡t❛✧✱ q✉❡ ❡st❛✈❛ ♥❛ ❝❛t❡❣♦r✐❛ ❞❡ ❡①✲♣r♦❢❡ss♦r❛ ❡ ❡s♣♦♥t❛♥❡❛♠❡♥t❡ s❡ t♦r♥♦✉ ♠ã❡ ♣♦r ❜ê♥çã♦✳ ▼✉✐t♦ ♦❜r✐❣❛❞♦ ♣❡❧♦s ❝♦♥s❡❧❤♦s✱ ❡♥s✐♥❛✲ ♠❡♥t♦s✱ ❡①❡♠♣❧♦ ❡ ♣❛r❝❡r✐❛ s❡♠♣r❡✳ ❚❛♠❜é♠ t❡ ❛♠♦ ♠✉✐t♦✦

❆♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦ q✉❡ ❤♦❥❡ sã♦ ❛♠✐❣♦s ❞♦s ♠❛✐s ❝❤❡❣❛❞♦s✱ ❆♥❛ ▼❛r②✱ ❆✉s❡♥✐r✱ ❉❛♥✐❡❧❛✱ ❉❡✐❞❡✱ ❊❞✐♠❛r✱ ❊❞✐✈❛♥ ❡ ▼❛ _{❏♦sé✱ ●✐❧t♦♥ ❡ ▲✉❝✐❡♥❡✱ ●✐✈❛❧❞♦✱ ■❣♦r✱}

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♦❜r✐❣❛❞♦✳

❆♦s ❛♠✐❣♦s ❊❞✐✈❛♥✱ ▲✉í③❛✱ ❙❛♥❞r❛ ❡ ❱❛♥❞❡rs♦♥✳ ❊st❡s ❡♥t❡♥❞❡r❛♠ ♠✐♥❤❛ ❛✉sê♥❝✐❛ ♥♦s ♥♦ss♦s ❡♥❝♦♥tr♦s✱ ❞✉r❛♥t❡ ❛s s❡♠❛♥❛s ♠❛✐s ❛t❛r❡❢❛❞❛s ❞♦ ♠❡str❛❞♦ ❡ ❢♦r❛♠ r❡s♣♦♥sá✈❡✐s ♣♦r ♠❡ ❞✐str❛✐r ❡♠ t❛♥t❛s ♦✉tr❛s t❡♥s❛s✳ ▼❡✉ ♠✉✐t♦ ♦❜r✐❣❛❞♦✳ ❆♦s ❛♠✐❣♦s ❋❡r♥❛♥❞❛✱ ❍❡❧❡♥❛ ❡ ❘♦♠ár✐♦ ♣❡❧♦s ✧❝❛❢❡③✐♥❤♦s✧❝♦♠♣❛rt✐❧❤❛❞♦s ♥♦ r❡t♦r♥♦ ❞❛ ✉♥✐✈❡rs✐❞❛❞❡✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❇r✉♥❛✱ ❈í♥t❤✐❛✱ ❊✇❡rt♦♥✱ ❲❛❞s♦♥✱ ❏❛♥❞❡rs♦♥✱ ❏♦❤♥②✱ ❏✉❧✐❛♥❛✱ ▲✉❝❛s✱ ❘♦❜s♦♥ ❡ s✉❛s r❡s♣❡❝t✐✈❛s ❢❛♠í❧✐❛s✱ ❆♥♥❛✱ ❉✐❡❣♦✱ ●✐♥❛❧❞♦✱ ❏♦♥❡s ❏ú♥✐♦r✱ ▼❡❧q✉✐❛❞❡s✱ ❘❛❢❛❡❧✱ ❘♦♥✐❡❧❛✱ ❚✐❛❣♦✱ ❡ t❛♥t♦s ♦✉tr♦s ♣♦r ❡♥t❡♥❞❡r❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❛✉s❡♥t❛r✲♠❡ ♥♦s ♥♦ss♦s ❡♥❝♦♥tr♦s s♦❝✐❛✐s✱ ♥❛ ❛❝❛❞❡♠✐❛✱ ❛té ♠❡s♠♦ ♠❡ ❞✐st❛♥❝✐❛♥❞♦✱ ❡♠ ✈✐rt✉❞❡ ❞❛s ♦❜r✐❣❛çõ❡s ❞❡ tr❛❜❛❧❤♦ ❡ ♠❡str❛❞♦ s✐♠✉❧t❛♥❡❛♠❡♥t❡✳ ❖❜r✐❣❛❞♦ ♣♦r s❡ ♠❛♥t❡r❡♠ ♠❡✉s ❛♠✐❣♦s ❞✉r❛♥t❡ t♦❞♦ ♦ ♣r♦❝❡ss♦✳ ❱❛♠♦s ❝♦♠❡♠♦r❛r ✭❡ ❜❡❜❡♠♦r❛r t❛♠❜é♠✮ ❛❣♦r❛✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❚❏✬s ♣♦r t♦❞❛s ❛s ✈✐✈ê♥❝✐❛s ❛ ❛♣r❡♥❞✐③❛❞♦s✳ ❆♣r❡♥❞✐ ♠✉✐t♦ ❝♦♠ ✈♦❝ês✳ ❊ s❡ ❤♦❥❡ s♦✉ ❝♦st✉♠❡✐r❛♠❡♥t❡ ❡❧♦❣✐❛❞♦ ♣❡❧❛ ❝♦♥❞✉t❛ ❡ ❜♦♥s ♠♦❞♦s✱ ❛tr✐❜✉♦ ❡♠ ❣r❛♥❞❡ ♣❛rt❡ ❛ ❡❞✉❝❛çã♦ r❡❝✐❜❛ ♣♦r ✈♦❝ês✳ Pr✐♥❝✐♣❛❧♠❡♥t❡✱ à ♠✐♥❤❛ ❛✈ó ●✐❝é❧✐❛✱ t✐❛s ❆♥❛ ❆♥❣é❧✐❝❛ ❡ ❊❞❝é❧✐❛✱ ❡ ♠❡✉s ❡t❡r♥♦s ❛♥ç✐ã♦s ❆✐❞♦✱ ❏♦sé✱ ▼❛r❝❡❧♦✱ ❘✐✈❛❧❞♦ ❡ ❘♦♥❛❧❞♦✳

➚s ❡q✉✐♣❡s ❞✐r❡t✐✈❛s✱ ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s ❞♦ ❈♦❧é❣✐♦ ❊st❛❞✉❛❧ ❏♦ã♦ ❳❳■■■✱ ❡♠ ❘✐❜❡✐✲ ró♣♦❧✐s✲❙❊✱ ♣♦r t❡r❡♠ ❡♥t❡♥❞✐❞♦ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♠✐♥❤❛s ❛✉sê♥❝✐❛s✱ ❜❡♠ ❝♦♠♦ ♣♦r ♥ã♦ ❡st❛r tã♦ ❡♥❣❛❥❛❞♦ ♥❛s ❞❡♠❛✐s ❛t✐✈✐❞❛❞❡s ♣❡❞❛❣ó❣✐❝❛s✱ ❝♦♠♦ ♠❡ é ❝♦st✉♠❡✐r♦✳

❆♦s ♠❡✉s ❝♦❧❡❣❛s✲❛♠✐❣♦s ❞❛ ✐♥❡sq✉❡❝í✈❡❧ t✉r♠❛ P❘❖❋▼❆❚ ✷✵✶✹✱ ♠❡✉ s✐♥❣❡❧♦ ♠✉✐t♦ ♦❜r✐❣❛❞♦✳ ❆s ❞❛♠❛s ♣r✐♠❡✐r♦✿ ❣r❛❝✐♦s❛ ✧♠❡♥✐♥❛✧▼ô♥✐❝❛✱ ❝♦♠♣❛♥❤❡✐r❛ ❙❛♠✐❧❧② ✭❝♦♠ ✷ ❧✬s ❡ ✶② ❦❦❦✮✱ ♣❛r❝❡✐r❛ ❙✐♠♦♥❡✱ s♦✉ ♠✉✐t♦ ❣r❛t♦ ♣♦r t❛♥t❛s ❤♦r❛s ❞❡ ❡st✉❞♦s✱ s❡❣r❡❞♦s✱ ❛❧♠♦ç♦s ❡ ✈✐❛❣❡♥s✳ ❖ ❜♦♠ ❞❡ ❞❡s❡♥❤♦ ❆♥❞❡rs♦♥✱ ♦ ❞♦♥♦ ❞♦s ♠✐❧ ✈í♥❝✉❧♦s ❆✉❣✉st♦✱ ♦ ❛♠✐❣♦ ❆r✐♦♥❛❧❞♦✱ ♦ ❛rt✐st❛ ❣❡♦♠étr✐❝♦ ❉❥❡♥❛❧✱ ♦ ✐♥t❡❧✐❣❡♥tíss✐♠♦ ❊♠❡rs♦♥✱ ♦ ❡①❡♠♣❧♦ ❞❡ ✈✐❞❛ ●✐❧❞♦✱ ♦ ❡①tr♦✈❡rt✐❞♦ ▼❛r❝❡❧♦ ❡ P❛✉❧♦ ✭✦✮✳ P♦r ú❧t✐♠♦✱ ♠❛s ♥ã♦ ♠❡♥♦s ✐♠♣♦rt❛♥t❡✱ ❏❛✐❧s♦♥✱ ♣❛r❝❡✐r♦✱ s✐♥❝❡r♦✱ ❤♦♥❡st♦✱ í♥t❡❣r♦ ❡ ✧♦❣r♦✲❛♠á✈❡❧✧✱ s✉❛ ❛♠✐③❛❞❡ ❢♦✐ ✉♠ ❞♦s ♠❡❧❤♦r❡s ♣r❡s❡♥t❡s q✉❡ ❡ss❡ ♠❡str❛❞♦ ♣♦❞❡r✐❛ ♠❡ ❞❛r✳

