❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ▼■◆❆❙ ●❊❘❆■❙
■◆❙❚■❚❯❚❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦
P♦♥t♦ ❋✐①♦ ❡ ❛✉t♦✈❛❧♦r❡s ♣♦s✐t✐✈♦s ♣❛r❛
♦♣❡r❛❞♦r❡s ♥ã♦ ❧✐♥❡❛r❡s
❙✐♠♦♥❡ ❘❛s♦ ❏❛♠❡❧ ❊❞✐♠
❖r✐❡♥t❛❞♦r✿
❆♥tô♥✐♦ ❩✉♠♣❛♥♦
❆❣r❛❞❡❝✐♠❡♥t♦s
❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r ❛❣r❛❞❡ç♦ ❛♦s ❞✐r❡t♦r❡s ❞♦ ❝♦❧é❣✐♦ ▲♦②♦❧❛ q✉❡ ♠❡ ❞❡r❛♠ ❛ ♦♣♦rt✉♥✐❞❛❞❡✱ ✐♥❝❡♥t✐✈♦ ❡ ❛ tr❛♥q✉✐❧✐❞❛❞❡ ♣❛r❛ ❢❛③❡r ♦ ▼❡str❛❞♦✳ ❊♠ ❡s♣❡❝✐❛❧ ❛❣r❛❞❡ç♦ ❛♦ P❛❞r❡ ◆❡❧✲ s♦♥ q✉❡ ♠❡ ✈❛❧♦r✐③♦✉ ❞❡ ❢♦r♠❛ tã♦ ❞❡s♣r❡♥❞✐❞❛✳ ❊ ♠✐♥❤❛ ❛t✉❛❧ ❝♦♦r❞❡♥❛❞♦r❛ ▲✉❝✐❧❛ q✉❡ s❡♠♣r❡ ❛❝r❡❞✐t♦✉ ❡♠ ♠❡✉ tr❛❜❛❧❤♦✳
❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ♠❡str❛❞♦ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ Pr♦❢❡ss♦r ❍❛♠✐❧t♦♥ q✉❡ ❝♦♥s❡❣✉✐✉ ♠❡ ❡♥①❡r❣❛r ❛❧é♠ ❞❡ ♠✐♥❤❛s ♥♦t❛s ❡ ❝♦♠ ✐st♦ r❡❛s❝❡♥❞❡✉ ♠✐♥❤❛ ❞✐s♣♦s✐çã♦ ❞❡ s❡❣✉✐r ❡♠ ❢r❡♥t❡✳
❆❣r❛❞❡ç♦ ♠❡✉s ❝♦❧❡❣❛s✳ ❊♠ ❡s♣❡❝✐❛❧ à P❛trí❝✐❛✱ à ▲✉❛♥❛ ❡ ❛♦ ❋❧á✈✐♦ q✉❡ ♠❡ ❛❥✉❞❛r❛♠ ❛ ♣r❡❡♥❝❤❡r ❧❛❝✉♥❛s ❞♦ ♠❡✉ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ❢♦r♠❛ ✐rr❡str✐t❛ ❡ q✉❡ ❢♦r❛♠ ✐♥❝❛♥sá✈❡✐s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ ❡st✉❞♦✳ ❆♦ ▲❡❛♥❞r♦ ♣❡❧❛ s✉❛ ❞✐s♣♦s✐çã♦ ❡♠ ❛❥✉❞❛r s❡♠♣r❡✳
❆♦ ♠❡✉ ♠❛r✐❞♦ ❏♦r❣❡✱ q✉❡ ♠❡ ❛♣♦✐♦✉ ❞❡ ❢♦r♠❛ ❝❛r✐♥❤♦s❛ ❡ ❞♦❛♥❞♦ ♠❛✐s ❞♦ q✉❡ r❡❝❡❜❡♥❞♦✳ ❆♦s ♠❡✉s ✜❧❤♦s ❏♦r❣❡✱ ❋❡❧✐♣❡ ❡ ❇r❡♥♦ ♣♦r ❧✐❞❛r❡♠ ❜❡♠ ❝♦♠ ❛ ❡s❝❛ss❡③ ❞❡ ♠ã❡ ❛♦ ❧♦♥❣♦ ❞❡st❡s ✷ ❛♥♦s✳
❊✱ ✜♥❛❧♠❡♥t❡ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❆♥tô♥✐♦ ❩✉♠♣❛♥♦ q✉❡ é ✉♠ ♠❡str❡ ❧❡❣ít✐♠♦✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦
❊st❛ ❞✐ss❡rt❛çã♦ ❝♦♥st❛ ❞❡ ✺ ❝❛♣ít✉❧♦s✳
◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ❞❡✜♥✐çõ❡s ❡ s✐♠❜♦❧♦❣✐❛s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞❛s ♥♦s ❝❛♣ít✉❧♦s ♣♦st❡r✐♦r❡s✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ♦ ♣r✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ❞❡ ❇r♦✇❞❡r ❡ s✉❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❡♠ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt ❡ ❞❡ ❇❛♥❛❝❤✳
◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s t❡♦r❡♠❛s ❞❡ ♣♦♥t♦ ✜①♦ ❞❡ ❛♣❧✐❝❛çõ❡s ♥ã♦ ❡①♣❛♥s✐✈❛s ❡♠ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ r❡✢❡①✐✈♦✳
◆♦ q✉❛rt♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ✈❡rsã♦ ❞♦ t❡♦r❡♠❛ ❞♦ ♣♦♥t♦ ✜①♦ ❞❡ ❇❛♥❛❝❤ ♣❛r❛ ❢✉♥çõ❡s T :B[x0, r]→2X ❡ s✉❛ ❛♣❧✐❝❛çã♦ ♥❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✳
❋✐♥❛❧♠❡♥t❡✱ ♥♦ q✉✐♥t♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s r❡s✉❧t❛❞♦s q✉❡ ✐♥❞✐❝❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❛✉t♦✈❛❧♦r❡s ♣♦s✐t✐✈♦s ♣❛r❛ ♦♣❡r❛❞♦r❡s ♥ã♦ ❧✐♥❡❛r❡s✳ ❖s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s sã♦ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞♦s t❡♦r❡♠❛s ❞❡ ♣♦♥t♦ ✜①♦ ❛♣r❡s❡♥t❛❞♦s ♥♦ ❝❛♣ít✉❧♦ ✸✳
❊st❛ ❞✐ss❡rt❛çã♦ t❡✈❡ ❝♦♠♦ ❜❛s❡ ♦ ❛rt✐❣♦ ❞❡ ■❙❆❈&◆➱▼❊❚❍ ❬❄❪ ♣✉❜❧✐❝❛❞♦ ♥♦ ❏♦✉r♥❛❧
♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ❡♠ 2006❡ ❡♠ ♣❛rt❡ ❞♦ ❛rt✐❣♦ ❞❡ ❩❯▼P❆◆❖
❬❄❪ ♣✉❜❧✐❝❛❞♦ ♣❡❧❛ ❘❡✈✐st❛ ▼❛t❡♠át✐❝❛ ❯♥✐✈❡rs✐tár✐❛❡♠1996 ✳
❯♠ ❞♦s ♦❜❥❡t✐✈♦s ❞❡st❡ t❡①t♦ é ❛♣r❡s❡♥t❛r ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ❤♦♠♦t❡t✐❛ (
y′(t) =y(θt)
y(0) = 1
❝♦♠ θ ∈R✱ q✉❡ t❡rá tr❛t❛♠❡♥t♦ ❞✐st✐♥t♦ ♣❛r❛0< θ <1 ❡θ >0✳
P❛r❛ s♦❧✉❝✐♦♥❛r ♦ ❝❛s♦ ❡♠ q✉❡ 0< θ <1 ✉t✐❧✐③❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ P♦♥t♦ ❋✐①♦ ♣❛r❛
❙✉♠ár✐♦ ❙✉♠ár✐♦
P❛r❛ ♦ ♦✉tr♦ ❝❛s♦ ❛♣❧✐❝❛r❡♠♦s ✉♠❛ ❡①t❡♥sã♦ ❞♦ ♣r✐♥❝í♣✐♦ ❞❡ ❇❛♥❛❝❤ ❛♣r❡s❡♥t❛❞❛ ♣♦r ❩❯▼P❆◆❖ ❬❄❪✳
❖ t❡♦r❡♠❛ ❞❡ ♣♦♥t♦ ✜①♦ ❞❡ ❇❛♥❛❝❤ ♣❛r❛ ❝♦♥tr❛çõ❡s✱ ❞❡♥tr❡ ♦s ✈ár✐♦s t❡♦r❡♠❛s ❞❡ ♣♦♥t♦ ✜①♦ ❝♦♥❤❡❝✐❞♦s✱ é ✉♠ ❞♦s ♠❛✐s út❡✐s✳ ❆❧é♠ ❞❡ ❛ss❡❣✉r❛r ❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❢♦r♥❡❝❡ ✉♠ ♠ét♦❞♦ ✐t❡r❛t✐✈♦ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ✈❛❧♦r❡s ❛♣r♦①✐♠❛❞♦s ♣❛r❛ ♦ ♣♦♥t♦ ✜①♦ ❡♠ q✉❡stã♦✳
●❡r❛❧♠❡♥t❡ ❛ ❞✐✜❝✉❧❞❛❞❡ ❡♠ ❛♣❧✐❝❛r ♦ t❡♦r❡♠❛ ❞❡ ♣♦♥t♦ ✜①♦ ❞❡ ❇❛♥❛❝❤ é ❡♥❝♦♥tr❛r ❡s♣❛ç♦s ♦✉ ♥♦r♠❛s ❡♠ q✉❡ ♦ ♦♣❡r❛❞♦r ❞❡ ✐♥t❡r❡ss❡ s❡❥❛ ✉♠❛ ❝♦♥tr❛çã♦✳
❉✉r❛♥t❡ ❛s ú❧t✐♠❛s ❞é❝❛❞❛s ✈ár✐♦s ❛✉t♦r❡s ✐♥✈❡st✐❣❛r❛♠ ❛s ❛♣❧✐❝❛çõ❡s ♥ã♦ ❡①♣❛♥s✐✈❛s ❡ ❝❤❡❣❛r❛♠ à ❝♦♥❝❧✉sã♦ ❞❡ q✉❡ ❛tr✐❜✉✐♥❞♦ ❛♦ ❡s♣❛ç♦ ✉♠❛ ❡str✉t✉r❛ s✉✜❝✐❡♥t❡♠❡♥t❡ r✐❝❛ ❛ ❤✐♣ót❡s❡ ❞❡ ❝♦♥tr❛çã♦ ♣♦❞❡ s❡r ❡♥❢r❛q✉❡❝✐❞❛ ♣❛r❛ ♥ã♦ ❡①♣❛♥s✐✈✐❞❛❞❡✳ ❈♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ✉♠❛ ❛♣❧✐❝❛çã♦ F : C ⊂ X → C ♥ã♦ ❡①♣❛♥s✐✈❛✱ ❞❡✜♥✐❞❛ ❡♠ ✉♠ s✉❜❝♦♥❥✉♥t♦ C ❢❡❝❤❛❞♦✱
❧✐♠✐t❛❞♦ ❡ ❝♦♥✈❡①♦ ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦ ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ ✜①♦ ❡♠ C✳
❆♣r❡s❡♥t❛r❡♠♦s ♥♦✈♦s t❡♦r❡♠❛s ❞❡ ♣♦♥t♦ ✜①♦ ♣❛r❛ ❛♣❧✐❝❛çõ❡s ♥ã♦ ❡①♣❛♥s✐✈❛s ❡♠ ❡s✲ ♣❛ç♦s ❞❡ ❇❛♥❛❝❤✱ q✉❡ sã♦ ❣❡♥❡r❛❧✐③❛çõ❡s ❞♦ r❡s✉❧t❛❞♦ ❝❧áss✐❝♦ ❛♣r❡s❡♥t❛❞♦ ♣♦r ❇r♦✇❞❡r ❬❄❪ q✉❡ ✉t✐❧✐③❛ s✉❜❝♦♥❥✉♥t♦s✱ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❝♦♠♣❛❝t♦s✱ ❞♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ ❆q✉✐ ❝♦♠♣❛❝✐❞❛❞❡ ♥ã♦ s❡rá ❡♥✈♦❧✈✐❞❛✳
❖s r❡s✉❧t❛❞♦s r❡❧❛t✐✈♦s ❛♦s ❛✉t♦✈❛❧♦r❡s sã♦ ❜❛s❡❛❞♦s ♥❡st❡s ♥♦✈♦s t❡♦r❡♠❛s ❡ ❡♠ ✉♠❛ ✈❛r✐❛çã♦ ❞♦ t❡♦r❡♠❛ ❞❡ ♣♦♥t♦ ✜①♦ ❞❡ ❆❧t♠❛♥ ❬❄❪✳ ❖❜t❡♠♦s ❡♠ ♣❛rt✐❝✉❧❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ❛✉t♦✈❛❧♦r❡s ♣❛r❛ ❛♣❧✐❝❛çõ❡s ♥ã♦ ❡①♣❛♥s✐✈❛s✳
❈❛♣ít✉❧♦ ✶
Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦
◆❡st❡ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ✉♠ ♣❛r❛❧❡❧♦ ❞❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡♠ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt ❡ ❞❡ ❇❛♥❛❝❤✳ ❆♣r❡s❡♥t❛r❡♠♦s ♣r♦♣♦s✐çõ❡s r❡❧❛t✐✈❛s ❛ ❡s♣❛ç♦s ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦ ❡ ❡str✐✲ t❛♠❡♥t❡ ❝♦♥✈❡①♦✳ ❚❡r♠✐♥❛r❡♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❞♦ ♣r✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ❞❡ ❇r♦✇❞❡r✳