➚ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❡r❣✐♣❡✱ ❛tr❛✈és ❞♦ ❝♦r♣♦ ❞♦❝❡♥t❡ q✉❡ ♠✐♥✐str♦✉ ❛s ❛✉❧❛s ❞♦ ❈✉rs♦ ♥♦ ♣ó❧♦ ❞❡ ■t❛❜❛✐❛♥❛✱ ❆r❧ú❝✐♦✱ ➱❞❡r✱ ❘✐❝❛r❞♦✱ ▼❛rt❛✱ ❙❛♠✉❡❧✱ ♣❛ss❛♥❞♦ s✉❛s ❡①♣❡r✐ê♥❝✐❛s ❡ tr❛♥s♠✐t✐♥❞♦ ❝♦♥❤❡❝✐♠❡♥t♦s✳ Pr✐♥❝✐♣❛❧♠❡♥t❡ ❛♦ ❛♠✐❣♦ ❉r✳ ▼❛t❡✉s ❆❧❡❣r✐✱ q✉❡ ♠❡ ❛❝♦♠♣❛♥❤♦✉ ❞❡s❞❡ ❛ ❣r❛❞✉❛çã♦ ❡ t✐✈❡ ♦ ♣r✐✈✐❧é❣✐♦ ❞❡ tê✲❧♦ ❝♦♠♦ ♠❡♥t♦r ❡♠ t♦❞♦s ♦s ♣❡rí♦❞♦s ❞♦ ♠❡str❛❞♦✳ ❆♦s ♣r♦❢❡ss♦r❡s ❘❛❢❛❡❧ ❡ ❲❛❣♥❡r✱ ❝♦♠ ♦s q✉❛✐s t✐✈❡ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❛♣r❡♥❞❡r ♥❛s ✐♥ú♠❡r❛s ❞✐s❝✐♣❧✐♥❛s q✉❡ ❝✉rs❡✐ ❝♦♠ ❛♠❜♦s✱ ♥❛ ❣r❛❞✉❛çã♦✱ ❡ ♣❡❧❛s s✉❛s ❞✐❞át✐❝❛s í♠♣❛r❡s ❞❡✐①❛♥❞♦ ✉♠ ❡①❡♠♣❧♦ ♣❛r❛ s❡r s❡❣✉✐❞♦ ❞❡ ♣❡rt♦✳ ❆ ✈♦❝ês✱ ✉♠ ♠✉✐t♦ ♦❜r✐❣❛❞♦✳

➚ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛✲❙❇▼ ♣❡❧❛ ✐♠♣❧❛♥t❛çã♦ ❞♦ P❘❖❋▼❆❚✱ ♦ q✉❡ ♠❡ ♣♦ss✐❜✐❧✐t♦✉ ❛ r❡❛❧✐③❛çã♦ ❞❡ ✉♠ ♣r♦❥❡t♦ ♣❡ss♦❛❧✿ ❋❛③❡r ❛ Pós✲❣r❛❞✉❛çã♦✱ ♥♦ ♥í✈❡❧ ❞❡ ♠❡str❛❞♦❀ ❡ à ❈❆P❊❙ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ✜♥❛♥❝❡✐r♦✳

❆♦ ♣r♦❢❡ss♦r ❞♦✉t♦r ❆❧❡❥❛♥❞r♦ ❈❛✐❝❡❞♦ ❘♦q✉❡✱ ♣♦r t❡r ♠❡ ❛❝❡✐t❛❞♦ ❝♦♠♦ ♦r✐❡♥t❛♥❞♦✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❛♦ ❧♦♥❣♦ ❞♦s ❡st✉❞♦s✱ ♣❡❧❛s ✐♥str✉çõ❡s ❡ ❝rít✐❝❛s ❝♦♥str✉t✐✈❛s✳ ❊♥✜♠✱ ♠✉✐t♦ ♦❜r✐❣❛❞♦ ♣❡❧❛ ♣❛r❝❡r✐❛✳

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❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦✱ sã♦ tr❛t❛❞♦s ❛❧❣✉♥s ❡❧❡♠❡♥t♦s ❞❛ ❛♥á❧✐s❡ ❢✉♥❝✐♦♥❛❧ ❝♦♠♦ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✱ ❡s♣❛ç♦s ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt✱ ❡st✉❞❛♠♦s t❛♠❜é♠ sér✐❡s ❞❡ ❋♦✉r✐❡r ❡ ♥♦ ✜♥❛❧ ❝♦♥s✐❞❡r❛♠♦s ❜r❡✈❡♠❡♥t❡ ❛ ❡q✉❛çã♦ ❞❛ ❝♦r❞❛ ✈✐❜r❛♥t❡✳ ❈♦♠ ✐ss♦✱ ♣❡r❝❡❜❡✲s❡ q✉❡ ♥ã♦ s❡ ♣r❡❝✐s❛ ❞❡ ♠✉✐t❛ t❡♦r✐❛ ♣❛r❛ ❝♦♥s❡❣✉✐r♠♦s r❡s✉❧t❛❞♦s s✐❣♥✐✜❝❛t✐✈♦s✳

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❆❜str❛❝t

■♥ t❤✐s ✇♦r❦✱ ✇❡ ❛r❡ tr❡❛t❡❞ s♦♠❡ ❡❧❡♠❡♥ts ♦❢ ❢✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s s✉❝❤ ❛s ❇❛♥❛❝❤ s♣❛❝❡s✱ ✐♥♥❡r ♣r♦❞✉❝t s♣❛❝❡s ❛♥❞ ❍✐❧❜❡rt s♣❛❝❡s✱ ❛❧s♦ st✉❞✐❡❞ ❋♦✉r✐❡r s❡r✐❡s ❛♥❞ ❛t t❤❡ ❡♥❞ ❜r✐❡✢② ❝♦♥s✐❞❡r t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ✈✐❜r❛t✐♥❣ str✐♥❣✳ ❲✐t❤ t❤✐s✱ ②♦✉ r❡❛❧✐③❡ t❤❛t ②♦✉ ❞♦ ♥♦t ♥❡❡❞ ❛ ❧♦t ♦❢ t❤❡♦r② ✐♥ ♦r❞❡r t♦ ❣❡t s✐❣♥✐✜❝❛♥t r❡s✉❧ts✳

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■♥tr♦❞✉çã♦

❆té ❝❡rt♦ ♣♦♥t♦✱ ❛ ❛♥á❧✐s❡ ❢✉♥❝✐♦♥❛❧ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ❝♦♠♦ á❧❣❡❜r❛ ❧✐♥❡❛r ❡♠ ❡s♣❛ç♦s ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛ ❝♦♠❜✐♥❛❞❛ ❝♦♠ ❛ ❛♥á❧✐s❡✳ ❊ss❛ ❝♦♠❜✐♥❛çã♦ ♣❡r♠✐t❡ ❞❛r s❡♥t✐❞♦ ❛ ✐❞❡✐❛s t❛✐s ❝♦♠♦ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❝♦♥t✐♥✉✐❞❛❞❡✳ P♦r ✐ss♦✱ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ❜r❡✈❡♠❡♥t❡ r❡❝♦r❞❛r ❡ r❡s✉♠✐r ✈ár✐❛s ✐❞❡✐❛s ❡ r❡s✉❧t❛❞♦s q✉❡ sã♦ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦ ❡st✉❞♦ ❞♦ ❛♥á❧✐s❡ ❢✉♥❝✐♦♥❛❧✳ ❉❡♥tr❡ ❡ss❡s r❡s✉❧t❛❞♦s✱ t❡♠♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ á❧❣❡❜r❛ ❧✐♥❡❛r ❡ ✐❞❡✐❛s ❡❧❡♠❡♥t❛r❡s ❞❡ ❡s♣❛ç♦s ♠étr✐❝♦s✳ ◆❡st❡s ú❧t✐♠♦s tr❛t❛♠✲s❡ ❝♦♥❝❡✐t♦s ❛♥❛❧ít✐❝♦s✱ t❛✐s ❝♦♠♦ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ s❡q✉ê♥❝✐❛s ❡ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ❢✉♥çõ❡s✳ ◆♦s ❡s♣❛ç♦s ♠étr✐❝♦s ❡♠ ❣❡r❛❧ ♥❡♥❤✉♠❛ ♦✉tr❛ ❡str✉t✉r❛ é ✐♠♣♦st❛ ❛❧é♠ ❞❡ ✉♠❛ ♠étr✐❝❛✱ ❛ q✉❛❧ é ✉t✐❧✐③❛❞❛ ♣❛r❛ ❞✐s❝✉t✐r ❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❝♦♥t✐♥✉✐❞❛❞❡✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ❡ssê♥❝✐❛ ❞❡ ❛♥á❧✐s❡ ❢✉♥❝✐♦♥❛❧ é ❝♦♥s✐❞❡r❛r ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛ ✭❡s♣❛ç♦s ♠étr✐❝♦s✮ ❡ ❡st✉❞❛r ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ❛s ❡str✉t✉r❛s ❛❧❣é❜r✐❝❛s ❡ ♠étr✐❝❛s ❞❡ss❡s ❡s♣❛ç♦s✱ ❡s♣❡❝✐✜❝❛♠❡♥t❡ q✉❛♥❞♦ t❛✐s ❡s♣❛ç♦s sã♦ ♠étr✐❝♦s ❡ ♠étr✐❝♦s ❝♦♠♣❧❡t♦s✱ ♥❛ ❧✐t❡r❛t✉r❛ ♦s ❧✐✈r♦s ❞❡ ❛♥á❧✐s❡ ❢✉♥❝✐♦♥❛❧ ❝♦❜r❡♠ ❡st❡s tó♣✐❝♦s✳ ♣♦r ❡①❡♠♣❧♦ ♠❡♥❝✐♦♥❛♠♦s ♦s ❧✐✈r♦s ❬✷✱ ✹✱ ✺❪✳

❖✉tr❛ ❢❡rr❛♠❡♥t❛ ✐♠♣♦rt❛♥t❡ ✉s❛❞❛ ♥❡st❛ t❡♦r✐❛ é ❛ ✐♥t❡❣r❛❧ ❞❡ ▲❡❜❡s❣✉❡✳ ■st♦ ♣♦rq✉❡ ♠✉✐t♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❝♦♥s✐st❡♠ ❡♠ ❝♦♥❥✉♥t♦s ❞❡ ❢✉♥çõ❡s ✐♥t❡❣rá✈❡✐s✳ ❆ ✜♠ ❞❡ ✉s❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❡s♣❛❝✐❛✐s ♠étr✐❝❛s ❞❡s❡❥á✈❡✐s✱ t❛✐s ❝♦♠♦ ❝♦♠♣❧❡t❡s✱ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ✉s❛r ❛ ✐♥t❡❣r❛❧ ❞❡ ▲❡❜❡s❣✉❡ ♥♦ ❧✉❣❛r ❞❡ ❛ ✐♥t❡❣r❛❧ ❞❡ ❘✐❡♠❛♥♥✱ ♥♦r♠❛❧♠❡♥t❡ ❞✐s❝✉t✐❞❛ ♥♦s ❝✉rs♦s ❞❡ ❛♥á❧✐s❡✱ ✈❡r ♠❛✐s ❞❡t❛❧❤❡s ❡♠ ❬✶❪✳