✶✳✶ ❙❡♠✐✲ Pr♦❞✉t♦ ✲ ✐♥t❡r♥♦
❖ s❡♠✐✲♣r♦❞✉t♦✲ ✐♥t❡r♥♦ ❢♦✐ ❞❡✜♥✐❞♦ ♣♦r ●✳ ▲✉♠❡r ❛♦ t❡♥t❛r tr❛♥s❢❡r✐r ♦s ❛r❣✉♠❡♥t♦s ❞♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ♣❛r❛ s✐t✉❛çõ❡s ❡♠ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✳
❉❡✜♥✐çã♦ ✶✳✶✳✶ ✭❙❡♠✐✲ Pr♦❞✉t♦ ✲ ✐♥t❡r♥♦✮ ❙❡❥❛ (❊,k · k) ✉♠ ❡s♣❛ç♦ r❡❛❧ ❞❡ ❇❛♥❛❝❤
❛r❜✐trár✐♦✳ ❉✐③❡♠♦s q✉❡ ✉♠ s❡♠✐✲ ♣r♦❞✉t♦✲✐♥t❡r♥♦ é ❞❡✜♥✐❞♦ ❡♠ ❊ s❡ ♣❛r❛ q✉❛✐sq✉❡rx, y ∈
❊ ❝♦rr❡s♣♦♥❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ ❞❡♥♦t❛❞♦ ♣♦r [x, y] s❛t✐s❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s
♣❛r❛ x, y ∈❊ ❡ λ∈R✿
(S1) [x+y, z] = [x, z] + [y, z] (S2) [λx, y] =λ[x, y]
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦✶✳✷✳ ❙❡♠✐✲♣r♦❞✉t♦✲✐♥t❡r♥♦ ❡ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤
(S3) [x, x]>0 ♣❛r❛ x6= 0 (S4) |[x, y]|2 ≤[x, x][y, y]
■r❡♠♦s s✉♣♦r q✉❡ t♦❞♦ s❡♠✐✲♣r♦❞✉t♦✲✐♥t❡r♥♦ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛ ♥♦r♠❛ s❛t✐s❢❛③ ❛ ♣r♦✲ ♣r✐❡❞❛❞❡ ❞❡ ❤♦♠♦❣❡♥❡✐❞❛❞❡[x, λy] =λ[x, y]
P❛r❛ ▲✉♠❡r ❬❄❪✱ ♦ s❡♠✐✲♣r♦❞✉t♦ ✐♥t❡r♥♦ ❢♦r♥❡❝❡ ❛♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ✉♠❛ ❡str✉t✉r❛ s✉✲ ✜❝✐❡♥t❡ ♣❛r❛ s❡ ♦❜t❡r r❡s✉❧t❛❞♦s ❣❡r❛✐s ♥ã♦ tr✐✈✐❛✐s✳ ■r❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦ q✉❛❧ ♥♦ ❧✉❣❛r ❞❡ ✉♠❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r ❞❡✜♥✐r❡♠♦s ❛ ❢♦r♠❛[x, y] ❛ q✉❛❧ é ❧✐♥❡❛r ❡♠ ✉♠❛
❝♦♠♣♦♥❡♥t❡✱ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈❛ ❡ s❛t✐s❢❛③ ❛ ❞❡s✐❣✉❧❛❞❛❞❡ ❞❡ ❙❝❤✇❛r③✳ ❚❛❧ ❢♦r♠❛ ✐♥❞✉③ ✉♠❛ ♥♦r♠❛ ❞❛❞❛ ♣♦r kxk= ([x, x])12✳
✶✳✷ ❙❡♠✐✲♣r♦❞✉t♦✲✐♥t❡r♥♦ ❡ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤
❖ t❡♦r❡♠❛ ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❘✐❡s③ ❡ ♦ ❝♦r♦❧ár✐♦ ❞♦ t❡♦r❡♠❛ ❞❡ ❍❛♥❤✲❇❛♥❛❝❤ ❛♣✲ r❡s❡♥t❛❞♦s ❛❜❛✐①♦ ❡ ❝✉❥❛s ❞❡♠♦♥tr❛çõ❡s s❡ ❡♥❝♦♥tr❛♠ ❡♠ ❇❘❊❩■❙ ❍❊■▼ ❬❄❪ sã♦ ♣ré r❡q✉✐s✐t♦s ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦s t❡♦r❡♠❛s ❄❄ ❡ ❄❄✳
❚❡♦r❡♠❛ ✶✳✷✳✶ ✭❚❡♦r❡♠❛ ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❘✐❡s③✮ ❚♦❞♦ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❝♦♥tí♥✉♦
f :❍ →R✱ ❞❡✜♥✐❞♦ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❍✱ é r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦
y∈❍ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
f(x) =< x, y > ♣❛r❛ t♦❞♦ x∈❍✳ ❆✐♥❞❛ kfk=kyk✳
❈♦r♦❧ár✐♦ ✶✳✷✳✷ ✭❈♦r♦❧ár✐♦ ❞♦ t❡♦r❡♠❛ ❞❡ ❍❛♥✲❇❛♥❛❝❤✮ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ♥♦r✲
♠❛❞♦ ❡ s❡❥❛ x0 ∈ X✳ ❊♥tã♦ ❡①✐st❡ f ∈ X∗ ❡♠ q✉❡ X∗ é ♦ ❝♦♥❥✉♥t♦ ❞♦s ❢✉♥❝✐♦♥❛✐s
❧✐♥❡❛r❡s ❝♦♥tí♥✉♦s t❛❧ q✉❡ kfk=kx0k ❡ f(x0) = kx0k2
❆ s❡❣✉✐r ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞❡ ✉♠ t❡♦r❡♠❛ ❞❡✈✐❞♦ ❛ ●✳ ▲✉♠❡r✱q✉❡ ♥♦s ❣❛r❛♥t❡ q✉❡ t♦❞❛ ♥♦r♠❛ ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♣r♦✈é♠ ❞❡ ✉♠ s❡♠✐✲♣r♦❞✉t♦✲✐♥t❡r♥♦✳
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦✶✳✷✳ ❙❡♠✐✲♣r♦❞✉t♦✲✐♥t❡r♥♦ ❡ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤
❚❡♦r❡♠❛ ✶✳✷✳✸ ❯♠ s❡♠✐✲♣r♦❞✉t♦✲✐♥t❡r♥♦ ❡♠ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s❡♠♣r❡ ❞❡✜♥❡ ✉♠❛ ♥♦r♠❛✱ ❛ s❛❜❡r✱ ([x, x])12✳ ❚♦❞❛ ♥♦r♠❛ ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ X ♣r♦✈é♠ ❞❡ ✉♠ s❡♠✐✲♣r♦❞✉t♦✲
✐♥t❡r♥♦✱ q✉❡ ❡♠ ❣❡r❛❧ ♥ã♦ é ú♥✐❝♦✱ ♦✉ s❡❥❛✱ ❡①✐st❡♠ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ s❡♠✐✲♣r♦❞✉t♦s✲ ✐♥t❡r♥♦s q✉❡ ❣❡r❛♠ ❛ ♠❡s♠❛ ♥♦r♠❛✳
❉❡♠♦♥str❛çã♦✿ ❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r ✈❛♠♦s ♣r♦✈❛r q✉❡ kxk= ([x, x])12 é ✉♠❛ ♥♦r♠❛✳
❛) kxk= [x, x]12 = 0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x= 0, ♣♦✐s s❡✱ x6= 0 t❡♠♦s [x, x]>0
❜)
kx+yk2 = [x+y, x+y] =|[x, x+y] + [y, x+y]|
≤ [x, x]12[x+y, x+y] 1
2 + [y, y] 1
2[x+y, x+y] 1 2
= (kxk+kyk])kx+yk
kx+yk ≤ kxk+kyk
❝)
kλxk2 = |λ|[x, λx]
≤ |λ| ([λx, λx]12 [x, x] 1 2)
= |λ| kλxk kxk
kλxk ≤ |λ| kxk
P❛r❛ λ6= 0 kxk=
1 λλx ≤ 1
|λ|
kλ xk✳
❈♦♥❝❧✉✐♠♦s ❡♥tã♦ q✉❡ kλxk=|λ| kxk
P♦r ♦✉tr♦ ❧❛❞♦✱ s❡❥❛ X ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦ ❡X∗ s❡✉ ❞✉❛❧✳ P❛r❛ ❝❛❞❛ x∈X✱
❡①✐st❡ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❍❛❤♥✲❇❛♥❛❝❤ ♣❡❧♦ ♠❡♥♦s ✉♠ ❢✉♥❝✐♦♥❛❧ Wx ∈X∗ t❛❧ q✉❡ Wx(x) =
(x, Wx) = kxk2✳ ❉❛❞❛ q✉❛❧q✉❡r ❢✉♥çã♦ W : X → X∗ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r ❢❛❝✐❧♠❡♥t❡ q✉❡
[x, y] = (x, Wy) ❞❡✜♥❡ ✉♠ s❡♠✐✲♣r♦❞✉t♦ ✐♥t❡r♥♦✱ ♦✉ s❡❥❛✱ s❛t✐s❢❛③ ❛s q✉❛tr♦ ♣r♦♣r✐❡❞❛❞❡s ❞❛❞❛s ♥❛ ❞❡✜♥✐çã♦ ❄❄✳ ❉❡ ❢❛t♦✱
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦✶✳✸✳ ❙❡♠✐✲♣r♦❞✉t♦✲✐♥t❡r♥♦ ❡ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt
❜) [λx, y] = (λx, Wy) =λ(x, Wy) =λ[x, y] ❝) [x, x] = (x, Wx) = kxk2 >0✱ ∀ x6= 0 ❞) |[x, y]|2 =|(x, W
x)|2 ≤ kxk2kyk2 ≤(x, Wx)(y, Wy)≤[x, x] [y, y]
✷
❖❜s❡r✈❛çã♦ ✶✳✷✳✹ ❆ t♦♣♦❧♦❣✐❛ ❡♠ ✉♠ ❡s♣❛ç♦ ❝♦♠ s❡♠✐✲♣r♦❞✉t♦✲✐♥t❡r♥♦ s❡rá ❛ t♦♣♦❧♦❣✐❛ ✐♥❞✉③✐❞❛ ♣❡❧❛ ♥♦r♠❛ [x, x]12✳
✶✳✸ ❙❡♠✐✲♣r♦❞✉t♦✲✐♥t❡r♥♦ ❡ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt
❯♠❛ q✉❡stã♦ s✉r❣❡ ♥❛t✉r❛❧♠❡♥t❡✿ ◗✉❛♥❞♦ ✉♠ ❡s♣❛ç♦ ❝♦♠ s❡♠✐✲♣r♦❞✉t♦✲✐♥t❡r♥♦ é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt❄
❆ r❡s♣♦st❛ é ❞❛❞❛ ♣❡❧♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿
❚❡♦r❡♠❛ ✶✳✸✳✶ ❖ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❍ ♣♦❞❡ s❡r tr❛♥s❢♦r♠❛❞♦ ❡♠ ✉♠ ❡s♣❛ç♦ ❝♦♠ s❡♠✐✲ ♣r♦❞✉t♦✲✐♥t❡r♥♦ ❞❡ ✉♠❛ ú♥✐❝❛ ♠❛♥❡✐r❛✳ ❯♠ s❡♠✐✲♣r♦❞✉t♦✲✐♥t❡r♥♦ é ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ ❛ ♥♦r♠❛ ✐♥❞✉③✐❞❛ ♣♦r ❡❧❡ ✈❡r✐✜❝❛ ❛ r❡❣r❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦✱ ♦✉ s❡❥❛✱
kx+yk2+kx−yk2 = 2(kxk2+kyk2)
❝♦♠ x, y ∈H
❉❡♠♦♥str❛çã♦✿
❉❛❞♦ q✉❛❧q✉❡r s❡♠✐✲♣r♦❞✉t♦✲✐♥t❡r♥♦ ❡♠ ❍✱ ✜①❛❞♦ y6= 0✱ [x, y] é ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r
❧✐♠✐t❛❞♦ Wy ❡♠ ❍ ✳ P❡❧♦ t❡♦r❡♠❛ ❄❄✱ ❡①✐st❡ z ∈ ❍✱ t❛❧ q✉❡✱ Wy(x) = [x, y] =< x, z > ♦♥❞❡<·>r❡♣r❡s❡♥t❛ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ✉s✉❛❧✳ ❚❡♠♦s ❡♥tã♦ q✉❡kyk=kzk❡ ❛ ✐❣✉❛❧❞❛❞❡
❡str✐t❛ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ kyk2 =< y, z >=kykkzk q✉❡ ♥♦s ❞áz =λy ✱ ♠❛s ♥♦✈❛♠❡♥t❡
< y, λy >=kyk2 ❡♥tã♦ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡y=z✳ ✷
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ✶✳✹✳ ❊s♣❛ç♦s ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦s
✶✳✹ ❊s♣❛ç♦s ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦s
◆❡st❛ s❡çã♦ ✈❛♠♦s ❡st❛❜❡❧❡❝❡r ❛❧❣✉♠❛s ❡q✉✐✈❛❧ê♥❝✐❛s ♥❛ ❞❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦s ✉♥✐❢♦r♠❡✲ ♠❡♥t❡ ❝♦♥✈❡①♦s✱ ❡ ❛♣r❡s❡♥t❛r ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❡❧❡♠❡♥t❛r ❞❡st❡ ❡s♣❛ç♦ q✉❡ s❡rá ❛♣❧✐❝❛❞❛ ♥♦ ♣r✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ❞❡ ❇r♦✇❞❡r ❡♠ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✲ ✈❡①♦s✳ ❱❡r❡♠♦s t❛♠❜é♠ ❡q✉✐✈❛❧ê♥❝✐s ♥❛ ❞❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦s ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦s✳
❉❡✜♥✐çã♦ ✶✳✹✳✶ ✭❊s♣❛ç♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦✮ ❯♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ (❊,k · k)
é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ t♦❞♦ ε ∈ ]0,2[ ❡①✐st❡ δ(ε) ∈ ]0,1[ t❛❧
q✉❡ s❡♠♣r❡ q✉❡ kxk ≤ R, kyk ≤ R, kx−yk ≥ ε R✱ x, y ∈ E✱ R > 0 ❡♥tã♦ t❡♠♦s
x+y
2
≤(1−δε)R
●❡♦♠❡tr✐❝❛♠❡♥t❡✱ ❝♦♥✈❡①✐❞❛❞❡ ✉♥✐❢♦r♠❡ s✐❣♥✐✜❝❛ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r ✷ ♣♦♥t♦sx, y ♥❛
❢r♦♥t❡✐r❛ ❞❛ ❜♦❧❛ ✉♥✐tár✐❛✱ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ q✉❡ ✉♥❡ x ❛ y ❡♥❝♦♥tr❛✲s❡ ❞❡♥tr♦
❞❛ ❜♦❧❛ ❞❡ r❛✐♦r <1✱ ♦♥❞❡ r ❞❡♣❡♥❞❡ ❞❛ ❞✐stâ♥❝✐❛kx−yk✳
❖❜s❡r✈❛çã♦ ✶✳✹✳✷ ✶✮ ❖ ❡s♣❛ç♦ ❊ = R2 ♠✉♥✐❞♦ ❞❛ ♥♦r♠❛ ❞♦ ♠á①✐♠♦ ♥ã♦ é ✉♥✐✲
❢♦r♠❡♠❡♥❡ ❝♦♥✈❡①♦✳
✷✮❚♦❞♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦✳ ■st♦ s❡❣✉❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞❛ r❡❣r❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦✳
✸✮Lp(Ω) é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦ ♣❛r❛ 1< p <∞✱ ♦♥❞❡ Ω é ✉♠ ❞♦♠í♥✐♦ ❡♠ Rm✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ❢❛t♦ ❡♥❝♦♥tr❛✲s❡ ♥❛s ♣á❣✐♥❛s 92−94❞❡ ❬❄❪✳
✶✳✺ ❊s♣❛ç♦ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦
❉❡✜♥✐çã♦ ✶✳✺✳✶ ✭ ❊s♣❛ç♦ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦✮ ❯♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❊ é ❡str✐✲ t❛♠❡♥t❡ ❝♦♥✈❡①♦ s❡ ♣❛r❛ t♦❞♦ x, y ∈❊ x6=y ✱ kxk=kyk= 1✱ t❡♠♦s✿
kλ x+ (1−λ)yk<1
♣❛r❛ t♦❞♦ λ∈(0,1)✳
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ✶✳✺✳ ❊s♣❛ç♦ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦
❱❡r❡♠♦s q✉❡ t♦❞♦ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦ é ❡①tr✐t❛♠❡♥t❡ ❝♦♥✈❡①♦ ❡ q✉❡ ❡①✐st❡♠ ❡s♣❛ç♦s q✉❡ sã♦ ❡①tr✐t❛♠❡♥t❡ ❝♦♥✈❡①♦s ❡ ♥ã♦ sã♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦s✳
❖ ít❡♠ (ii) ❞❛❞♦ ❛❜❛✐①♦ ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ♣♦r ❘■❊❙❩ ❬❄❪ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ s✉❜st✐t✉✐r ❛
r❡❣r❛ ❞♦ ♣❛r❛❧❡♦❣r❛♠♦✱ ❡①✐st❡♥t❡ ❛♣❡♥❛s ❡♠ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt✱ q✉❡ ❛ss❡❣✉r❛ ❛ ❝♦♥✈❡r❣ê♥✲ ❝✐❛ ❞❡ s❡q✉ê♥❝✐❛s ♠✐♥✐♠✐③❛♥t❡s gn ❡♠ ❝♦♥❥✉♥t♦s ❝♦♥✈❡①♦sG✳ ❙❡q✉ê♥❝✐❛ ♠✐♥✐♠✐③❛♥t❡gn é q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ t❛❧ q✉❡ kgnk →µ ❡♠ q✉❡ µé ♦ ❡❧❡♠❡♥t♦ ❞❡ ♥♦r♠❛ ♠í♥✐♠❛ ❞❡ G✳
Pr♦♣♦s✐çã♦ ✶✳✺✳✷ ❆s s❡❣✉✐♥t❡s ❛✜r♠❛t✐✈❛s s♦❜r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ (❊,k · k) sã♦
❡q✉✐✈❛❧❡♥t❡s✿
(i) ❊ é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦
(ii) ❊①✐st❡ k > 0 t❛❧ q✉❡ ♣❛r❛ t♦❞♦ ǫ > 0 s❡ R ≤ kxk ≤ R+ε✱ R ≤ kyk ≤ R+ε ❡
kx+y
2 k ≥R ❡♥tã♦ kx−yk< k ε
❉❡♠♦♥str❛çã♦✿ i)⇒ii)
❱❛♠♦s s✉♣♦r ♣♦r ❛❜s✉r❞♦ q✉❡ ❡①✐st❡K >0t❛❧ q✉❡ ♣❛r❛ t♦❞♦ε >0t❡♠♦skx−yk> k ε✳
❈♦♠♦ ♦ ❡s♣❛ç♦ é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦✱ t♦♠❛♥❞♦K =R t❡♠♦s q✉❡kxk ≤R ≤R+ε✱
kxk ≤R≤R+ε ♣❛r❛ t♦❞♦ x, y ∈❊ ❡♥tã♦
x+y
2
≤(1−δ(ε))R ≤R ♦ q✉❡ é ❛❜s✉r❞♦✳
ii)⇒i)
❱❛♠♦s s✉♣♦r ♣♦r ❛❜s✉r❞♦ q✉❡
x+y
2
≥(1−δ(ε))R✱ t♦♠❡ R1 = R −δ(ε)R < R✱
❧♦❣♦✱ R = R1 +ε✳ ❈♦♠♦ ♣♦r ❤✐♣ót❡s❡ R1 ≤ kxk ≤ R1+ε✱ R1 ≤ kyk ≤ R1+ε ❡♥tã♦
kx−yk< k εs❡ t♦♠❛r♠♦sk =R1 t❡♠♦s kx−yk< k ε <(R−ε)ε < Rε♦ q✉❡ ❝♦♥tr❛r✐❛
❛ ❤✐♣ót❡s❡ ❞♦ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦✳ ✷
Pr♦♣♦s✐çã♦ ✶✳✺✳✸ ❆s s❡❣✉✐♥t❡s ❛✜r♠❛t✐✈❛s s♦❜r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ (❊,k · k) sã♦
❡q✉✐✈❛❧❡♥t❡s✿
(i) E é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ✶✳✺✳ ❊s♣❛ç♦ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦
(ii) ❙❡ ❛s s❡q✉ê♥❝✐❛s (xn),(yn) ⊆ E sã♦ t❛✐s q✉❡ kxnk = kynk = 1 ❡ kxn+ynk → 2 ❡♥tã♦ kxn−ynk →0
❉❡♠♦♥str❛çã♦✿ i)⇒ii
❱❛♠♦s s✉♣♦r ♣♦r ❛❜s✉r❞♦ q✉❡ ❡①✐st❡ ε >0 t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ δ = 1
n✱ ❡①✐st❡♠ kxnk =
kynk = 1 ❝♦♠ kxn−ynk ≥ ε ♠❛s ❝♦♠♦ ❊ é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦ ❡♥tã♦ kxn+ynk ≤
2(1− 1
n)❡ ✐st♦ ❝♦♥tr❛r✐❛ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡kxn+ynk →2✳ ▲♦❣♦✱ t❡♠♦s ✉♠❛ ❝♦♥tr❛❞✐çã♦✳
ii)⇒i)
❱❛♠♦s s✉♣♦r ♣♦r ❛❜s✉r❞♦ q✉❡ ❊ ♥ã♦ é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦✳ ❊♥tã♦✱ ❡①✐st❡♠kxnk=
kynk = 1 ❝♦♠ kxn−ynk ≥ ε ♠❛s kxn+ynk ≥ 2(1− 1n) n ∈ N✳ ▲♦❣♦✱ kxn+ynk → 2 ❡ ✐st♦ ✐♠♣❧✐❝❛ q✉❡kxn−ynk →0♦ q✉❡ ❣❡r❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ✷ ❖❜s❡r✈❛çã♦ ✶✳✺✳✹ ✭Pr♦♣r✐❡❞❛❞❡s ❞❡ δ(ε)✮ ◆❛ ❞❡✜♥✐çã♦ ❄❄ ❞❡ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡♠❡♥t❡