◆❡st❡ ❡st✉❞♦✱ t❡♥t❛r❡♠♦s ♠♦❞❡❧❛r ❡ ❝♦♠♣r❡❡♥❞❡r ♦ ♣r♦❜❧❡♠❛ ❞❛ ❝♦r❞❛ ✈✐❜r❛♥t❡✱ ♦ q✉❛❧ é ✉♠ s✐st❡♠❛ ❡st✉❞❛❞♦ ♣❡❧♦s ❢ís✐❝♦s ❡ ♠❛t❡♠át✐❝♦s ♥❛ ❤✐stór✐❛ ❞❛ ❝✐ê♥❝✐❛❀ t❛❧ ♣r♦❜❧❡♠❛ ❞❛t❛ ♣❛r❛❧❡❧❛♠❡♥t❡ ❝♦♠ à ❡s❝♦❧❛ ♣✐t❛❣ór✐❝❛ ✭s❡❝✳ ❱■ ❛✳❝✳✮✳ ❙❡♥❞♦ ✜①❛s ❛s ❞✉❛s ❡①tr❡♠✐❞❛❞❡s ❞❛ ❝♦r❞❛✱ ♦♥❞❡ ♣õ❡✲s❡ ❡♠ ✈✐❜r❛çã♦ ❛❢❛st❛♥❞♦ ✉♠ ❞♦s s❡✉s ♣♦♥t♦s ❞❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡stá✈❡❧ ❬✼❪✳

❆s ❝♦r❞❛s ✈✐❜r❛♥t❡s sã♦ ✉♠ t❡♠❛ ✐♠♣♦rt❛♥t❡ ♥❛ ❋ís✐❝❛✱ ❛❧❣✉♥s ❞♦s ♣r✐♠❡✐r♦s ❡st✉❞♦s ♣♦❞❡♠ s❡r ✈✐st♦s ❡♠ ❬✻❪✳ P♦r ❡①❡♠♣❧♦✱ ♥♦ r❡❢❡r❡♥t❡ à ♠ús✐❝❛ ❛s ❝♦r❞❛s ❞♦s ✐♥str✉♠❡♥t♦s ♠✉s✐❝❛✐s sã♦ ❝♦r❞❛s ✈✐❜r❛♥t❡s✱ ♣❡r♠✐t✐♥❞♦ ❛ss✐♠ ♦ ❡st✉❞♦ ❞❛s ❝♦r❞❛s ✈✐❜r❛♥t❡s ❛ ❝♦♠♣r❡✲ ❡♥sã♦ ❞♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦s ✐♥str✉♠❡♥t♦s ❞❡ ❝♦r❞❛ ✭❣✉✐t❛rr❛✱ ♣✐❛♥♦✱ ❤❛r♣❛✱ ✈✐♦❧✐♥♦✱ ✈✐♦❧❛✱ ✈✐♦❧♦♥❝❡❧♦✱ ❝♦♥tr❛❜❛✐①♦✱ ❡t❝✮✳

❆s ❡q✉❛çõ❡s ❞❛ ❝♦r❞❛ ✈✐❜r❛♥t❡ sã♦ ♠♦❞❡❧❛❞❛s ♣♦r ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s q✉❡ ❛❞♠✐t❡♠ ❝♦♠♦ s♦❧✉çã♦ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ❢✉♥çõ❡s ❝❤❛♠❛❞❛s ❡①♣❛♥sõ❡s ❞♦ s❡♥♦ ♦✉ ❞♦ ❝♦ss❡♥♦ ❞❡ ❋♦✉r✐❡r✳ ❆s sér✐❡s ❞❡ ❋♦✉r✐❡r✱ ♣♦r s✉❛ ✈❡③✱ é ❛ s♦♠❛ ❞❡ t❡r♠♦s ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ♦rt♦♥♦r♠❛❧ q✉❡ ❢♦r♠❛♠ ❜❛s❡ ♣❛r❛ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❞❛❞♦ ❬✸✱ ✺❪✳

P♦r t❛✐s ♠♦t✐✈♦s✱ ♥♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ t✐t✉❧❛❞♦ ♣r❡❧✐♠✐♥❛r❡s✱ ❡♥✉♥❝✐❛r❡♠♦s ❛❧❣✉♥s r❡✲ s✉❧t❛❞♦s s♦❜r❡ ❡s♣❛ç♦s ♠étr✐❝♦s ❡ t❡♦r✐❛ ❞❛ ♠❡❞✐❞❛✳ ◆♦ s❡❣✉♥❞♦✱ ❞❛r❡♠♦s ❛t❡♥çã♦ ❛♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ♥♦r♠❛❞♦s✱ ❡ ❞❡♣♦✐s ♦s ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✳

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❊♠ s❡❣✉✐❞❛✱ ♥♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ ❞❡✜♥✐r❡♠♦s ❡st✉❞❛r❡♠♦s ♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✱ ♣❛r❛ ❞❡♣♦✐s tr❛t❛r ❛ ❝♦♠♣❧❡t❡s ❞❡ t❛✐s ❡s♣❛ç♦s ❡ ❡st✉❞❛r ♦s ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt✳

P♦r ✜♠✱ ♥♦ ú❧t✐♠♦ ❝❛♣ít✉❧♦ s❡rá tr❛t❛❞♦ ❛♦ r❡s♣❡✐t♦ ❞❛ ❡q✉❛çã♦ ❞❛ ❝♦r❞❛ ✈✐❜r❛♥t❡✳ ❆♥t❡s ❞❡ ❝❤❡❣❛r ♥❛ ♠♦❞❡❧❛❣❡♠ ❞❡ss❛ ❡q✉❛çã♦✱ ✐r❡♠♦s ❞❡♠♦♥str❛r q✉❡ ❛s ♣❛r❝❡❧❛s ❞❛s ❡①♣❛♥sõ❡s ❞❡ ✉♠❛ ❢✉♥çã♦f ❞♦ s❡♥♦ ❡ ❞♦ ❝♦ss❡♥♦ ❞❡ ❋♦✉r✐❡r sã♦ ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦

♦rt♦♥♦r♠❛❧✳

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❈❛♣ít✉❧♦ ✶

Pr❡❧✐♠✐♥❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦✱ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r✱ ❊s♣❛ç♦s ▼étr✐✲ ❝♦s ❡ ■♥t❡❣r❛❧ ❞❡ ▲❡❜❡sq✉❡✱ ♥❡❝❡ssár✐♦s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❞❡♠♦♥str❛çã♦ ❞❡ r❡s✉❧t❛❞♦s ♥♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s✳

❚❡♦r❡♠❛ ✶✳✶ ❙❡❥❛ (M, d) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❡ s❡❥❛ A_⊂M.

✭❛✮ A é ❢❡❝❤❛❞♦ ❡ é ✐❣✉❛❧ à ✐♥t❡rs❡çã♦ ❞❛s ❝♦❧❡çõ❡s ❞❡ t♦❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s ❢❡❝❤❛❞♦s ❞❡

▼ q✉❡ ❝♦♥té♠ ❆ ✭❆ss✐♠ A é ♦ ♠❡♥♦r ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ q✉❡ ❝♦♥t❡♠ A✮❀

✭❜✮ ❆ é ❢❡❝❤❛❞♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ A=A❀

✭❝✮ ❆ é ❢❡❝❤❛❞♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ {xn} ❡♠ ❆ q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ ✉♠

❡❧❡♠❡♥t♦ x_∈M✱ ❡♥tã♦ x_∈A❀

✭❞✮ x_∈A s❡✱ ❡ s♦♠❡♥t❡ s❡✱ inf_{d(x, y);y _∈A_}= 0❀

❚❡♦r❡♠❛ ✶✳✷ ❙✉♣♦♥❤❛ q✉❡ (M, d) é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❡ s❡❥❛ A_⊂M. ❊♥tã♦✿

✭❛✮ ❙❡ ❆ é ❝♦♠♣❧❡t♦✱ ❡♥tã♦ ❡❧❡ é ❢❡❝❤❛❞♦❀

✭❜✮ ❙❡ ▼ é ❝♦♠♣❧❡t♦✱ ❡♥tã♦ ❆ é ❝♦♠♣❧❡t♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❡ ❢♦r ❢❡❝❤❛❞♦❀ ✭❝✮ ❙❡ ❆ é ❝♦♠♣❛❝t♦✱ ❡♥tã♦ ❡❧❡ é ❢❡❝❤❛❞♦ ❡ ❧✐♠✐t❛❞♦❀

✭❞✮ ❚♦❞♦ s✉❜❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ ❡ ❧✐♠✐t❛❞♦ ❞❡ Fn é ❝♦♠♣❛❝t♦✳

❙❡❥❛CF(M) ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s f :M →F❝♦♥t✐♥✉❛s✳ ◆ós ♦♠✐t✐r❡♠♦s F❡ s✐♠♣❧✐✜❝❛✲

r❡♠♦s ❛ ❡s❝r✐t❛ C(M)✳

❚❡♦r❡♠❛ ✶✳✸ ❖ ❡s♣❛ç♦ ♠étr✐❝♦ C(M) é ❝♦♠♣❧❡t♦✳

❉❡✜♥✐çã♦ ✶✳✶ ❙✉♣♦♥❤❛ q✉❡ f é ✉♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ ❡ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ b t❛❧ q✉❡ f(x)_≤b ❡♠ q✉❛s❡ t♦❞♦s ♦s ♣♦♥t♦s✳ ❊♥tã♦ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ s✉♣r❡♠♦ ❡ss❡♥❝✐❛❧ ❞❡ f ♣♦r

esssupf = inf_{b:f(x)_≤b ❛✳❡✳ _}.

(13)

❉❡✜♥✐çã♦ ✶✳✷ ❉❡✜♥✐♠♦s ♦s ❡s♣❛ç♦s

Lp₍_X_{) =}

{f :f é ♠❡♥s✉rá✈❡❧ ❡ (R

X|f|

p_dµ₎1p <∞},1≤p <∞;

L∞₍_X_{) =}_{_f _:_f _{é ♠❡♥s✉rá✈❡❧ ❡} _sup_|_f_|_<_∞}_.

◗✉❛♥❞♦ X = [a, b]_⊂R é ✉♠ ✐♥t❡r✈❛❧♦ ❧✐♠✐t❛❞♦ ❡ ₁_≤_p≤ ∞✱ ♥ós ❡s❝r❡✈❡♠♦s _Lp_[_{a, b}_]_✳

❚❡♦r❡♠❛ ✶✳✹ ❙✉♣♦♥❤❛ q✉❡ 1_≤p_{≤ ∞}✳ ❊♥tã♦ ♦ ❡s♣❛ç♦ ♠étr✐❝♦ Lp₍_X₎ _{é ❝♦♠♣❧❡t♦✳ ❊♠}

♣❛rt✐❝✉❧❛r✱ ♦ ❡s♣❛ç♦ ❞❛s s❡q✉ê♥❝✐❛s lp _{é ❝♦♠♣❧❡t♦✳}

❚❡♦r❡♠❛ ✶✳✺ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍♦❧❞❡r

❙❡❥❛♠ 1< p, q < _∞ ❝♦♥❥✉❣❛❞♦s ❞❡ ▲❡❜❡s❣✉❡✱ ♦✉ s❡❥❛✱ 1

p +

1

q = 1✳

❙❡❥❛♠ {an} ❡ {bn} s❡q✉ê♥❝✐❛s ❞❡ ♥ú♠❡r♦s r❡❛✐s ♦✉ ❝♦♠♣❧❡①♦s✳ ❊♥tã♦✿

|

N

X

n=1

anbn| ≤( N

X

n=1

|an|p)

1

p ·(

N

X

n=1

|bn|q)

1

q.

❚❡♦r❡♠❛ ✶✳✻ ❇❡♣♣♦ ▲❡✈✐ ✭♦✉ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ▼♦♥ót♦♥❛✮ ❙❡❥❛ (X,P

, µ)✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ _hfnin∈N✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s r❡❛✐s ✐♥t❡❣rá✈❡✐s

❡♠ ❳ t❛✐s q✉❡

f(x) = lim

n_→∞fn(x), µ ✲ qt♣✳ ❡♠ X.

❙✉♣♦♥❤❛ q✉❡ ❛ s❡q✉ê♥❝✐❛ é ♠♦♥ót♦♥❛ ❝r❡s❝❡♥t❡✳ ❙❡ supn_∈N R

fndµ < ∞✱ ❡♥tã♦ f é ✐♥t❡✲

❣rá✈❡❧ ❡

Z

f dµ= lim

n→∞

Z

fndµ.

▲❡♠❛ ✶✳✶ ✭❞❡ ❋❛t♦✉✮

❙❡❥❛ fn:E →R ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s ♥ã♦ ♥❡❣❛t✐✈❛s✱ ❡♥tã♦✿

Z

( lim

n→∞inffn)dµ≤nlim→∞inf

Z

fndµ.

(14)

❈❛♣ít✉❧♦ ✷

❊s♣❛ç♦s ◆♦r♠❛❞♦s

◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❝♦♠♣r❡❡♥❞❡r ❛s ❙ér✐❡s ❞❡ ❋♦✉r✐❡r ❡ ❛❧❣✉♠❛s ❞❡ s✉❛s ❛♣❧✐❝❛çõ❡s✳

✷✳✶ ❊①❡♠♣❧♦s ❞❡ ❊s♣❛ç♦s ◆♦r♠❛❞♦s

❖s ❡s♣❛ç♦s ♥♦r♠❛❞♦s sã♦ ❡str✉t✉r❛s ♠❛✐s r✐❝❛s q✉❡ ♦s ❡s♣❛ç♦s ♠étr✐❝♦s✱ ✐st♦ é✱ sã♦ ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s q✉❡ ♣♦ss✉❡♠ ❞✉❛s ♦♣❡r❛çõ❡s ❢❡❝❤❛❞❛s ❞❡✜♥✐❞❛s s♦❜r❡ ❡❧❡✳ ❯♠❛ ❞❡❧❛s é ❛ s♦♠❛ ❞❡ ✈❡t♦r❡s✱ ❡ ❛ ♦✉tr❛ ♦ ♣r♦❞✉t♦ ♣♦r ✉♠ ❡s❝❛❧❛r✱ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳

▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ q✉❛♥❞♦ ♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s R2 ❡ R3 sã♦ r❡♣r❡s❡♥t❛❞♦s ♥♦ s❡♥t✐❞♦

✉s✉❛❧✱ t❡♠♦s ❛ ✐❞❡✐❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ✈❡t♦r ❡♠ R2 ❡ R3 ❛ss♦❝✐❛❞♦ ❛ ❝❛❞❛ ✈❡t♦r✳

❊st❛ é ❝❧❛r❛♠❡♥t❡ ✉♠❛ ✈❛♥t❛❣❡♠ q✉❡ ♥♦s ❞á ✉♠❛ ❝♦♠♣r❡❡♥sã♦ ♠❛✐s ❛♣r♦❢✉♥❞❛❞❛ ❞❡ss❡s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ◗✉❛♥❞♦ ♥ós ♠✉❞❛♠♦s ♣❛r❛ ♦✉tr♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ✭♣♦ss✐✈❡❧♠❡♥t❡ ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛✮✱ ♣♦❞❡♠♦s t❡r ❛ ❡s♣❡r❛♥ç❛ ❞❡ ♦❜t❡r ♠❛✐s ❞❡t❛❧❤❡s s♦❜r❡ ❡ss❡s ❡s♣❛ç♦s s❡ ♣✉❞❡r♠♦s✱ ❞❡ ❛❧❣✉♠ ♠♦❞♦✱ ❛tr✐❜✉✐r ❛❧❣♦ s❡♠❡❧❤❛♥t❡ ❛♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ✈❡t♦r ♣❛r❛ ❝❛❞❛ ✈❡t♦r ♥♦ ❡s♣❛ç♦✳

❈♦♥s❡q✉❡♥t❡♠❡♥t❡ ♦❧❤❛♠♦s ♣❛r❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❛①✐♦♠❛s q✉❡ sã♦ s❛t✐s❢❡✐t♦s ♣❛r❛ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ✈❡t♦r ❡♠R2 _❡_R3_{✳ ❊st❡ ❝♦♥❥✉♥t♦ ❞❡ ❛①✐♦♠❛s ✈❛✐ ❞❡✜♥✐r ❛ ✧♥♦r♠❛✧❞❡}

✉♠ ✈❡t♦r✱ ❡ ❛♦ ❧♦♥❣♦ ❞❡st❛ ❞✐ss❡rt❛çã♦ ♥ós ✈❛♠♦s ❝♦♥s✐❞❡r❛r ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡s♣❛ç♦s ✈❡t♦✲ r✐❛✐s ♥♦r♠❛❞♦s✳ ◆❡st❡ ❝❛♣ít✉❧♦ ♥ós ✐♥✈❡st✐❣❛r❡♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s ❡❧❡♠❡♥t❛r❡s ❞❡ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ♥♦r♠❛❞♦s✳

❉❡✜♥✐çã♦ ✷✳✶ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ R✳ ❯♠❛ ♥♦r♠❛ ❡♠ _X é ✉♠❛ ❢✉♥çã♦

k·k:X _→R t❛❧ q✉❡ ♣❛r❛ t♦❞♦s _{x, y} _∈_X ❡ _α _∈R✱

✭✐✮ kx_{k ≥}0❀

✭✐✐✮ kx_k= 0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x= 0❀

✭✐✐✐✮ kαx_k=_|α_|._kx_k❀

✭✐✈✮ kx+y_{k ≤ k}x_k+_ky_k✳

(15)

❈♦♠♦ ✉♠❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦❧❤❛r ❛s ♥♦r♠❛s✱ ✐♠♣❧✐❝❛♠♦s q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ✈❡t♦r ❡♠ R2 ❡ R3 s❛t✐s❢❛③ ♦s ❛①✐♦♠❛s ❞❡ ✉♠❛ ♥♦r♠❛✳ ■st♦ s❡rá ✈❡r✐✜❝❛❞♦ ♥♦ ❡①❡♠♣❧♦

✷✳✷✱ ♠❛s ✈❛❧❡ ♠❡♥❝✐♦♥❛r q✉❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ✐t❡♠ ✭✐✈✮ ❞❛ ❞❡✜♥✐çã♦ ✷✳✶ é ❝❤❛♠❛❞❛ ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ ✉♠❛ ✈❡③ q✉❡✱ ❡♠R2✱ ❞✐③❡♠♦s s✐♠♣❧❡s♠❡♥t❡ q✉❡ ❛ ♠❡❞✐❞❛ ❞❡ ✉♠

❧❛❞♦ ❞♦ tr✐❛♥❣✉❧♦ é s❡♠♣r❡ ♠❡♥♦r q✉❡ ❛ s♦♠❛ ❞❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ❞♦s ♦✉tr♦s ❞♦✐s✳

❊①❡♠♣❧♦ ✷✳✶ ❆ ❢✉♥çã♦ k·k: Rn _→_R _{❞❡✜♥✐❞❛ ♣♦r} _k₍_x

1, ..., xn)k = (Pn_j₌₁|xj|2)

1

2 é ✉♠❛

♥♦r♠❛ ❡♠ Rn ❝❤❛♠❛❞❛ ❞❡ ♥♦r♠❛ ✉s✉❛❧ ✭♦✉ ❝❛♥ô♥✐❝❛✮ ❡♠ Rn✳

◆ã♦ ❞❛r❡♠♦s ❛ s♦❧✉çã♦ ❞♦ ❡①❡♠♣❧♦ ✷✳✶ ♣♦✐s ♦ ❣❡♥❡r❛❧✐③❛r❡♠♦s ♥♦ ❡①❡♠♣❧♦ ✷✳✸✳ ❈♦♠♦Fn

é t❛❧✈❡③ ♦ ❡s♣❛ç♦ ♥♦r♠❛❞♦ ♠❛✐s s✐♠♣❧❡s ❞❡ ✈✐s✉❛❧✐③❛r✱ q✉❛♥❞♦ t♦❞❛s ❛s ♥♦✈❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ♥♦r♠❛❞♦s sã♦ ✐♥tr♦❞✉③✐❞❛s ♣♦st❡r✐♦r♠❡♥t❡✱ ❡❧❡ ♣♦❞❡ s❡r út✐❧ ♣❛r❛ t❡♥t❛r ✈❡r ♦ q✉❡ s✐❣♥✐✜❝❛ ♣r✐♠❡✐r♦ ♥♦ ❡s♣❛ç♦Fn ♠❡s♠♦ q✉❡ ❡❧❡ t❡♥❤❛ ❞✐♠❡♥sã♦ ✜♥✐t❛✳

❊①❡♠♣❧♦ ✷✳✷ ❙❡❥❛X ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ s♦❜r❡R❝♦♠ ❜❛s❡_{_e₁_{, e}₂_{, ..., e}_n}✳

◗✉❛❧q✉❡r x_∈X ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ Pn

j=1λj.ej ♣❛r❛ ú♥✐❝♦sλ1, λ2, ..., λn∈R✳ ❊♥tã♦ ❛

❢✉♥çã♦k·k:X _→R ❞❡✜♥✐❞❛ ♣♦r _k_x_k_{= (}Pn

j=1|λj|2)

1

2 é ✉♠❛ ♥♦r♠❛ ❡♠ X✳

❙♦❧✉çã♦✿ ❙❡❥❛♠ x = Pn

j=1λj.ej✱ y = Pn_j₌₁µj.ej✱ ✈❡t♦r❡s ❞❡ X ❡ α ∈ F✳ ❊♥tã♦✱ αx =

Pn

j=1αλj.ej ❡✿

✭✐✮ kx_k = (Pn

j=1|λj|2)

1

2 ≥ 0 ♣♦r s❡r ❛ r❛✐③ q✉❛❞r❛❞❛ ❞❡ ✉♠❛ s♦♠❛ ❞❡ ♥ú♠❡r♦s ♥ã♦

♥❡❣❛t✐✈♦s✳

✭✐✐✮ ❙❡ x = 0✱ ❡♥tã♦ _kx_k = 0✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ _kx_k = 0 ❡♥tã♦ (Pn

j=1|λj|2)

1 2 = 0✳

❉♦♥❞❡ s❡❣✉❡ q✉❡λj = 0 ♣❛r❛ 1≤j ≤n✳ ▲♦❣♦✱ x= 0✳

✭✐✐✐✮

kαx_k=

( n X j=1

|αλj|2)

1 2

=_|α_|.(

n

X

j=1

|λj|2)

1

2 =|α| kxk

✭✐✈✮

kx+y_k2 =

n

X

j=1

|λj+µj|2 = n

X

j=1

|λj|2+ n

X

j=1

λjµj+ n

X

j=1

λjµj + n

X

j=1

|µj|2

=

n

X

j=1

|λj|2 + 2 n

X

j=1

Re(λjµj) + n

X

j=1

|µj|2

≤

n

X

j=1

|λj|2 + 2 n

X

j=1

|λj||µj|+ n

X

j=1

|µj|2

=

n

X

j=1

|λj|2 + 2( n

X

j=1

|λj|2)

1 2 ·(

n

X

j=1

|µj|2)

1 2 +

n

X

j=1

|µj|2

= _kx_k2+ 2_kx_{k k}y_k+_ky_k2

= (_kx_k+_ky_k)2.