❝♦♥✈❡①♦ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❛ ❢✉♥çã♦δ: [0,2]→[0,1]❝♦♠ δ(2) = 1✳ ■st♦ ♥ã♦ ❣❡r❛ ♥❡♥❤✉♠❛
❝♦♥tr❛❞✐çã♦✱ ❝♦♠♦ ✈❡r❡♠♦s ❛ s❡❣✉✐r✳ ❉❡ ❢❛t♦✱ s❡ kxk ≤ R✱ kyk ≤ R ❡ kx−yk ≥ 2R✱
❡♥tã♦ s❡❣✉❡ q✉❡ kx+2yk = 0✳ P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s kxk ≤ R✱ kyk ≤ R✱ kx−yk ≥ 2R
❡ kx+2yk > 0✳ ❊♥tã♦ kxk ≤ R✱ k −yk ≤ R ❡ kx− (−y)k ≥ 0 ❡ ♣❡❧❛ ❞❡✜♥✐çã♦ ❄❄
kx+(2−y)k< R✱ ❝♦♥tr❛❞✐③❡♥❞♦ kx−yk ≥2R✳
❱❛♠♦s t♦♠❛r δ(0) = 0 ❡ ❛❧é♠ ❞✐ss♦ s✉♣♦r q✉❡ ❛ ❢✉♥çã♦ δ: [0,2]→[0,1] é ♠♦♥ót♦♥❛✱
❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡ ❡ ❝♦♠ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ δ(ε)→0 q✉❛♥❞♦ ε →0✳
❙❡ δ(·) ♥ã♦ ♣♦ss✉✐r t❛❧ ♣r♦♣r✐❡❞❛❞❡✱ ❡♥tã♦ t♦♠❡ δ2✱ ❡♠ q✉❡
δ1(ε) = sup
η∈[0,ε]
δ(η), δ2 =
εδ1(ε)
2
◆❛ ❞❡✜♥✐çã♦ ❄❄ ♣♦❞❡♠♦s s✉❜st✐t✉✐r δ ♣♦r δ1 ❡ δ2✳ ❖❜s❡r✈❡ q✉❡ δ1 ≤ δ2 ♣❛r❛ 0 < ε ≤ 1✳
❆❣♦r❛ δ2 ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡s❡❥❛❞❛✳
Pr♦♣♦s✐çã♦ ✶✳✺✳✺ ❙❡❥❛♠ ❊ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦ ❡
η : [0,1] → [0,2] ❛ ❢✉♥çã♦ ✐♥✈❡rs❛ ❞❛ ❢✉♥çã♦ δ(·)✳ ❙❡ kz −xk ≤ R✱ kz −yk ≤ R✱
z−
x+y
2
≥r ❡♠ q✉❡ 0 < r < R✱ ❡♥tã♦✱ kx−yk ≤R η(
R−r R )
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ✶✳✻✳ ❊s♣❛ç♦s ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦s
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ ε =η(R−r
R )❡ kx−yk> R ε✳ ❊♥tã♦✱ δ(ε) =δ(η( R−r
R )) = R−r
R ✳ ❙❡ kx−yk> R ε ❡♥tã♦✱ ♣❡❧♦ ❢❛t♦ ❞♦ ❡s♣❛ç♦ s❡r ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦ t❡♠♦s✿
z−x+z−y
2
≤ (1−δ(ε))R
<
1− R−r
R
R=r
♦ q✉❡ ❝♦♥tr❛r✐❛ ❛ ❤✐♣ót❡s❡✳ ✷
✶✳✻ ❊s♣❛ç♦s ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦s
Pr♦♣♦s✐çã♦ ✶✳✻✳✶ ❆s ❛✜r♠❛t✐✈❛s ❛❜❛✐①♦ r❡❧❛t✐✈❛s ❛♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❊ sã♦ ❡q✉✐✈❛✲ ❧❡♥t❡s✿
(i) ❊ é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦✱ ♦✉ s❡❥❛✱ s❡ ♣❛r❛ t♦❞♦ x, y ∈ ❊ x 6= y ✱ kxk = kyk = 1✱
t❡♠♦s✱ kλ x+ (1−λ)yk<1 ♣❛r❛ t♦❞♦ λ∈(0,1)✳
(ii) ❈❛❞❛ f ♥ã♦ ♥✉❧♦ ❡♠ ❊∗ ❛ss✉♠❡ s✉♣r❡♠♦ ♥♦ ♠á①✐♠♦ ❡♠ ✉♠ ♣♦♥t♦ ❞❛ ❜♦❧❛ ✉♥✐tár✐❛✳
(iii) ❙❡ x6=y ❡ kxk=kyk= 1 ❡♥tã♦ kx+yk<2
(iv) ❆ ❢r♦♥t❡✐r❛ ❞❛ ❜♦❧❛ ✉♥✐tár✐❛ ♥ã♦ ❝♦♥té♠ s❡❣♠❡♥t♦s r❡t♦s
❉❡♠♦♥str❛çã♦✿
i)⇒ii)
❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❛❧❣✉♠ f ∈ ❊∗ ❡①✐st❛♠ ✷ ✈❡t♦r❡s x1 6= x2 ❝♦♠ kx1k = kx2k = 1 ❡
f(x1) = f(x2) = kfk✳
P❛r❛ λ∈(0,1)t❡♠♦s✿
kfk kλ x1+ (1−λ)x2k ≥f(λ x1 + (1−λ)x2) =λ f(x1) + (1−λ)f(x2) = kfk
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ✶✳✻✳ ❊s♣❛ç♦s ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦s
▲♦❣♦✱ ❝♦♠♦ kfk 6= 0 t❡♠♦s kλ x1+ (1−λ)x2k ≥ 1 ♦ q✉❡ ❝♦♥tr❛r✐❛ ❛ ❤✐♣ót❡s❡ ❞♦
❡s♣❛ç♦ s❡r ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦✳
ii)⇒iii)
❙❡❥❛♠ x, y ∈ ❊✱ t❛✐s q✉❡ x 6= y ,kxk = kyk = 1 ❡ kx+yk = 2✳ P❡❧♦ t❡♦r❡♠❛ ❞❡
❍❛❤♥✲❇❛♥❛❝❤ ❡①✐st❡ f ∈ ❊∗ t❛❧ q✉❡ kfk = 1 ❡ f(x+2y) = kx+y
2 k = 1 ❝♦♥s❡q✉❡♥t❡♠❡♥t❡
f(x) +f(y) = 2✳ ❈♦♠♦ f(x) ≤ 1 ❡ f(y)≤ 1✱ s❡❣✉❡ q✉❡f(x) = f(y) = kfk = 1 ♦ q✉❡ é
✉♠❛ ❝♦♥tr❛❞✐çã♦✳
iii)⇒iv)
❈♦♠♦ ♣♦r ❤✐♣ót❡s❡ kx + yk < 2 t❡♠♦s
x+y
2
< 1✳ ❚♦♠❛♥❞♦ λ = 12 t❡♠♦s
kλ x+ (1−λ)yk<1✱ ♦✉ s❡❥❛ ❛ ❢r♦♥t❡✐r❛ ❞❛ ❜♦❧❛ ✉♥✐tár✐❛ ♥ã♦ ❝♦♥té♠ s❡❣♠❡♥t♦s r❡t♦s✳
iv)⇒i)
❱❛♠♦s s✉♣♦r ♣♦r ❛❜s✉r❞♦ q✉❡ ❊ ♥ã♦ é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦✱ ♦✉ s❡❥❛✱ ❡①✐st❡λ0 ∈(0,1)
t❛❧ q✉❡ kλ0x+ (1−λ0)yk = 1✳ ❚❡♠♦s q✉❡ ♣r♦✈❛r q✉❡ ♦ s❡❣♠❡♥t♦ [x, y] ❡stá ❝♦♥t✐❞♦ ♥❛
❜♦❧❛ ✉♥✐tár✐❛ ♣❛r❛ ♦❜t❡r♠♦s ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❚♦♠❡ λ0 < λ <1
λ0x+ (1−λ0)y=
λ0
λ [λ x+ (1−λ)y] +
1−λ0
λ
y
❖❜t❡♠♦s✿
1 =λ0x+ (1−λ0)yk ≤
λ0
λ kλ x+ (1−λ)yk+
1− λ0
λ
kyk
λ0
λ kλ x+ (1−λ)yk ≥1− kyk+
1− λ0
λ
kyk
kλ x+ (1−λ)yk ≥1⇒ kλ x+ (1−λ)yk= 1
❖ ❝❛s♦ 0< λ < λ0 é ❛♥á❧♦❣♦✳ ✷
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ✶✳✻✳ ❊s♣❛ç♦s ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦s
❖❜s❡r✈❛çã♦ ✶✳✻✳✷ ❯♠ ❡s♣❛ç♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦ é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦✳
❉❡ ❢❛t♦✱ t♦♠❡ x, y ∈ ❊, x 6= y ❝♦♠ kxk = kyk = 1❀ ❡♥tã♦ ❡①✐st❡ ε > 0 t❛❧ q✉❡
kx −yk ≥ ε ⇒ kx+yk ≤2(1−δ)<2 ❡♥tã♦✱ ♣❡❧♦ ít❡♠ ✭✐✐✐✮ ❞❛ ♣r♦♣♦s✐çã♦ ❄❄✱ ❊ é
❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦✳
❊①❡♠♣❧♦ ✶✳✻✳✸ ❖s ❡s♣❛ç♦s lp e Lp ❝♦♠ 1 < p < ∞ sã♦ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦s✱ ♣♦✐s sã♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦s✳ ❯s❛♥❞♦ ❛ ❝♦♥✈❡①✐❞❛❞❡ ❞❛ ❢✉♥çã♦ xp ❝♦♠ p >1 ♠♦str❛✲s❡ ❢❛❝✐❧♠❡♥t❡ q✉❡ ❡st❡s ❡s♣❛ç♦s sã♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦s✳ ❊st❛ ❞❡♠♦♥str❛çã♦ s❡ ❡♥❝♦♥tr❛ ❡♠ ❬❄❪✳
❊①❡♠♣❧♦ ✶✳✻✳✹ ❖s ❡s♣❛ç♦s l1 e L∞ ♥ã♦ sã♦ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦s✳
❉❡ ❢❛t♦✱ t♦♠❡ e1 = (1,0...) ❡ e2 = (0,1...) ❡♥tã♦✱ ke1k = ke2k = 1 ❡ ke1 +e2k = 2✱
❧♦❣♦✱ l1 ♥ã♦ é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦✳
❈♦♥s✐❞❡r❡ x=e1+e2 ❡ y =e1−e2✱ ❡♥tã♦✱ kxk∞=kyk∞= 1 ❡ kx+yk∞ = 2✱ ✐st♦ é✱
l∞ ♥ã♦ é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦✳
❈♦♥s❡q✉❡♥t❡♠❡♥t❡ ❝♦♠ ❛ ♥♦r♠❛kxk1 =
n X
i=1
|xi|❡kxk∞= max
1≤i≤n|xi|♥ã♦ é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦✳
❖❜s❡r✈❛çã♦ ✶✳✻✳✺ ❊①✐st❡♠ ❡s♣❛ç♦s q✉❡ sã♦ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦s ❡ ♥ã♦ sã♦ ✉♥✐❢♦r♠❡✲ ♠❡♥t❡ ❝♦♥✈❡①♦s✳ P♦r ❡①❡♠♣❧♦✱ s❡ k · k ❞❡♥♦t❛ ❛ ♥♦r♠❛ ✉s✉❛❧ ❡♠ C([0,1]) ✭❡s♣❛ç♦ ❞❛s
❢✉♥çõ❡s ❝♦♥tí♥✉❛s ♥♦ ✐♥t❡r✈❛❧♦ [0,1]✮ ❡♥tã♦✱
|||x|||=kxk+
Z 1
0 |
x(t)|2
1 2
❞❡✜♥❡ ✉♠❛ ♥♦r♠❛ ❡q✉✐✈❛❧❡♥t❡ ❡♠ C([0,1]) ❛ q✉❛❧ é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦ ❡ ♥ã♦ é ✉♥✐✲
❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦✳
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦✶✳✼✳ ❊①✐stê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦ ❞❡ ♥♦r♠❛ ♠í♥✐♠❛
❈♦♥s✐❞❡r❛♥❞♦ ♦ ❢❛t♦ ❞❡ q✉❡ ❛ ♥♦r♠❛ ✉s✉❛❧ ❡♠ C([0,1]) é ❛ ♥♦r♠❛ ❞♦ s✉♣✱ t❡♠♦s✿
|||x||| = kx(t)k+
Z 1
0
|x(t)|2
1 2
≤ kxk+
Z 1
0 |
x|2
1 2
= 2kfk
▲♦❣♦✱ ❛s ♥♦r♠❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s✳
C([0,1]) ❝♦♠ ❛ ♥♦r♠❛ ||| · ||| é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✳ ❉❡ ❢❛t♦✱
|||(1−λ)x+λ y||| = k(1−λ)x+λ yk+k(1−λ)x+λ yk2
< (1−λ)kxk+λkyk+ (1−λ)kxk2+λkyk2
= (1−λ)|||x|||+λ|||y|||= 1
❖❜s❡r✈❛çã♦ ✶✳✻✳✻ ◆♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❛ ❢✉♥çã♦ ♥♦r♠❛ é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✳
k(1−λ)x+λ yk<(1−λ)kxk+λkyk ♣❛r❛ t♦❞♦ x6=y
❙❛❜❡♠♦s q✉❡ ♦ ❡s♣❛ç♦ C([0,1]) ❝♦♠ ❛ ♥♦r♠❛ ❞♦ s✉♣ ♥ã♦ é r❡✢❡①✐✈♦✱ ♣♦rt❛♥t♦✱ ♥ã♦ é
r❡✢❡①✐✈♦ ♥❛ ♥♦r♠❛||| · |||✱ ♣♦✐s ❡❧❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s✳ ▲♦❣♦✱ ♥ã♦ é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦✳
✶✳✼ ❊①✐stê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦ ❞❡ ♥♦r♠❛ ♠í♥✐♠❛
❙❛❜❡♠♦s q✉❡ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ✉♠ s✉❜❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦ ❢❡❝❤❛❞♦ ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞❡ ♥♦r♠❛ ♠í♥✐♠❛✱ ❝♦♠♦ ❡♥✉♥❝✐❛❞♦ ♥♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r✳
❚❡♦r❡♠❛ ✶✳✼✳✶ ❚♦❞♦ ❝♦♥❥✉♥t♦ C ⊆ ❍ ❝♦♥✈❡①♦ ❡ ❢❡❝❤❛❞♦✱❡♠ q✉❡ ❍ é ✉♠ ❡s♣❛ç♦ ❞❡
❍✐❧❜❡rt✱ ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ x0 ∈C t❛❧ q✉❡ kx0k ≤ kxk ♣❛r❛ t♦❞♦ x∈C
❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❘■❊❙❩ &❇❊▲❆ ❬❄❪
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ✶✳✽✳ ❚❡♦r❡♠❛ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦
❊♠ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ ❛ ✉♥✐❝✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦ ❞❡ ♥♦r♠❛ ♠í♥✐♠❛ ♦❝♦rr❡ s❡ ♦ ❡s♣❛ç♦ ❢♦r ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦✳
Pr♦♣♦s✐çã♦ ✶✳✼✳✷ ❚♦❞♦ s✉❜❝♦♥❥✉♥t♦ C✱ ♥ã♦ ✈❛③✐♦✱ ❢❡❝❤❛❞♦✱ ❝♦♥✈❡①♦✱ ❞❡ ✉♠ ❡s♣❛ç♦ ❞❡
❇❛♥❛❝❤ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦ ❊ ❝♦♥té♠ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞❡ ♠❡♥♦r ♥♦r♠❛✳
❉❡♠♦♥str❛çã♦✿ ❉❛❞♦s x, y ∈C✱ ❝♦♠♦C é ❝♦♥✈❡①♦✱ x+y
2 ∈C✳
❚♦♠❡ α = inf{kxk;x ∈ C}✳ ❉❛❞♦ ε > 0✱ t❡♠♦s α ≤ kxk ≤ α+ǫ, α ≤ kyk ≤ α+ǫ✱
❡♥tã♦✱
x+y
2
> α✳ ❈♦♠♦ ♦ ❡s♣❛ç♦ é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦ ❡♥tã♦ ❡①✐st❡ k >0t❛❧ q✉❡
kx−yk ≤k ε (I)✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ✐♥❢ t❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛(yn)∈C t❛❧ q✉❡
kynk → α q✉❛♥❞♦ n → ∞✳ ❙✉❜st✐t✉✐♥❞♦ ♥❛ ❞❡s✐❣✉❛❧❞❛❞❡ (I) x e y ♣♦r yn e ym t❡♠♦s
kyn−ymk< k ε♣❛r❛ t♦❞♦m, n > n0✱ ❡♥tã♦(yn) é s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✳ ❈♦♠♦ ♦ ❡s♣❛ç♦ é ❝♦♠♣❧❡t♦ ❡①✐st❡ R∈ ❊ t❛❧ q✉❡yn→R✱ ✐st♦ é kynk →0 q✉❛♥❞♦ n → ∞✳ ❈♦♠♦ ❛ ♥♦r♠❛ é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♠ ❊ s❡❣✉❡ q✉❡
lim
n→∞kynk=kn→∞lim ynk=kRk=α
❆ ✉♥✐❝✐❞❛❞❡ é tr✐✈✐❛❧✱ ♣♦✐s s❡ kx −yk ≤ k ε ♣❛r❛ t♦❞♦ ε t♦♠❛♥❞♦ ε = 0 t❡♠♦s✿
kx−yk ≤0 ❡ ♣♦rt❛♥t♦✱x=y✳
✷
✶✳✽ ❚❡♦r❡♠❛ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦
❉❡✜♥✐çã♦ ✶✳✽✳✶ ✭❆♣❧✐❝❛çã♦ ♥ã♦ ❡①♣❛♥s✐✈❛✮ ❙❡❥❛ A ⊆ ❳ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ ✉♠ ❡s✲
♣❛ç♦ ♥♦r♠❛❞♦ X✳ ❯♠❛ ❛♣❧✐❝❛çã♦ T :A→❳ é ♥ã♦ ❡①♣❛♥s✐✈❛✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ t♦❞♦ x, y ∈A t❡♠♦s✿
kT(x)−T(y)k ≤ kx−yk.
❖❜s❡r✈❛çã♦ ✶✳✽✳✷ ❙❡ t♦♠❛r♠♦s ❊ =Rn ❝♦♠ ❛ ♥♦r♠❛ ❡✉❝❧✐❞✐❛♥❛✱ r♦t❛çõ❡s ❡ ♣r♦❥❡çõ❡s
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ✶✳✽✳ ❚❡♦r❡♠❛ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦
❉❡✜♥✐çã♦ ✶✳✽✳✸ ✭❆♣❧✐❝❛çã♦ s❡♠✐❢❡❝❤❛❞❛✮ ❙❡❥❛C ⊆❊ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✱ ❢❡❝❤❛❞♦✱
❝♦♥✈❡①♦✱ ✐❧✐♠✐t❛❞♦✳ ❯♠❛ ❛♣❧✐❝❛çã♦ h:C −→❊ é s❡♠✐❢❡❝❤❛❞❛ ❡♠ ❈ s❡ ♣❛r❛ q✉❛❧q✉❡r s❡✲
q✉ê♥❝✐❛{xn} ⊆C ❢r❛❝❛♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡ ❛ ✉♠ ❡❧❡♠❡♥t♦x0 ∈C (xn⇀ x0)❝♦♠ {h(xn)} ❝♦♥✈❡r❣❡♥t❡ ❡♠ ♥♦r♠❛ ♣❛r❛ ✉♠ ❡❧❡♠❡♥t♦y0 (h(xn)→y0)✱ t❡♠♦s q✉❡x0 ∈C ❡h(x0) =y0✳
❖✉ s❡❥❛✱ s❡ xn⇀ x0 ❡ h(xn)→y0 ❡♥tã♦ x0 ∈C ❡ h(x0) =y0✳
❖ t❡♦r❡♠❛ ❛❜❛✐①♦ ♥♦s ❢♦r♥❡❝❡ ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠ ♦♣❡r❛❞♦r s❡♠✐✲❢❡❝❤❛❞♦ ♥ã♦ tr✐✈✐❛❧✳ ■r❡♠♦s s❡♣❛r❛r ❡st❡ t❡♦r❡♠❛ ❡♠ ❞♦✐s ❝❛s♦s✳ ❊♠ ✉♠ ❞❡❧❡s ✈❛♠♦s ❝♦♥s✐❞❡r❛r X ✉♠
❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❡ ✈❡r❡♠♦s q✉❡ ❛ ❞❡♠♦♥str❛çã♦ é s✐♠♣❧❡s✱ ❞❡✈✐❞♦ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ❣❡♦♠❡tr✐❛ ♥❡st❡ ❡s♣❛ç♦✳ ◆♦ ♦✉tr♦ ❝❛s♦ t♦♠❛r❡♠♦s X ❝♦♠♦ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳
❚❡♦r❡♠❛ ✶✳✽✳✹ ❙❡❥❛C✉♠ s✉❜❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦✱ ❢❡❝❤❛❞♦ ❞❡ ✉♠ ❡s♣❛ç♦X✉♥✐❢♦r♠❡♠❡♥t❡
❝♦♥✈❡①♦✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❛♣❧✐❝❛çã♦ F : C → X s❡❥❛ ♥ã♦ ❡①♣❛♥s✐✈❛✳ ❚❡♠♦s✱ ❡♥tã♦ q✉❡ I−F é s❡♠✐❢❡❝❤❛❞❛✳
❉❡♠♦♥str❛çã♦✿ P❛r❛ ♣r♦✈❛r♠♦s q✉❡I −F é s❡♠✐❢❡❝❤❛❞❛✱ t❡♠♦s q✉❡ ♣r♦✈❛r q✉❡ ❞❛❞♦s xn⇀ x0 ∈C ❡xn−F(xn)→y0✱ ❡♥tã♦✱ F(x0) =x0−y0✳
❈♦♠♦ F é ♥ã♦ ❡①♣❛♥s✐✈❛ t❡♠♦s q✉❡✿
kF(xn)−F(x0)k ≤ kxn−x0k = kxn−F(xn)−y0+F(xn)−x0+y0k
≤ kxn−F(xn)−y0k+kF(xn)−(x0−y0)k
≤ kF(xn)−(x0−y0)k+an ❝♦♠ an →0✳
❆ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ é ✈á❧✐❞❛ ♣❛r❛ q✉❛❧q✉❡r ❡s♣❛ç♦ ♥♦r♠❛❞♦ ✈✐st♦ q✉❡ ♦s ú♥✐❝♦s ❛r❣✉✲ ♠❡♥t♦s ✉t✐❧✐③❛❞♦s ❢♦r❛♠ ❛ ♥ã♦ ❡①♣❛♥s✐✈✐❞❛❞❡ ❞❡ F ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✳
❆ ♣❛rt✐r ❞❛q✉✐ ❛ ❞❡♠♦♥str❛çã♦ s❡ ❜✐❢✉r❝❛ ❡♠ ❞♦✐s ❝❛s♦s ♦s q✉❛✐s s❡rã♦ ✈✐st♦s ❛ s❡❣✉✐r✳
10 ❝❛s♦✮ X é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt
❚♦♠❡ x6=x0 ✳❚❡♠♦s✿
kxn−xk2 = kxn−x0+x0−xk2
= kxn−x0k2+ 2hxn−x0, x0−xi+kx0 −xk2.