P♦rt❛♥t♦✱ kx+y_{k ≤ k}x_k+_kyk✳

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❊①❡♠♣❧♦ ✷✳✸ ❙❡❥❛ ❙ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ q✉❛❧q✉❡r ❡ s❡❥❛ ❳ ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ s♦❜r❡ F✳ ❙❡❥❛ Fb(S, X) ♦ s✉❜❡s♣❛ç♦ ❧✐♥❡❛r ❞❡ F(S, X) ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s f : S → X t❛❧ q✉❡

{kf(x)_k;x_∈S_} é ❧✐♠✐t❛❞♦✳ ▼♦str❡ q✉❡ _Fb(S, X) t❡♠ ✉♠❛ ♥♦r♠❛ ❞❡✜♥✐❞❛ ♣♦r

kf_kb = sup{kf(s)k;s∈S}.

❙♦❧✉çã♦✿ ❙❡❥❛♠ f, g_{∈ F}b(S, X)❡ α∈F✳

✭✐✮ kf_kb = sup{kf(s)k;s ∈S} ≥0✳

✭✐✐✮ ❙❡ f = 0✱ ❡♥tã♦ f(s) = 0✱ ♣❛r❛ t♦❞♦ s _∈ S✳ ❉❛í✱ _kf(s)_k = 0✱ ♣❛r❛ t♦❞♦ s _∈ S ❡✱

❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ kf_kb = 0

P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ kf_kb = sup{kf(s)k;s ∈ S} = 0✱ ❡♥tã♦ kf(s)k = 0✱ ♣❛r❛ t♦❞♦

s _∈S✳ ❆ss✐♠✱ f(s) = 0✱ ♣❛r❛ t♦❞♦ s _∈S ❡✱ ❡♥tã♦ f = 0 ✳

✭✐✐✐✮ kαf_kb = sup{kαf(s)k;s∈S}=|α| ·sup{kf(s)k;s∈S}=|α| · kfkb✳

✭✐✈✮ ◆♦t❡ q✉❡✱ kf(s) +g(s)_{k ≤ k}f(s)_k+_kg(s)_{k ≤ k}f(s)_kb+kg(s)kb✱ ♣❛r❛ t♦❞♦ s ∈ S✳

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ kf +g_kb = sup{kf(s) +g(s)k;s ∈S} ≤ kf(s)kb+kg(s)kb

❊①❡♠♣❧♦ ✷✳✹ ❙❡❥❛ M ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❛❝t♦ ❡ s❡❥❛ _CF(M) ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡

❢✉♥çõ❡s ❝♦♥tí♥✉❛s s♦❜r❡F❞❡✜♥✐❞❛s ❡♠ _M✳ ❊♥tã♦ ❛ ❢✉♥çã♦ _k·k_:_C_F₍_M₎_→R ❞❡✜♥✐❞❛ ♣♦r

kf_k= sup_{|f(x)_|:x_∈M_}é ✉♠❛ ♥♦r♠❛ ❡♠ _CF(M)❝❤❛♠❛❞❛ ❞❡ ♥♦r♠❛ ✉s✉❛❧ ❡♠CF(M)✳

❙♦❧✉çã♦✿ ❙❡❥❛♠ f, g_{∈ C}F(M) ❡ α∈F✳

✭✐✮ kf_k= sup_{|f(x)_|:x_∈M_{} ≥}0✳

✭✐✐✮ ❙❡ f é ❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ ♥✉❧❛✱ ❡♥tã♦f(x) = 0✱ ♣❛r❛ t♦❞♦ x_∈M✳ ❊✱ ❞❛í✱

kf_k= sup_{|f(x) :x_∈M_}= 0.

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡kf_k= 0✱ ❡♥tã♦ sup_|f(x)_|:x_∈M = 0. P♦r ✐ss♦✱ f(x) = 0✱ ♣❛r❛ t♦❞♦

x_∈M✳ ▲♦❣♦✱ ❛ ❢✉♥çã♦ é ♥✉❧❛✳

✭✐✐✐✮ kαf_k= 0✱ ❡♥tã♦

sup_{|αf(x)_|:x_∈M_}=_|α_|sup_{|f(x) :x_∈M_}=_|α_{| k}f_k.

✭✐✈✮ ❙❡ y_∈M✱ ❡♥tã♦

|(f+g)(y)_{| ≤ |}f(y)_|+_|g(y)_{| ≤ k}f_k+_kg_k.

P♦rt❛♥t♦✱ k(f+g)(y)_k=sup_{|(f +g)(x)_|:x_∈M_{} ≤ k}f_k+_kg_k.

❊①❡♠♣❧♦ ✷✳✺ P❛r❛ ❝❛❞❛ n _∈N✱ s❡❥❛ _f_n _{: [0}_,_1]_→ R ❞❡✜♥✐❞❛ ♣♦r _f_n₍_x_{) =} _xn_{✳ ❊♥❝♦♥tr❡}

❛ ♥♦r♠❛ fn ♥♦s s❡❣✉✐♥t❡s ❝❛s♦s✿

✭❛✮ ♥♦ ❡s♣❛ç♦ ♥♦r♠❛❞♦ CR([0,1])❀

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✭❜✮ ♥♦ ❡s♣❛ç♦ ♥♦r♠❛❞♦ L1_[0_,_1]_✳

❙♦❧✉çã♦✿

❛✮ ❯s❛♥❞♦ ❛ ♥♦r♠❛ ❝❛♥ô♥✐❝❛✱ t❡♠♦s✿

kfnk= sup{kfn(x)k;x∈[0,1]}= 1.

❜✮ ❈♦♠♦ f é ❝♦♥tí♥✉❛✱ ♣♦r ▲❡❜❡s❣✉❡✱

kfnk= (

Z 1

0 |

fn|(x)dx)

1

n = Z 1

0 |

xn_|dx = [x

n+1

n+ 1] = 1

n+ 1.

◆♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦✱ ♠♦str❛r❡♠♦s q✉❡ ❛❧❣✉♥s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❞❡ ❢✉♥çõ❡s ✐♥t❡❣rá✈❡✐s ❞❡✜♥✐❞♦s ♥❛s ♣r❡❧✐♠✐♥❛r❡s t❡♠ ♥♦r♠❛✳ ❘❡❝♦r❞❛r❡♠♦s q✉❡ s❡ (X,P

, µ) é ✉♠ ❡s♣❛ç♦ ❞❡

♠❡❞✐❞❛ ❡ 1_≤p < _∞✱ ❡♥tã♦ ♦s ❡s♣❛ç♦s Lp₍_X₎_{❢♦r❛♠ ✐♥tr♦❞✉③✐❞♦s ♥❛ ❞❡✜♥✐çã♦ ✶✳✷✳}

❊①❡♠♣❧♦ ✷✳✻ ❙❡❥❛ (X,P

, µ) ✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛✳

✭✐✮ ❙❡ 16p < ∞✱ ❡♥tã♦

kf_k_p = (

Z

x|

f_|pdµ)1p

é ❛ ♥♦r♠❛ ❡♠ Lp₍_X₎ _{❝❤❛♠❛❞❛ ❞❡ ♥♦r♠❛ ✉s✉❛❧ ❡♠} _Lp₍_X₎_❀

✭✐✐✮ kf_k_∞=esssup_{|f(x)_|:x_∈X_} é ❛ ♥♦r♠❛ ❡♠ L∞₍_X₎ _{❝❤❛♠❛❞❛ ❛ ♥♦r♠❛ ✉s✉❛❧ ❡♠}

L∞₍_X₎_✳

❆ ♥♦t❛çã♦ ❡s♣❡❝í✜❝❛ ✐♥tr♦❞✉③✐❞❛ ♥❛s ♣r❡❧✐♠✐♥❛r❡s ♣❛r❛ ♦ ❝❛s♦ ❞❡ ♠❡❞✐❞❛s ❝♦♥tá✈❡✐s ❡♠

N✳ ❘❡❧❡♠❜r❛♥❞♦ q✉❡ _lp _{é ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ t♦❞❛s ❛s s❡q✉ê♥❝✐❛s}

{xn} ❡♠ F t❛✐s q✉❡

P∞

n=1|xn|p <∞♣❛r❛ 1≤p <∞ ❡ l∞ ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ t♦❞❛s ❛s s❡q✉ê♥❝✐❛s ❧✐♠✐t❛❞❛s

❡♠F✳ P♦rt❛♥t♦✱ s❡ ❧❡✈❛r♠♦s ❛ ♠❡❞✐❞❛ ❞❡ ❝♦♥t❛❣❡♠ ❡♠ N♥♦ ❊①❡♠♣❧♦ ✷✳✻ ♥ós ❞❡❞✉③✐♠♦s

q✉❡lp _♣❛r❛ ₁

≤p <_∞ ❡l∞ _{sã♦ ❡s♣❛ç♦s ♥♦r♠❛❞♦s✳}

P❛r❛ ❝♦♠♣❧❡t❛r♠♦s ♥♦ss❛ ❞❡✜♥✐çã♦ ❞❡ ♥♦r♠❛ ♥❡ss❡s ❡s♣❛ç♦s✱ ✈❡❥❛♠♦s ♦ ❡①❡♠♣❧♦ ✷✳✼✳

❊①❡♠♣❧♦ ✷✳✼ ✳

✭✐✮ ❙❡ 1 6 x < _∞✱ ❡♥tã♦ _kxnk_p = (P∞_n₌₁|xn|p)

1

pé ❛ ♥♦r♠❛ ❡♠ lp ❝❤❛♠❛❞❛ ❞❡ ♥♦r♠❛

✉s✉❛❧ ❡♠ lp_❀

✭✐✐✮ kxnk_p =sup{kxnk:n∈N} é ✉♠❛ ♥♦r♠❛ ✉s✉❛❧ ❡♠ lp✳

❉❡ ❛❣♦r❛ ❡♠ ❞✐❛♥t❡✱ s❡ ❡s❝r❡✈❡r♠♦s q✉❛✐sq✉❡r ❞♦s ❡s♣❛ç♦s ♥♦s ❡①❡♠♣❧♦s ✷✳✶✱ ✷✳✺✱ ✷✳✻ ❡ ✷✳✼ s❡♠ ♠❡♥❝✐♦♥❛r ❡①♣❧✐❝✐t❛♠❡♥t❡ ❛ ♥♦r♠❛✱ ❛ss✉♠✐r❡♠♦s q✉❡ ❛ ♥♦r♠❛ ✉s❛❞❛ é ❛ ♥♦r♠❛ ✉s✉❛❧ ❞❡ss❡s ❡s♣❛ç♦s✳

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❊①❡♠♣❧♦ ✷✳✽ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ❛ ♥♦r♠❛_k·k ❡ s❡❥❛S ✉♠ s✉❜❡s♣❛ç♦ ❧✐♥❡❛r

❞❡ X✳ ❙❡❥❛ _k·k_S ❛ r❡str✐çã♦ ❞❡ _k·k ❛ S✳ ❊♥tã♦ _k·k_S é ✉♠❛ ♥♦r♠❛ ❡♠ S✳

❊①❡♠♣❧♦ ✷✳✾ ❙❡❥❛♠ X✱ Y ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ♥♦r♠❛❞♦s s♦❜r❡ F ❡ _Z ₌_X_×_Y ♦ ♣r♦❞✉t♦

❝❛rt❡s✐❛♥♦ ❞❡ X ❡ Y✳ ❙❡ (_k·k)1 é ✉♠❛ ♥♦r♠❛ ❡♠ X ❡ (k·k)2 é ✉♠❛ ♥♦r♠❛ ❡♠ Y✱ ❡♥tã♦

k(x, y)_k=_kx_k₁+_ky_k₂ ❞❡✜♥❡ ✉♠❛ ♥♦r♠❛ ❡♠ Z✳

❙♦❧✉çã♦✿ ❙❡❥❛♠ (x, y)❡ (a, b)_∈Z✳

✭❛✮ k(x, y)_k=_kx_k₁+_ky_k₂ _≥0✱ ♣♦✐s_kx_k₁ _≥0 ❡_ky_k₂ _≥0❀

✭❜✮ ❙❡ k(x, y)_k= 0, ❡♥tã♦ _kx_k₁+_ky_k₂ = 0, ❧♦❣♦ _kx_k₁ =_{− k}y_k₂, ♣♦rt❛♥t♦ x=y = 0❀

✭❝✮ kα(x, y)_k=_k(αx, αy)_k=_kαx_k₁+_kαy_k₂ =_|α_{| k}x_k₁+_|α_{| k}y_k₂ =_|α_|(_kx_k₁+_ky_k₂)

✭❞✮k(x, y) + (a, b)_k=_k(a+x, b+y)_k=_ka+x_k₁+_kb+y_k₂ =_ka_k₁+_kx_k₁+_kb_k₂+_ky_k₂

≤ kx_k₁+_ky_k₂+_ka_k₁+_kb_k₂ =_k(x, y)_k+_k(a, b)_k

▲♦❣♦✱ k(x, y)_k=_kx_k₁+_ky_k₂ ❞❡✜♥❡ ✉♠❛ ♥♦r♠❛ ❡♠Z✳

❊①❡♠♣❧♦ ✷✳✶✵ ❙❡❥❛ ❳ ✉♠ ❡s♣❛ç♦ ❧✐♥❡❛r ♥♦r♠❛❞♦✳ ❙❡ x _∈ X _{0_} ❡ r > 0✱ ❡♥❝♦♥tr❡

α_∈R✱ t❛❧ q✉❡ _k_αx_k₌_r.

❙♦❧✉çã♦✿ ❙❡ α=_± r

kx_k✱ ❡♥tã♦ kαxk=|α| · kxk=r.

❈♦♠♦ ✈✐♠♦s ♥♦s ❡①❡♠♣❧♦s ✷✳✷ ❛ ✷✳✼ ❡ ✷✳✾✱ ❡①✐st❡♠ ♠✉✐t♦s ❡s♣❛ç♦s ♥♦r♠❛❞♦s ❞✐❢❡r❡♥t❡s ❡ ✐ss♦ ❡①♣❧✐❝❛✱ ❡♠ ♣❛rt❡✱ ♣♦rq✉❡ ♦ ❡st✉❞♦ ❞❡ ❡s♣❛ç♦s ♥♦r♠❛❞♦s é ✐♠♣♦rt❛♥t❡✳ ❯♠❛ ✈❡③ q✉❡ ❛ ♥♦r♠❛ ❞❡ ✉♠ ✈❡t♦r é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ✈❡t♦r ❡♠R3✱ ♥ã♦ é

❞❡ s✉r♣r❡❡♥❞❡r q✉❡ ❝❛❞❛ ❡s♣❛ç♦ ♥♦r♠❛❞♦ s❡❥❛ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❞❡ ❢♦r♠❛ ♠✉✐t♦ ♥❛t✉r❛❧✳

▲❡♠❛ ✷✳✶ ❙❡❥❛ ❳ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ♥♦r♠❛ k·k ✳ ❙❡ d:X_×X _→R ❡stá ❞❡✜♥✐❞❛

♣♦r d(x, y) = _kx₋y_k✱ ❡♥tã♦ (X, d) é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛x, y, z _∈X✳ ❯s❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♥♦r♠❛✱ t❡♠♦s✿

✭❛✮ d(x, y) = _kx₋y_{k ≥}0❀

✭❜✮ d(x, y) = 0 _{↔ k}x₋y_k= 0_↔x₋y= 0_↔x=y❀

✭❝✮ d(x, y) = _kx₋y_k=_k(₋1)(y₋x)_k=_{| −}1_{| k}y₋x_k=_ky₋x_k=d(y, x)❀

✭❞✮ d(x, z) = _kx₋z_k=_kx₋y+y₋z_{k ≤ k}x₋y_k+_ky₋z_k=d(x, y) +d(y, z)✳

▲♦❣♦✱ d s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s ❞❡ ✉♠❛ ♠étr✐❝❛✳

❙❡Xé ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ♥♦r♠❛_k·k❡dé ❛ ♠étr✐❝❛ ❞❡✜♥✐❞❛ ♣♦rd(x, y) =_kx₋yk✱

❡♥tã♦ (X, d)❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ♠étr✐❝♦ ❡ d é ❝❤❛♠❛❞❛ ❞❡ ♠étr✐❝❛ ❛ss♦❝✐❛❞❛ à k·k✳

❙❡♠♣r❡ q✉❡ ✉s❛♠♦s ✉♠❛ ♠étr✐❝❛ ♦✉ ✉♠ ❝♦♥❝❡✐t♦ ❞❡ ❡s♣❛ç♦ ♠étr✐❝♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♥✈❡r❣ê♥❝✐❛✱ ❝♦♥t✐♥✉✐❞❛❞❡ ♦✉ ❝♦♠♣❧❡t❡s✱ ❡♠ ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦✱ ❡♥tã♦ s❡♠♣r❡ ✐r❡♠♦s ✉s❛r ❛ ♠étr✐❝❛ ❛ss♦❝✐❛❞❛ ❝♦♠ ❛ ♥♦r♠❛ ✉s✉❛❧❀ ♠❡s♠♦ q✉❡ ✐ss♦ ♥ã♦ ❡st❡❥❛ ❡①♣❧✐❝✐t❛❞♦✳ ❆s ♠étr✐❝❛s ❛ss♦❝✐❛❞❛s ❝♦♠ ❛s ♥♦r♠❛s ✉s✉❛✐s ❥á ❡st❛♠♦s ❢❛♠✐❧✐❛r✐③❛❞♦s✳

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❊①❡♠♣❧♦ ✷✳✶✶ ❆s ♠étr✐❝❛s ❛ss♦❝✐❛❞❛s ❝♦♠ ❛s ♥♦r♠❛s ✉s✉❛✐s ♥♦s ❡s♣❛ç♦s ❛❜❛✐①♦ sã♦ ❛s ♠étr✐❝❛s ✉s✉❛✐s ✭t❛♠❜é♠ ❝❤❛♠❛❞❛s ❝❛♥ô♥✐❝❛s ♦✉ ✉s✉❛✐s✮✳

✭❛✮ Fn_❀

✭❜✮ CF(M)✱ ♦♥❞❡ ▼ é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❛❝t♦❀

✭❝✮ Lp₍_X₎ _♣❛r❛ ₁

≤x <_∞✱ ♦♥❞❡ (X,P

, µ) é ✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛❀

✭❞✮ lp_{✱ ♦♥❞❡} ₍_X,P

, µ) é ✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛✳

❙♦❧✉çã♦✿

✭❛✮ ❙❡ x, y _∈Fn_{✱ ❡♥tã♦}_d₍_{x, y}_{) =}_k_x₋_y_k_{= (}Pn

j=1|xj−yj|2)

1

2 ❡ ❞❛í✱dé ❛ ♠étr✐❝❛ ✉s✉❛❧

❡♠ Fn✳

✭❜✮ ❙❡ f, g_{∈ C}F(M)✱ ❡♥tã♦ d(x, y) =kf −gk= sup{|f(x)−g(x)|:x∈M} ❡ ❞❛í✱d é ❛

♠étr✐❝❛ ✉s✉❛❧ ❡♠CF✳

✭❝✮ ❙❡ f, g _∈ Lp₍_X₎_{✱ ❡♥tã♦} _d₍_{x, y}_{) =}

kf ₋g_k = (R

X|f −g|

p_dx₎1p ❡ ❞❛í✱ d é ❛ ♠étr✐❝❛

✉s✉❛❧ ❡♠ Lp₍_X₎_✳

✭❞✮ ❙❡ f, g _∈ L∞₍_X₎_{✱ ❡♥tã♦} _d₍_{x, y}_{) =} _k_f₋_g_k ₌_ess_sup_{|_f₍_x₎₋_g₍_x₎_|_: _x _∈_X_} _{❡ ❞❛í✱}

d é ❛ ♠étr✐❝❛ ✉s✉❛❧ ❡♠L∞₍_X₎_✳

❯s❛♥❞♦ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ❝♦♥t❛❣❡♠ ❡♠Ns❡❣✉❡ q✉❡✱ ❛s ♠étr✐❝❛s ❛ss♦❝✐❛❞❛s ❝♦♠ ❛s ♥♦r♠❛s