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ✶✳✽✳ ❚❡♦r❡♠❛ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦
❈♦♠♦ xn⇀ x0 ❡♥tã♦ ∃ n0 ∈Nt❛❧ q✉❡ ∀ n ≥n0
2hxn−x0, x0−xi ≤ kxn−x0kkx0−xk
t❡♥❞❡ ♣❛r❛ ③❡r♦✳ ▲♦❣♦ 2hxn−x0, x0−xi+kx0−xk2 é ♣♦s✐t✐✈♦✳ P♦rt❛♥t♦✱
kxn−xk2 >kxn−x0k2+k
❱❛♠♦s s✉♣♦r ♣♦r ❛❜s✉r❞♦ q✉❡ F(x0) 6= x0−y0✳ P❡❧❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❛♥t❡r✐♦r❡s ❝♦♥✲
❝❧✉✐♠♦s✿
kF(xn)−(x0−y0)k2+k ≤ kF(xn)−F(x0)k2
≤ kF(xn)−(x0−y0)k2+an
▲♦❣♦✱an≥k✱ ❛❜s✉r❞♦ ♣♦✐san →0✳ ❊♥tã♦✱ F(x0) = x0−y0✱ ♦✉ s❡❥❛✱I−F é s❡♠✐❢❡❝❤❛❞❛✳
✷
❆♥t❡s ❞❡ ♣❛ss❛r♠♦s ♣❛r❛ ♦ s❡❣✉♥❞♦ ❝❛s♦ ✈❛♠♦s ❞❛r ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ♣❛r❛ ❛ ❞❡s✐❣✉❛❧❞❛❞❡
kxn−xk2 >kxn−x0k2+k
♥♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳
❯t✐❧✐③❛♥❞♦ ❛ r❡t❛ r❡❛❧ ❡ ♣r♦♣r✐❡❞❛❞❡s ✉s✉❛✐s ❞♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt X✱ ♦❜s❡r✈❛r❡♠♦s
q✉❡ s❡ ✉♠❛ s❡q✉ê♥❝✐❛ xn ❝♦♥✈❡r❣❡ ❢r❛❝❛♠❡♥t❡ ♣❛r❛ ✉♠ ❡❧❡♠❡♥t♦ x0 ∈ X ✱ ❡♥tã♦ ♣❛r❛
n > n0 ∈N♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ ❡ x0 ♣❡rt❡♥❝❡rã♦ ❛ ✉♠ ♠❡s♠♦ s❡♠✐✲❡s♣❛ç♦✳
❊st❡ ❢❛t♦ ♥♦s ❧❡✈❛rá ❛ ❝♦♥❝❧✉✐r ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✳
❚♦♠❡ x 6= x0✱ ❡♥tã♦✱ v = x0 − x é ✉♠ ✈❡t♦r ♥ã♦ ♥✉❧♦ ❡ ❝♦♠♦ xn ⇀ x0 t❡♠♦s
q✉❡ yn = xn − x ⇀ x0−x = v✳ ❉❡✜♥❛ ♦ ❢✉♥❝✐♦♥❛❧ f(z) =< z, v >✳ ❈♦♥s✐❞❡r❡
M =kerf = [v]⊥ ❡ z
0 t❛❧ q✉❡ α=f(z0) =λ f(v) =λkvk2 ❝♦♠ λ= 34✳
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ✶✳✽✳ ❚❡♦r❡♠❛ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦
v λnv
z0
❋✐❣✉r❛ ✶✳✶✿ ❋✐❣✉r❛ ✶
0 f(z0) = 34kvk2
f(yn)
kvk2 =f(v)
❋✐❣✉r❛ ✶✳✷✿ ❋✐❣✉r❛ ✷
❆✜r♠❛♠♦s q✉❡ M +z0 =f−1(α)✳ ❉❡ ❢❛t♦✱ t♦♠❡ w ∈f−1(α)✱ ♠❛s w=w−z0+z0 ❡
f(w−z0) = 0, portanto, w−z0 ∈M✱ ❧♦❣♦✱ w∈M+z0✱ ♦✉ s❡❥❛✱ f−1(α)⊂M +z0✳
P♦r ♦✉tr♦ ❧❛❞♦✱ t♦♠❡ y ∈ M +z0✱ ❡♥tã♦✱ ❡①✐st❡ z ∈ M t❛❧ q✉❡ y =z+z0✱ ♣♦rt❛♥t♦✱
f(y) =f(z+z0) =f(z) +f(z0) =α✱ ❧♦❣♦✱y ∈f−1(α)❡ ✐st♦ ✐♠♣❧✐❝❛ q✉❡M+z0 ⊂f−1(α)✳
❈♦♥❝❧✉✐♠♦s ❛ss✐♠ q✉❡ M +z0 =f−1(α)✳
❆❣♦r❛ ✈❛♠♦s ♠♦str❛r q✉❡ yn ❡ v ❡stã♦ ♥♦ ♠❡s♠♦ s❡♠✐✲❡s♣❛ç♦ ❡♠ r❡❧❛çã♦ ❛ f−1(α)✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ yn⇀ v ❡♥tã♦ f(yn)→f(v) =kvk2✱ ❧♦❣♦✱ ❡①✐st❡ n0 ∈N t❛❧ q✉❡ ♣❛r❛ t♦❞♦
n > n0✱f(yn)> 34kvk2 > α✳
▲❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦ ❢❛t♦ ♠❡♥❝✐♦♥❛❞♦ ❛❝✐♠❛✱ ✈❛♠♦s ♣❛rt✐r ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡s❡❥❛❞❛✳
❉❡❝♦♠♣♦♥❞♦yn♦rt♦❣♦♥❛❧♠❡♥t❡ t❡♠♦s✿ yn =zn+λnv✱ ♣♦rt❛♥t♦✱yn−v =zn+λnv−v ❡ kynk2 =kznk2+λ2nkvk2✱ ❧♦❣♦✱
kyn−vk2 = kznk2+kλnv−vk2
= kynk2−λ2nkvk
2+ (λ
n−1)2kvk2
= kynk2−2λnkvk2+kvk2
▼❛s✱ f(yn) = f(zn+λnv) = λnf(v) > 34f(v) ❡ ❝♦♠♦ f(v) = kvk2 > 0 t❡♠♦s q✉❡
λn > 34 ♣❛r❛ t♦❞♦ n > n0✳
P♦rt❛♥t♦✱ kyn−vk2 < kynk2− 12kvk2✱ ♦✉ s❡❥❛✱ kynk2 >kyn−vk2 +12kvk2 ♣❛r❛ t♦❞♦
n > n0✳
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ✶✳✽✳ ❚❡♦r❡♠❛ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦
❚♦♠❛♥❞♦ k = 12kvk2 ♦❜t❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡s❡❥❛❞❛
kxn−xk2 >kxn−x0k2+k
2a ❝❛s♦✮✿ E é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦✳
❆ ❢❛❧t❛ ❞❛ ♦rt♦❣♦♥❛❧✐❞❛❞❡ ♥♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ♥♦s r❡♠❡t❡ ❛ ✉♠❛ ❞❡♠♦♥str❛çã♦ ♥❛❞❛ ❣❡♦♠étr✐❝❛ ❝♦♠ ♠✉✐t♦s ❛rt✐❢í❝✐♦s ❛❧❣é❜r✐❝♦s✳
P❛r❛ ❡st❛ ❞❡♠♦♥str❛çã♦ ❡s❝♦❧❤❡r❡♠♦s ❛ ❢✉♥çã♦δ : [0,2]→[0,1]❝♦♠ ❛s ❝❛r❛❝t❡ríst✐❝❛s
❛♣r❡s❡♥t❛❞❛s ♥❛ ♦❜s❡r✈❛çã♦ ❄❄✳
❱❛♠♦s s✉♣♦r q✉❡ ❞❛❞♦s x0 6= x1 ∈ C ❡ 0 < t < 1✳ ❚❡♠♦s q✉❡ ♣❛r❛ t♦❞♦ ε ∈ ]0,1[
❡①✐st❡ a(ε) > 0 t❛❧ q✉❡ s❡♠♣r❡ q✉❡ kF x0 −x0k ≤ ε✱ kF x1 −x1k ≤ ε ❡♥tã♦ ♣❛r❛ t♦❞♦
xt=tx0+ (1−t)x1 ⇒ kF xt−xtk ≤a(ε)✳ ❊♥tã♦✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ✉♠ i= 0,1 t❛❧ q✉❡✿
xi−
xt+F xt
2
≥ kxi −xtk
❉❡ ❢❛t♦✱ ✈❛♠♦s s✉♣♦r ♣♦r ❛❜s✉r❞♦ q✉❡ ❡①✐st❡ t0 ∈]0,1[t❛❧ q✉❡✿
xi−
xt0 +F xt0
2
≥ kxi−xt0k
kx1−x0k ≤
x1−
xt0 +F xt0
2 + x0−
xt0 +F xt0
2
< kx1−xt0k+kx0−xt0k
= kx1−(t0x0+ (1−t0)x1)k+kx0−(t0x0+ (1−t0)x1)k
= t0(kx1−x0k) +kx1−x0k −t0(kx1−x0k)
= kx1−x0k
P♦rt❛♥t♦✱ kx1−x0k<kx1−x0k✱ ❛❜s✉r❞♦✳
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ✶✳✽✳ ❚❡♦r❡♠❛ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦
kF xt−xik = kF xt−F xi+F xi−xik
≤ kF xt−F xik+kF xi−xik
≤ kxt−xik+kF xi−xik
≤ r+ε
❚♦♠❡✱ ♥❛ ♣r♦♣♦s✐çã♦ ❄❄✱ x=xt✱ y=F xt ❡ z =xi ♦❜t❡♠♦s ❡♥tã♦✿
kF xt−xtk ≤ sup r∈[0,d(C)]
(r+ε)η( ε
r+ε) = a(ε)
❊♠ q✉❡ d(C) é ♦ ❞✐â♠❡tr♦ ❞❡ C✳
❈♦♠ a(ε)❞❡✜♥✐❞♦ ❞❡st❛ ❢♦r♠❛ t❡♠♦s✱ a(ε) =ε η(1)✱ ♦✉ s❡❥❛✱ a(ε) = 2ε ♣❛r❛ r = 0✳
❆❧é♠ ❞✐ss♦✱ t♦♠❛♥❞♦ ♦ s✉♣r❡♠♦ s❡♣❛r❛❞❛♠❡♥t❡ s♦❜r❡ ♦s ✐♥t❡r✈❛❧♦s [0,√ε−ε[❡ [√ε−
ε, d(C)]✱ t❡♠♦s✱ ♣❡❧❛ ♠♦♥♦t✐❝✐❞❛❞❡ ❞❡η(·) q✉❡✿
a(ε)≤♠❛①{√εη(1),(d(C) +ε)η(√ε)} →0
q✉❛♥❞♦ ε→0
❈♦♠♦ a(ε)≥2ε, kF xt−xtk ≤ a(ε) ❡ a(ε)→0 q✉❛♥❞♦ ε →0✱ ♦❝♦rr❡ ♣❛r❛ ♦s ❝❛s♦s ❡♠ q✉❡ x1 6=x2 t= 0,1 ❡ x0 =x1✳
❱❛♠♦s ♠♦str❛r q✉❡ ♣❛r❛ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ (xn)❡♠ C✱ s❡ xn⇀ x❡(I−F)(xn)→0 q✉❛♥❞♦ n→ ∞✱ ❡♥tã♦✱ x∈C ❡(I−F)(x) = 0✳
❙❛❜❡♠♦s q✉❡ t♦❞♦ ❢❡❝❤❛❞♦ ❝♦♥✈❡①♦ é ❢r❛❝❛♠❡♥t❡ ❢❡❝❤❛❞♦✱ ❧♦❣♦✱ x∈C✳
P❛r❛ ε0 ∈(0,1)❡s❝♦❧❤❡♠♦s ❛ s❡q✉ê♥❝✐❛ (εn) t❛❧ q✉❡
εn≤εn−1 ❡ a(εn)≤εn−1 ♣❛r❛ t♦❞♦ n∈N
❈❛♣ít✉❧♦ ✶✳ Pr✐♥❝í♣✐♦ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦ ✶✳✽✳ ❚❡♦r❡♠❛ ❞❡ s❡♠✐✲❢❡❝❤❛♠❡♥t♦
■st♦ é ♣♦ssí✈❡❧ ♣♦rq✉❡ a(ε) → 0 q✉❛♥❞♦ ε → 0✳ ❊s❝♦❧❤❡♥❞♦ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ s❡
♥❡❝❡ssár✐♦✱ t❡♠♦s✿
kF xn−xnk ≤εn ♣❛r❛ t♦❞♦ n ∈N
❊♥tã♦✱
kF y−yk ≤ε0 ♣❛r❛ t♦❞♦ y∈co{xn :n∈N} ❉❡ ❢❛t♦✱ ❛♣❧✐❝❛♥❞♦ ♠ét♦❞♦ ❞❡ ✐♥❞✉çã♦ ✜♥✐t❛ t❡♠♦s✿
❙❡❥❛ y1 ∈ co{xm, xn}✱ ❡♠ q✉❡ 1 ≤ m < n✳ ❚❡♠♦s q✉❡ kF xm − xmk ≤ εm ❡
kF xn−xnk ≤εn ❡ εn≤εm ❡♥tã♦✱ kF y1−y1k ≤a(εn)≤εm−1 ≤ε0
❙❡❥❛ y2 ∈ co{xk, xm, xn} 1 ≤ k < m < n t❡♠♦s q✉❡ y2 ∈ co{xk, y1}✳ ▲♦❣♦✱ kF y1−
y1k ≤εm−1✳ ▼❛s εm−1 ≤εk✱ ❡♥tã♦✱kF xk−xkk ≤εk ❡ kF y1−y1k ≤εk✳
❚❡♠♦s ♣♦rt❛♥t♦✱ kF y2−y2k ≤a(εk)≤εk−1 ≤ε0✳
❙❡ xn ⇀ x q✉❛♥❞♦ n → ∞✱ ❡♥tã♦ x ∈ co{xn: n∈N}✳ ▲♦❣♦✱ kF x−xk ≤ ε0✳ ❈♦♠♦
ε0 ❢♦✐ t♦♠❛❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡ ♣❡q✉❡♥♦✱ F x−x = 0✳ ❈♦♠♦ ♣❛r❛ ✉♠ y ✜①❛❞♦ F x+y
t❛♠❜é♠ é ♥ã♦ ❡①♣❛♥s✐✈❛✱ t❡♠♦s ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✱ ♦✉ s❡❥❛✱I−F é s❡♠✐✲❢❡❝❤❛❞❛✳
❈❛♣ít✉❧♦ ✷
❚❡♦r❡♠❛s ❞❡ ♣♦♥t♦ ✜①♦ ♣❛r❛ ❛♣❧✐❝❛çõ❡s
♥ã♦ ❡①♣❛♥s✐✈❛s
●❡r❛❧♠❡♥t❡ ❛ ❞✐✜❝✉❧❞❛❞❡ ❡♠ ❛♣❧✐❝❛r ♦ t❡♦r❡♠❛ ❞♦ ♣♦♥t♦ ✜①♦ ❞❡ ❇❛♥❛❝❤ é ❡♥❝♦♥tr❛r ❡s♣❛ç♦s ♦✉ ♥♦r♠❛s ❡♠ q✉❡ ♦ ♦♣❡r❛❞♦r ❞❡ ✐♥t❡r❡ss❡ s❡❥❛ ✉♠❛ ❝♦♥tr❛çã♦✳ ◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s r❡s✉❧t❛❞♦s ♥♦s q✉❛✐s ♦ t❡♦r❡♠❛ ❞❡ ♣♦♥t♦ ✜①♦ ❞❡ ❇❛♥❛❝❤ ♣❛r❛ ♦♣❡r❛❞♦r❡s
K−❝♦♥tr❛t✐✈♦s ♣♦❞❡ s❡r ❡st❡♥❞✐❞♦ ♣❛r❛ ♦♣❡r❛❞♦r❡s ♥ã♦ ❡①♣❛♥s✐✈♦s ❡♠ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤
r❡✢❡①✐✈♦✳
◆❛ ❞❡✜♥✐çã♦ ❛❜❛✐①♦ ❝♦♥s✐❞❡r❡ X∗ ♦ ❡s♣❛ç♦ ❞✉❛❧ ❞♦ ❡s♣❛ç♦ ♥♦r♠❛❞♦ X ❡X∗∗ ♦ ❡s♣❛ç♦
❜✐❞✉❛❧ ❞❡ X✳
❉❡✜♥✐çã♦ ✷✳✵✳✺ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦✳ ❉❡✜♥❛ ϕ : X → X∗∗ t❛❧ q✉❡ ϕ(x) = Λx ❡♠ q✉❡Λx:X∗ →R ❡ Λx(f) = f(x)✳ ❈♦♠♦ Λx ∈X∗∗, ❡♥tã♦✱ é ❧✐♥❡❛r ♣❛r❛ t♦❞♦x∈X✳ ❆ ❢✉♥çã♦ϕ ❝❤❛♠❛✲s❡ ❛♣❧✐❝❛çã♦ ❞❡ r❡✢❡①✐✈✐❞❛❞❡✳ ❙✉❛ ✐♠❛❣❡♠ϕ(X)⊂X∗∗ é ✉♠❛ ❜✐❥❡çã♦ ✐s♦♠étr✐❝❛✳ ❉✐③❡♠♦s q✉❡ X é r❡✢❡①✐✈♦ s❡ ϕ(X) =X∗∗✳
❖❜s❡r✈❛çã♦ ✷✳✵✳✻ ❚♦❞♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦ é r❡✢❡①✐✈♦✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ❢❛t♦ s❡ ❡♥❝♦♥tr❛ ❡♠ ❨❖❙■❉❆ ❬❄❪✳
❈❛♣ít✉❧♦ ✷✳ ❚❡♦r❡♠❛s ❞❡ ♣♦♥t♦ ✜①♦ ♣❛r❛ ❛♣❧✐❝❛çõ❡s ♥ã♦ ❡①♣❛♥s✐✈❛s
❙❡❥❛ (❊,k · k) ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡G:❊×❊−→R ✉♠❛ ❛♣❧✐❝❛çã♦ q✉❡ s❛t✐s❢❛③ ❛s
s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿
(g1) G(λx, y) =λ G(x, y)
(g2) kxk2 ≤G(x, x) ♣❛r❛ q✉❛❧q✉❡r x∈❊
(g3) G(x+y, z) = G(x, z) +G(y, z)
(g4) |G(x, y)| ≤Mkxkkyk
◆♦ t❡♦r❡♠❛ ❄❄ só s❡rã♦ ❡①✐❣✐❞❛s ❛s ❝♦♥❞✐çõ❡sg1 ❡g2 ♣❛r❛ ❛ ❛♣❧✐❝❛çã♦G✳ ❏á ♥♦ t❡♦r❡♠❛
❄❄ q✉❡ é ✉♠❛ ✈❛r✐❛çã♦ ❞♦ t❡♦r❡♠❛ ❄❄ s❡rã♦ ❡①✐❣✐❞❛s ❛s ✹ ❝♦♥❞✐çõ❡s ♣❛r❛ ❛ ❛♣❧✐❝❛çã♦ G✳
❚❡♦r❡♠❛ ✷✳✵✳✼ ❙❡❥❛ (❊,k · k) ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ r❡✢❡①✐✈♦ ❡ C ⊆❊ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦
✈❛③✐♦✱ ♥ã♦ ❧✐♠✐t❛❞♦✱ ❢❡❝❤❛❞♦✱ ❝♦♥✈❡①♦ ❝♦♠ 0∈C✳ ❚♦♠❡ f :C −→❊ ✉♠❛ ❛♣❧✐❝❛çã♦ ♥ã♦
❡①♣❛♥s✐✈❛ t❛❧ q✉❡ f(C)⊆C ❡ I−f é s❡♠✐❢❡❝❤❛❞❛✳ ❙❡
lim sup
kxk→∞
G(f(x), x)
kxk2 <1
❡♥tã♦ f ♣♦ss✉✐ ♣♦♥t♦ ✜①♦ ❡♠ ❈✳
❉❡♠♦♥str❛çã♦✿ ❆✜r♠❛♠♦s q✉❡ f é ✉♠❛ ❛♣❧✐❝❛çã♦ ❧✐♠✐t❛❞❛ ✭❉✐③❡♠♦s q✉❡ ✉♠❛ ❛♣❧✐✲
❝❛çã♦ f é ❧✐♠✐t❛❞❛ q✉❛♥❞♦ ✐♠❛❣❡♠ ❞❡ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦ ❢♦r ❧✐♠✐t❛❞♦✮✳ ❉❡ ❢❛t♦✱ ✜①❛♥❞♦ f(x0) t❡♠♦s✿
kf(x)−f(x0)k ≤ kx−x0k
≤ kx−0k+kx0−0k
≤ 2M
♣♦✐s D⊆C é ❧✐♠✐t❛❞♦✱ ♦✉ s❡❥❛✱ ❡①✐st❡ M > 0✱ t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ x∈D
kxk ≤M ❡ ✜①❛❞♦x0 ∈D t❡♠♦s q✉❡ kx−x0k ≤M✳ P♦rt❛♥❞♦✱ ❝♦♠♦
❈❛♣ít✉❧♦ ✷✳ ❚❡♦r❡♠❛s ❞❡ ♣♦♥t♦ ✜①♦ ♣❛r❛ ❛♣❧✐❝❛çõ❡s ♥ã♦ ❡①♣❛♥s✐✈❛s
kf(x)k ≤ kf(x)−f(x0)k+kf(x0)k✱ ❝♦♥❝❧✉✐♠♦s q✉❡✱ kf(x)k ≤M +K✳
❙❡❥❛ λn ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♥t✐❞❛ ❡♠ ]0,1[✱ t❛❧ q✉❡✱ lim
n→0λn = 0✳ P❛r❛ ❝❛❞❛ n ∈ N
❝♦♥s✐❞❡r❡ ❛ ❛♣❧✐❝❛çã♦ fn:C →E ❞❡✜♥✐❞❛ ♣♦r✿
fn(x) = (1−λn)f(x)
❈♦♠♦ C é ❝♦♥✈❡①♦✱ 0∈C ❡f(C)⊆C✱ t❡♠♦s q✉❡✱ fn(x)∈C ∀x∈C✱ ❧♦❣♦✱ fn(C)⊆C ❉❛❞♦sx , y ∈C t❡♠♦s✿
kfn(x)−fn(y)k = k(1−λn)f(x)− (1−λn)f(y)k
= |(1−λn)| kf(x)−f(y)k
≤ (1−λn)kx−yk ❊♥tã♦✱ fn é ✉♠❛ ❝♦♥tr❛çã♦✳