✉s✉❛✐s ❡♠ lp _❡ _l∞ _{sã♦ t❛♠❜é♠ ♠étr✐❝❛s ✉s✉❛✐s ♥❡ss❡s ❡s♣❛ç♦s✳ ❈♦♥❝❧✉✐r❡♠♦s ❡st❛ s❡ssã♦}

❝♦♠ ✐♥❢♦r♠❛çõ❡s ❜ás✐❝❛s s♦❜r❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ s❡q✉ê♥❝✐❛s ❡♠ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ♥♦r♠❛❞♦s✳

❚❡♦r❡♠❛ ✷✳✶ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ R ❝♦♠ ♥♦r♠❛ _k·k✳ ❙❡❥❛♠ _{_x_n_} ❡ _{_y_n_}

s❡q✉ê♥❝✐❛s ❡♠ X q✉❡ ❝♦♥✈❡r❣❡♠ ♣❛r❛ x, y _∈ X✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ _{αn} ✉♠❛ s❡q✉ê♥❝✐❛

q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ α_∈R✳ ❊♥tã♦✿

✭❛✮ |kx_{k − k}y_{k| ≤ k}x₋yk❀

✭❜✮ limn→∞kxnk=kxk❀

✭❝✮ limn_→∞(xn+yn) =x+y❀

✭❞✮ limn_→∞αnxn =αx✳

❉❡♠♦♥str❛çã♦✿

✭❛✮ ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ t❡♠♦s✿

Se_kx_k=_kx₋y+y_{k ≤ k}x₋y_k+_ky_k, ❡♥tã♦ _kx_{k − k}y_{k ≤ k}x₋y_k ❡ Se_ky_k=_ky₋x+x_{k ≤ k}y₋x_k+_kx_k, ❡♥tã♦ _ky_{k − k}x_{k ≤ k}y₋x_k

❉♦♥❞❡ ❝♦♥❝❧✉✐✲s❡ q✉❡ | kx_{k − k}y_{k | ≤ k}x₋y_k✳

✭❜✮ ❚❡♠♦s q✉❡✱

lim

n_→∞xn=x

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❡

| kx_{k − k}xnk | ≤ kx−xnk,

♣❛r❛ t♦❞♦n_∈N✳ ▲♦❣♦✱

lim

n_→∞kxnk=kxk.

✭❝✮ P♦r ❤✐♣ót❡s❡✱ xn →x ❡yn→y✳ ◆♦t❡ q✉❡✿

kxn+yn−(x+y)k=kxn−x+yn−yk ≤ kxn−xk+kyn−yk,

♣❛r❛ t♦❞♦n_∈N✳ ❆ss✐♠✱ ❝♦♥❝❧✉✐✲s❡ q✉❡

lim

n→∞(xn+yn) = x+y.

✭❞✮ P♦r {αn}s❡r ❝♦♥✈❡r❣❡♥t❡✱ s❛❜❡♠♦s q✉❡ ❡❧❛ é ❧✐♠✐t❛❞❛✳ ❊♥tã♦✱ ❡①✐st❡ k >0 t❛❧ q✉❡

|αn| ≤k✱ ♣❛r❛ t♦❞♦ n∈N✳ ❚❛♠❜é♠✱

kαnxn−αxk=kαnxn−αnx+αnx−αxk ≤ |αn| kxn−xk+kxk · kαn−αk

≤k_kxn−xk+kxk · kαn−αk,

♣❛r❛ t♦❞♦n_∈N✳ ▲♦❣♦✱

lim

n→∞αnxn =αx.

❯♠❛ ♠❛♥❡✐r❛ ❞✐❢❡r❡♥t❡ ❞❡ ✐♥❞✐❝❛r ♦s r❡s✉❧t❛❞♦s ❞♦ t❡♦r❡♠❛ ✷✳✶ ✐t❡♥s ✭❜✮✱ ✭❝✮ ❡ ✭❞✮ é q✉❡ ❛ ♥♦r♠❛✱ ❛ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s✳ ■ss♦ ♣♦❞❡ s❡r ✈✐st♦ ✉s❛♥❞♦ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❝♦♥t✐♥✉✐❞❛❞❡ s❡q✉❡♥❝✐❛❧✳

❊①❡♠♣❧♦ ✷✳✶✷ ❙❡❥❛♠ ❳ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ♥♦r♠❛k·k1 ❡ ❨ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠

♥♦r♠❛ k · k2✳ ❙❡❥❛ Z =X×Y ❝♦♠ ♥♦r♠❛ ❞♦ ❡①❡♠♣❧♦ ✷✳✾✳ ❙❡❥❛ (xn, yn) ✉♠❛ s❡q✉ê♥❝✐❛

❡♠ ❩✳

✭❛✮ ▼♦str❡ q✉❡ (xn, yn)❝♦♥✈❡r❣❡ ♣❛r❛ (x, y) ❡♠ ❩ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ {xn} ❝♦♥✈❡r❣❡ ♣❛r❛

① ❡♠ ❳ ❡ {yn} ❝♦♥✈❡r❣❡ ♣❛r❛ ② ❡♠ ❨✳

✭❜✮ ▼♦str❡ q✉❡ (xn, yn) é ❞❡ ❈❛✉❝❤② ❡♠ Z s❡✱ ❡ s♦♠❡♥t❡ s❡✱ {xn} ❡♠ X ❡ {yn} ❡♠ Y

❢♦r❡♠ ❞❡ ❈❛✉❝❤②✳

❙♦❧✉çã♦✿

❛✮ ❉❛❞♦ ǫ > 0. ❙✉♣♦♥❤❛♠♦s q✉❡ (xn, yn)→(x, y)∈ Z✱ q✉❛♥❞♦ n → ∞✳ ❊♥tã♦✱ ❡①✐st❡

N _∈ N t❛❧ q✉❡ _|₍_x_n ₋_{x, y}_n ₋_y₎_k ₌ _k₍_x_n_{, y}_n₎₋₍_{x, y}₎_{k ≤} _ǫ✱ q✉❛♥❞♦ _n _≥ _N✳ ❆ss✐♠✱

kxn−xk1 ≤ k(xn, yn)−(x, y)k ≤ǫ ❡ kyn−yk2 ≤ k(xn, yn)−(x, y)k ≤ǫ✱ q✉❛♥❞♦ n≥N✳

❉❛í✱{xn} ❡{yn} ❝♦♥✈❡r❣❡♠ ♣❛r❛ x∈X ❡ y∈Y✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛♠♦s q✉❡ xn →x ❡ yn →y✳ ❊♥tã♦✱ ❡①✐st❡ N1, N2 ∈ N t❛❧ q✉❡

kxn−xk1 ≤ ₂ǫ✱ q✉❛♥❞♦ n≥N1 ❡kyn−yk2 ≤ ₂ǫ✱ q✉❛♥❞♦n≥N2✳ ❙❡❥❛N0 = max{N1, N2}✳

❊♥tã♦✱ k(xn, yn) + (x, y)k = k(xn+x, yn +y)k = kxn−xk1 +kyn−yk2 ≤ ǫ₂ + ǫ₂ = ǫ✱

q✉❛♥❞♦n _≥N0✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱(xn, yn)❝♦♥✈❡r❣❡ ♣❛r❛ (x, y)❡♠ ❩✳

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❜✮ ❉❛❞♦ ǫ > 0. ❙✉♣♦♥❤❛♠♦s q✉❡ (xn, yn) → (xm, ym) ∈ Z✱ q✉❛♥❞♦ n → ∞✳ ❊♥tã♦✱

❡①✐st❡ N _∈N t❛❧ q✉❡ _|₍_x_n₋_x_m_{, y}_n₋_y_m₎_k₌_k₍_x_n_{, y}_n₎₋₍_x_m_{, y}_m₎_{k ≤} _ǫ✱ q✉❛♥❞♦ _n _≥_N✳

❆ss✐♠✱ kxn−xmk1 ≤ k(xn, yn)−(xm, ym)k ≤ǫ ❡ kyn−ymk2 ≤ k(xn, yn)−(xm, ym)k ≤ǫ

q✉❛♥❞♦n _≥N✳ ❉❛í✱_{xn}❡ {yn} ❝♦♥✈❡r❣❡♠ ♣❛r❛ xm ∈X ❡ ym ∈Y✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛♠♦s q✉❡ xn → xm ❡ yn → ym✳ ❊♥tã♦✱ ❡①✐st❡ N1, N2 ∈ N

t❛❧ q✉❡ kxn −xmk1 ≤ ǫ₂✱ q✉❛♥❞♦ n ≥ N1 ❡ kyn −ymk2 ≤ ₂ǫ✱ q✉❛♥❞♦ n ≥ N2✳ ❙❡❥❛

N0 =max{N1, N2}✳ ❊♥tã♦✱k(xn, yn) + (xm, ym)k=k(xn+xm, yn+ym)k=kxn−xmk1+

kyn −ymk2 ≤ ǫ₂ + ǫ₂ = ǫ✱ q✉❛♥❞♦ n ≥ N0✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ (xn, yn) ❝♦♥✈❡r❣❡ ♣❛r❛

(xm, ym)❡♠ Z.

✷✳✷ ❊s♣❛ç♦s ◆♦r♠❛❞♦s ❞❡ ❉✐♠❡♥sã♦ ❋✐♥✐t❛

❖s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ♠❛✐s s✐♠♣❧❡s ❞❡ s❡ ❡st✉❞❛r sã♦ ♦s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✱ ❡♥tã♦ ✉♠ ❧✉❣❛r ♥❛t✉r❛❧ ♣❛r❛ ❝♦♠❡ç❛r ♦ ♥♦ss♦ ❡st✉❞♦ ❞❡ ❡s♣❛ç♦s ♥♦r♠❛❞♦s é ❝♦♠ ❡s♣❛ç♦s ♥♦r♠❛❞♦s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✳ ❱✐♠♦s ♥♦ ❡①❡♠♣❧♦ ✷✳✷ q✉❡ t❛✐s ❡s♣❛ç♦s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ t❡♠ ✉♠❛ ♥♦r♠❛✱ ♠❛s ❡st❛ ♥♦r♠❛ ❞❡♣❡♥❞❡ ❞❛ ❜❛s❡ ❡s❝♦❧❤✐❞❛✳ ■ss♦ s✉❣❡r❡ q✉❡ ♣♦❞❡ ❤❛✈❡r ❞✐❢❡r❡♥t❡s ♥♦r♠❛s ❡♠ ❝❛❞❛ ❡s♣❛ç♦ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✳ ▼❡s♠♦ ❡♠ R2 ❥á ✈✐♠♦s q✉❡ ❡①✐st❡♠ ♣❡❧♦