❆♣❧✐❝❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❝♦♥tr❛çã♦ ❞❡ ❇❛♥❛❝❤✱ ♦❜t❡♠♦s ✉♠ ❡❧❡♠❡♥t♦xn ∈C✱ t❛❧ q✉❡✱
fn(xn) = xn✳ ❱❛♠♦s ♣r♦✈❛r q✉❡ ❛ s❡q✉ê♥❝✐❛ xn é ❧✐♠✐t❛❞❛ ♣❛r❛ q✉❡ ♣♦ss❛♠♦s ❛♣❧✐❝❛r ❛ ❤✐♣ót❡s❡ ❞❡ s❡♠✐❢❡❝❤❛♠❡♥t♦ ❞❡ I −f✳ ❊♥tã♦✱ s✉♣♦♥❤❛♠♦s ♣♦r ❛❜s✉r❞♦ q✉❡ (xn) ♥ã♦ é ❧✐♠✐t❛❞❛✱ ♦✉ s❡❥❛✱ q✉❡ ❡①✐st❡ s✉❜s❡q✉ê♥❝✐❛ (xni) ❞❡(xn)✱ t❛❧ q✉❡✱ kxnik → ∞✳
❈♦♠♦ lim sup
kxk→∞
G(f(x), x)
kxk2 < 1✱ ❡♥tã♦✱ ❡①✐st❡♠ β ∈ (0,1) ❡ ρ0 > 0✱ t❛✐s q✉❡✱ ♣❛r❛ t♦❞♦
kxk> ρ0
G(f(x), x)
kxk2 ≤β <1
▲♦❣♦✱
G(f(x), x)≤βkxk2
❈❛♣ít✉❧♦ ✷✳ ❚❡♦r❡♠❛s ❞❡ ♣♦♥t♦ ✜①♦ ♣❛r❛ ❛♣❧✐❝❛çõ❡s ♥ã♦ ❡①♣❛♥s✐✈❛s
P❛r❛ n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❡ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ (g2) t❡♠♦s✿
kxnk2 ≤ G(xn, xn)
= G[(1−λn)f(xn), xn]
= (1−λn)G(f(xn), xn)
≤ (1−λn)βkxnk2
❉✐✈✐❞✐♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ♣♦r kxnk2 t❡♠♦s✿
1≤(1−λn)β ❈♦♠♦ λn→0 ❡♥tã♦ β ≥1 ❛❜s✉r❞♦✱ ♣♦✐s✱ β∈(0,1)✳
P♦rt❛♥t♦ (xn) é ❧✐♠✐t❛❞❛✳ ❈♦♠♦xn = (1−λn)f(xn) t❡♠♦s✿
kxn−f(xn)k=λnkf(xn)k →0 q✉❛♥❞♦ n→ ∞
❖ ❡s♣❛ç♦ ❊ s❡♥❞♦ r❡✢❡①✐✈♦ ❡ (xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛✱ ♦ t❡♦r❡♠❛ ❞❡ ❊❜❡r❧❡✐♥✲ ❙❤♠✉❧②❛♥ ♥♦s ❣❛r❛♥t❡ ✭❡✈❡♥t✉❛❧♠❡♥t❡ t♦♠❛♥❞♦ s✉❜s❡q✉ê♥❝✐❛✮ q✉❡ (xn) ❝♦♥✈❡r❣❡ ❢r❛❝❛✲ ♠❡♥t❡ ❛x0 ∈C✱ ♣♦✐s ♣♦r ❤✐♣ót❡s❡C é ❝♦♥✈❡①♦ ❡ ❢♦rt❡♠❡♥t❡ ❢❡❝❤❛❞♦✱ ♣♦rt❛♥t♦✱ ❢r❛❝❛♠❡♥t❡
❢❡❝❤❛❞♦✳ ❈♦♠♦ I−f é s❡♠✐❢❡❝❤❛❞❛✱ t❡♠♦s✿
x0−f(x0) = 0 ⇒f(x0) =x0
❖✉ s❡❥❛✱ f ♣♦ss✉✐ ♣♦♥t♦ ✜①♦ ❡♠ C✳ ✷
❖ t❡♦r❡♠❛ s❡❣✉✐♥t❡ é ✉♠❛ ✈❛r✐❛çã♦ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✳ ❙✉❛ ❞❡♠♦♥str❛çã♦ s❡❣✉❡ ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✳
❚❡♦r❡♠❛ ✷✳✵✳✽ ❙❡❥❛ (❊,k · k)✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ r❡✢❡①✐✈♦ ❡ C ⊆E ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦
✈❛③✐♦✱ ♥ã♦ ❧✐♠✐t❛❞♦✱ ❢❡❝❤❛❞♦✱ ❝♦♥✈❡①♦✳ ❚♦♠❡ f : C −→ ❊ ✉♠❛ ❛♣❧✐❝❛çã♦ ♥ã♦ ❡①♣❛♥s✐✈❛
t❛❧ q✉❡ f(C)⊆C ❡ I−f é s❡♠✐❢❡❝❤❛❞❛✳ ❙❡ ♣❛r❛ ❛❧❣✉♠x0 ∈C
lim sup
kxk→∞
G(f(x)−x0, x)
kxk2 <1
❈❛♣ít✉❧♦ ✷✳ ❚❡♦r❡♠❛s ❞❡ ♣♦♥t♦ ✜①♦ ♣❛r❛ ❛♣❧✐❝❛çõ❡s ♥ã♦ ❡①♣❛♥s✐✈❛s
❡♥tã♦ f ♣♦ss✉✐ ♣♦♥t♦ ✜①♦ ❡♠ ❈✳
❉❡♠♦♥str❛çã♦✿ ❆✜r♠❛♠♦s q✉❡ f é ✉♠❛ ❛♣❧✐❝❛çã♦ ❧✐♠✐t❛❞❛ ✭❉✐③❡♠♦s q✉❡ ✉♠❛ ❛♣❧✐✲
❝❛çã♦ f é ❧✐♠✐t❛❞❛ q✉❛♥❞♦ ✐♠❛❣❡♠ ❞❡ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦ ❢♦r ❧✐♠✐t❛❞♦✮✳ ❉❡ ❢❛t♦✱ ✜①❛♥❞♦ f(x1) t❡♠♦s✿
kf(x)−f(x1)k ≤ kx−x1k
≤ kxk+kx1k
≤ 2M
♣♦✐s D⊆C é ❧✐♠✐t❛❞♦✱ ♦✉ s❡❥❛✱ ❡①✐st❡ M > 0✱ t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ x∈D
kxk ≤M ❡ ✜①❛❞♦x1 ∈D t❡♠♦s q✉❡ kx−x1k ≤M✳ P♦rt❛♥❞♦✱ ❝♦♠♦
kf(x)k ≤ kf(x)−f(x1)k+kf(x1)k✱ ❝♦♥❝❧✉✐♠♦s q✉❡✱ kf(x)k ≤M +K✳
❙❡❥❛ λn ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♥t✐❞❛ ❡♠ (0,1)✱ t❛❧ q✉❡✱ limx→0λn = 0✳ P❛r❛ ❝❛❞❛ n ∈ N ❝♦♥s✐❞❡r❡ ❛ ❛♣❧✐❝❛çã♦ fn:C →E ❞❡✜♥✐❞❛ ♣♦r✿
fn(x) = (1−λn)f(x) +λnx0
❈♦♠♦ C é ❝♦♥✈❡①♦✱ f(C)⊆C✱ t❡♠♦s q✉❡✱ fn(x)∈C ∀x∈C✱ ❧♦❣♦✱ fn(C)⊆C ❉❛❞♦sx , y ∈C t❡♠♦s✿
kfn(x)−fn(y)k = k(1−λn)f(x) +λnx0− (1−λn)f(y)−λnx0k
= |(1−λn)| kf(x)−f(y)k
≤ (1−λn)kx−yk ❈♦♠♦ (1−λn)<1❡♥tã♦✱ fn é ✉♠❛ ❝♦♥tr❛çã♦ ✳
❆♣❧✐❝❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❝♦♥tr❛çã♦ ❞❡ ❇❛♥❛❝❤✱ ♦❜t❡♠♦s ✉♠ ❡❧❡♠❡♥t♦xn ∈C✱ t❛❧ q✉❡✱
fn(xn) = xn✳
❱❛♠♦s ♣r♦✈❛r q✉❡ ❛ s❡q✉ê♥❝✐❛ xn é ❧✐♠✐t❛❞❛ ♣❛r❛ q✉❡ ♣♦ss❛♠♦s ❛♣❧✐❝❛r ❛ ❤✐♣ót❡s❡ ❞❡ s❡♠✐❢❡❝❤❛♠❡♥t♦ ❞❡ I −f✳ ❊♥tã♦✱ s✉♣♦♥❤❛♠♦s ♣♦r ❛❜s✉r❞♦ q✉❡ (xn) ♥ã♦ é ❧✐♠✐t❛❞❛✱ ♦✉ s❡❥❛✱ q✉❡ ❡①✐st❡ s✉❜s❡q✉ê♥❝✐❛ (xni)❞❡ (xn)✱ t❛❧ q✉❡✱kxnik → ∞✳
❈❛♣ít✉❧♦ ✷✳ ❚❡♦r❡♠❛s ❞❡ ♣♦♥t♦ ✜①♦ ♣❛r❛ ❛♣❧✐❝❛çõ❡s ♥ã♦ ❡①♣❛♥s✐✈❛s
❈♦♠♦ lim sup
kxk→∞
G(f(x)−x0, x)
kxk2 < 1✱ ❡♥tã♦✱ ❡①✐st❡♠ β ∈(0,1) ❡ ρ0 >0✱ t❛✐s q✉❡✱ ♣❛r❛
t♦❞♦kxk> ρ0
G(f(x)−x0, x)
kxk2 ≤β <1
▲♦❣♦✱
G(f(x)−x0, x)≤βkxk2
P❛r❛ n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❡ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ (g2) t❡♠♦s✿
kxnk2 ≤ G(xn, xn)
= G[(1−λn)f(xn) +λnx0, xn]
= (1−λn)G(f(xn)−x0, xn) +G(x0, xn)
≤ (1−λn)βkxk2+Mkx0kkxnk
❉✐✈✐❞✐♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ♣♦r kxnk2 t❡♠♦s✿
1≤(1−λn)β
❈♦♠♦ λn → 0 ❡♥tã♦ β ≥ 1 ❛❜s✉r❞♦✱ ♣♦✐s✱ β ∈ (0,1)✳ P♦rt❛♥t♦ (xn) é ❧✐♠✐t❛❞❛✳ ❈♦♠♦
xn= (1−λn)f(xn) +λnx0 t❡♠♦s✿
kxn−f(xn)k=λnkf(Xn)−x0k →0
q✉❛♥❞♦ n→ ∞ ✭❥á q✉❡ f é ❧✐♠✐t❛❞❛✮✳
❖ ❡s♣❛ç♦ ❊ s❡♥❞♦ r❡✢❡①✐✈♦ ❡ (xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛✱ ♦ t❡♦r❡♠❛ ❞❡ ❊❜❡r❧❡✐♥✲ ❙❤♠✉❧②❛♥ ♥♦s ❣❛r❛♥t❡ ✭❡✈❡♥t✉❛❧♠❡♥t❡ t♦♠❛♥❞♦ s✉❜s❡q✉ê♥❝✐❛✮ q✉❡ (xn) ❝♦♥✈❡r❣❡ ❢r❛❝❛✲ ♠❡♥t❡ ❛x∗ ∈C✱ ♣♦✐s ♣♦r ❤✐♣ót❡s❡C é ❝♦♥✈❡①♦ ❡ ❢♦rt❡♠❡♥t❡ ❢❡❝❤❛❞♦✱ ♣♦rt❛♥t♦✱ ❢r❛❝❛♠❡♥t❡ ❢❡❝❤❛❞♦✳ ❈♦♠♦ I−f é s❡♠✐❢❡❝❤❛❞❛✱ t❡♠♦s✿
x∗−f(x∗) = 0 ⇒f(x∗) =x∗
❖✉ s❡❥❛✱ f ♣♦ss✉✐ ♣♦♥t♦ ✜①♦ ❡♠ C✳ ✷