♠❡♥♦s ❞✉❛s ♥♦r♠❛s✿

✭❛✮ ❛ ♥♦r♠❛ ✉s✉❛❧ ❞❡✜♥✐❞❛ ♥♦ ❡①❡♠♣❧♦ ✷✳✶❀

✭❜✮ ❛ ♥♦r♠❛ k(x, y)_k=_|x_|+_|y_|,❞❡✜♥✐❞❛ ♥♦ ❡①❡♠♣❧♦ ✷✳✼✳

P❛r❛ ♠♦str❛r ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❡ss❛s ❞✉❛s ♥♦r♠❛s✱ é ✐♥str✉t✐✈♦ ❡s❜♦ç❛r ♦ ❝♦♥❥✉♥t♦ {(x, y) _∈ R2 _: _k₍_{x, y}₎_k _{= 1}_} ♣❛r❛ ❝❛❞❛ ♥♦r♠❛✳ ◆♦ ❡♥t❛♥t♦✱ ♠❡s♠♦ q✉❛♥❞♦ t❡♠♦s ❞✉❛s

♥♦r♠❛s s♦❜r❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ s❡ ❛s ♥♦r♠❛s ♥ã♦ sã♦ ♠✉✐t♦ ❞✐❢❡r❡♥t❡s✱ é ♣♦ssí✈❡❧ q✉❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❡s♣❛ç♦s ♠étr✐❝♦s ♣♦❞❡♠ s❡r ❛s ♠❡s♠❛s ♣❛r❛ ❛♠❜❛s ❛s ♥♦r♠❛s✳ ❆ ❛✜r♠❛çã♦ ♠❛✐s ♣r❡❝✐s❛ ❞♦ q✉❡ s❡ ❡♥t❡♥❞❡ ♣♦r ✧♥ã♦ ♠✉✐t♦ ❞✐❢❡r❡♥t❡✧é ❞❛❞❛ ♥❛ ❞❡✜♥✐çã♦ ❛ s❡❣✉✐r✳

❉❡✜♥✐çã♦ ✷✳✷ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ s❡❥❛♠ _k·k₁ ❡ _k·k₂ ❞✉❛s ♥♦r♠❛s ❡♠ X✳ ❆

♥♦r♠❛ k·k2 é ❡q✉✐✈❛❧❡♥t❡ ❛ ♥♦r♠❛ k·k1 s❡ ❡①✐st❡♠ m, M > 0 t❛✐s q✉❡ ♣❛r❛ t♦❞♦ x ∈ X✱

m_kx_k₁ _{≤ k}x_k₂ _≤M_kx_k₁✳

❚❡♥❞♦ ❡♠ ✈✐st❛ ❛ t❡r♠✐♥♦❧♦❣✐❛ ✉s❛❞❛✱ ♥ã♦ ❞❡✈❡ s❡r s✉r♣r❡s❛ ❞❡✜♥✐r ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ♥♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ♥♦r♠❛s ❡♠ ❳✱ ❝♦♠♦ ✐r❡♠♦s ♠♦str❛r ❛ s❡❣✉✐r✳

▲❡♠❛ ✷✳✷ ❙❡❥❛X ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ s❡❥❛♠_k·k₁✱_k·k₂ ❡_k·k₃ três ♥♦r♠❛s ❡♠ X✳ ❙❡❥❛♠

k·k2 ❡q✉✐✈❛❧❡♥t❡ ❛ ♥♦r♠❛ k·k1 ❡ k·k3 ❡q✉✐✈❛❧❡♥t❡ ❛ ♥♦r♠❛ k·k2✳ ❊♥tã♦

✭❛✮ k·k1 é ❡q✉✐✈❛❧❡♥t❡ ❛ ♥♦r♠❛ k·k2❀

✭❜✮ k·k3 é ❡q✉✐✈❛❧❡♥t❡ ❛ ♥♦r♠❛ k·k1✳

(22)

❉❡♠♦♥str❛çã♦✿ P♦r ❤✐♣ót❡s❡✱ ❡①✐st❡♠ m, M > 0 t❛✐s q✉❡ m_kx_k₁ _{≤ k}x_k₂ _≤ M_kx_k₁ ❡ k, K >0 t❛✐s q✉❡ k_kx_k₂ _{≤ k}x_k₃ _≤K_kx_k₂✱ ♣❛r❛ t♦❞♦ x_∈X✳ ❆ss✐♠✱

✭❛✮ 1

M kxk2 ≤ kxk1 ≤ 1

mkxk2✱ ♣❛r❛ t♦❞♦ x∈X❀

✭❜✮ mk_kx_k₁ _{≤ k}x_k₃ _≤M K_kx_k₁✱ ♣❛r❛ t♦❞♦ x_∈X✳

❆❣♦r❛ ♠♦str❛r❡♠♦s q✉❡ ❡♠ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ❞✉❛s ♥♦r♠❛s ❡q✉✐✈❛❧❡♥t❡s✱ ❛s ♣r♦✲ ♣r✐❡❞❛❞❡s ❞❡ ❡s♣❛ç♦s ♠étr✐❝♦s sã♦ ❛s ♠❡s♠❛s ♣❛r❛ ❛♠❜❛s ❛s ♥♦r♠❛s✳

❊①❡♠♣❧♦ ✷✳✶✸ ❙❡❥❛ P ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ✭❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛✮ ❞♦s ♣♦❧✐♥ô♠✐♦s ❞❡✜♥✐❞♦s ❡♠ ❬✵✱✶❪✳ ❯♠❛ ✈❡③ q✉❡ P é ✉♠ s✉❜❡s♣❛ç♦ ❧✐♥❡❛r ❞❡ CF([0,1])✱ ❡❧❡ t❡♠ ✉♠❛ ♥♦r♠❛kpk1 =

sup_{|p(x)_|;x _∈[0,1]_} ❡✱ ✉♠❛ ✈❡③ q✉❡ _P é ✉♠ s✉❜❡s♣❛ç♦ ❧✐♥❡❛r ❞❡ L1_[0_,_1]_{✱ ❡❧❡ t❡♠ ♦✉tr❛}

♥♦r♠❛kp_k2 =R₀1|p(x)|dx✳ ▼♦str❡ q✉❡ k · k1 ❡ k · k2 ♥ã♦ sã♦ ❡q✉✐✈❛❧❡♥t❡s ❡♠ P✳

❙♦❧✉çã♦✿ ❙✉♣♦♥❤❛♠♦s q✉❡k·k1 ❡k·k2s❡❥❛♠ ❡q✉✐✈❛❧❡♥t❡s ❡♠P.❆ss✐♠✱ ❡①✐st❡♠m, M >0

t❛✐s q✉❡ m_kp_k1 ≤ kpk2 ≤ Mkpk1, ♣❛r❛ t♦❞♦ p ∈ P✳ ❈♦♠♦ m > 0✱ ❡①✐st❡ n ∈ N t❛❧ q✉❡ 1

n < m. ❙❡❥❛ pn : [0,1] → R ❞❛❞❛ ♣♦r pn(x) = x

n+1_{✳ ❊♥tã♦} _k_p

nk1 = 1n+1 = 1 ❡

kp_k2 = R1

0 x

n+1_dx ₌ xn

n =

1

n✳ ▲♦❣♦✱ m = mkpk1 ≤ kpk2 =

1

n✳ ❈♦♥tr❛❞✐çã♦✦ P♦rt❛♥t♦✱

k · k1 ❡ k · k2 ♥ã♦ sã♦ ❡q✉✐✈❛❧❡♥t❡s ❡♠ P✳

▲❡♠❛ ✷✳✸ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ _k·k ❡ _k·k₁ ♥♦r♠❛s ❡♠ X✱ ❡ d ❡ d1 ❛s ♠étr✐❝❛s

❞❡✜♥✐❞❛s ♣♦r d1(x, y) = kx−yk₁✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛ k > 0 t❛❧ q✉❡ kxk ≤ kkxk₁ ♣❛r❛

t♦❞♦ x_∈X✳ ❙❡❥❛ _{xn} ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ X✳

✭❛✮ ❙❡ {xn} ❝♦♥✈❡r❣❡ ♣❛r❛ x✱ ♥♦ ❡s♣❛ç♦ ♠étr✐❝♦ (X, d1)✱ ❡♥tã♦ {xn} ❝♦♥✈❡r❣❡ ♣❛r❛ x✱

♥♦ ❡s♣❛ç♦ ♠étr✐❝♦ (X, d)❀

✭❜✮ ❙❡ {xn} é ❞❡ ❈❛✉❝❤② ♥♦ ❡s♣❛ç♦ ♠étr✐❝♦(X, d1)✱ ❡♥tã♦ {xn} é ❞❡ ❈❛✉❝❤② ♥♦ ❡s♣❛ç♦

♠étr✐❝♦ (X, d)✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ǫ >0✳

✭❛✮ P♦r ❤✐♣ót❡s❡✱ ❡①✐st❡ N _∈ N t❛❧ q✉❡ _k_x_n₋_x_k _< ǫ

k✱ q✉❛♥❞♦ n ≥ N✳ ▲♦❣♦✱ q✉❛♥❞♦

n_≥N✱

kxn−xk ≤kkxn−xk₁, ǫ.

P♦rt❛♥t♦✱ {xn} ❝♦♥✈❡r❣❡ ♣❛r❛ x✱ ♥♦ ❡s♣❛ç♦ ♠étr✐❝♦ (X, d1)✳

✭❜✮ P♦r ❤✐♣ót❡s❡✱ ❡①✐st❡ N _∈ N t❛❧ q✉❡ _k_x_n₋_x_m_k_< ǫ

k✱ q✉❛♥❞♦ n, m≥ N✳ ❙❡❣✉❡ q✉❡✱

q✉❛♥❞♦n, m_≥N✱

kxn−xmk ≤kkxn−xmk₁ < ǫ.

P♦rt❛♥t♦✱ {xn} é ❞❡ ❈❛✉❝❤② ♥♦ ❡s♣❛ç♦ ♠étr✐❝♦(X, d)✳

❈♦r♦❧ár✐♦ ✷✳✶ ❙❡❥❛X ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ s❡❥❛♠ _k·k❡ _k·k₁ ♥♦r♠❛s ❡q✉✐✈❛❧❡♥t❡s ❡♠ X✳

❙❡❥❛♠d❡d1 ❛s ♠étr✐❝❛s ❞❡✜♥✐❞❛s ♣♦r d(x, y) =kx−yk ❡d1(x, y) = kx−yk₁✳ ❙❡❥❛ {xn}

✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ X✳

✭❛✮ {xn} ❝♦♥✈❡r❣❡ ♣❛r❛ x✱ ♥♦ ❡s♣❛ç♦ ♠étr✐❝♦ (X, d1) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ {xn} ❝♦♥✈❡r❣❡

♣❛r❛ x✱ ♥♦ ❡s♣❛ç♦ ♠étr✐❝♦ (X, d1)❀