Real_Analysis_Exercises by dap ndfk

Ask seic Pragmatik c An lushc IWAKEIM IWANNHS

Aj na, Dekèmbrioc 2008

'Askhsh 1 'Estw φ : [0, +∞) → R aÔxousa sun rthsh tètoia ¸ste φ(t) = 0 ⇔ t = 0 kai φ(t + s) ≤ φ(t) + φ(s), ∀t, s ≥ 0. An ρ metrik sèna sÔnolo X , deÐxte ìti kai h τ = φ ◦ ρ eÐnai epÐshc metrik .

'Askhsh 2 'Estw ρ metrik sèna sÔnolo X . DeÐxte ìti oi ρ1 = min(ρ, 1) kai ρ3 = ρa , 0 < a ≤ 1 eÐnai metrikèc sto X .

ρ , ρ2 = 1+ρ

'Askhsh 3 'Estw X 6= ∅ sÔnolo kai f : R → (a, b) ⊆ R suneq c kai gnhsÐwc aÔxousa. OrÐzoume th sun rthsh ρ : X × X → R me ρ(x, y) = |f (x) − f (y)|, ∀x, y ∈ X. DeÐxte ìti h ρ eÐnai metrik sto X .

'Askhsh 4 'Estw X 6= ∅ sÔnolo. OrÐzoume th sun rthsh ¯ ¯ ¯ x ¯ y ¯ , ∀x, y ∈ X. ρ : X × X → R me ρ(x, y) = ¯¯ − 1 + |x| 1 + |y| ¯ DeÐxte ìti h ρ eÐnai metrik sto X .

'Askhsh 5 'Estw a,b ∈ R me a<b, X = {f : [α, β] → R|∃f 0 kai eÐnai suneq c}

kai ρ : X × X → R, me ρ(f, g) = sup{|f 0 (x) − g 0 (x)| : x ∈ [α, β]} + |f (β) − g(β)|, ∀f, g ∈ X . DeÐxte ìti h ρ eÐnai metrik sto X . Z β 'Askhsh 6 'Estw R([α, β]) = {f : [α, β] → R : ∃ f (x)dx} kai α

C([α, β]) = {f : [α, β] → R : f suneq c}. Exet ste an oi akìloujec sunart seic eÐnai metrikèc sta sÔnola R([α, β]), C([α, β]) : ρ(f, g) = sup{|f (x) − g(x)| : x ∈ [α, β]} Z β τ (f, g) = |f (x) − g(x)|dx α

µZ

σ(f, g) = µZ

|f (x) − g(x)|2 dx

α β

d(f, g) =

|f (x) − g(x)| dx α

¶ 21

¶ p1 , p ≥ 1.

'Askhsh 7 'Estw X to sÔnolo ìlwn twn pragmatik¸n akolouji¸n kai β = (βn ) mia akoloujÐa me ∞ X

∞ X

βn < ∞. Gia x = (xn ) kai y = (yn ) jètoume

n=1

|xn − yn | . ApodeÐxte ìti o (X, d) eÐnai metrikìc q¸roc kai 1 + |x − y | n n n=1 upologÐste th di metrì tou.

d(x, y) =

βn ·

'Askhsh 8 'Estw (X, ρ) metrikìc q¸roc kai A = {a > 0|ρa metrik }. DeÐxte ìti to A eÐnai upodi sthma tou (0, +∞).

'Askhsh 9 Ston R2 , poièc eÐnai oi sfaÐrec S((0, 0), r), r > 0, wc proc tic

metrikèc p ρ(x, y) = max{|x1 − y1 |, |x2 − y2 |}, ρ1 = |x1 − y1 | + |x2 − y2 | kai ρ2 = |x1 − y1 |2 + |x2 − y2 |2 , gia x = (x1 , x2 ), y = (y1 , y2 );

'Askhsh 10 DeÐxte ìti se k je metrikì q¸ro (X, ρ) ta peperasmèna sÔnola

eÐnai kleist .

'Askhsh 11 'Estw (X, ρ) metrikìc q¸roc, x ∈ X kai r > 0. Tìte S(x, r) ⊆ e r), en¸ den isqÔei genik h isìthta. Eidik stouc EukleÐdeiouc q¸rouc isqÔei S(x, h isìthta.

'Askhsh 12 'Estw (X, ρ) metrikìc q¸roc kai A, B ⊆ X mh ken .

DeÐxte ìti: (i) δ(A) = δ(A), ìpou δ(A) = sup{ρ(x, y)|x, y ∈ A}. (ii) An A ⊆ B , tìte d(x, B) ≤ d(x, A), ∀x ∈ X (iii) d(x, A) = d(x, A). (iv) d(A, B) = d(A, B) = d(A, B), ìpou d(A, B) = inf{ρ(x, y)|x ∈ A, y ∈ B}.

'Askhsh 13 'Estw (X, ρ) metrikìc q¸roc kai A, B ⊆ X mh ken me A sumpagèc.

Tìte: (i) Up rqoun x, y ∈ A tètoia ¸ste δ(A) = ρ(x, y). (ii) Up rqei a ∈ A tètoio ¸ste d(A, B) = d(a, B). (iii) An to B eÐnai sumpagèc, tìte up rqoun a ∈ A kai b ∈ B tètoia ¸ste d(A, B) = ρ(a, b). (iv) To (iii) den isqÔei an to A eÐnai kleistì kai ìqi anagkaÐa sumpagèc.

'Askhsh 14 'Estw C([0, 1]) = {f : [0, 1] → R|f suneq c}. Jètoume ρ1 (f, g) = sup{|f (x) − g(x)| : x ∈ [0, 1]}, Z 1 ρ2 (f, g) = |f (x) − g(x)|dx, 0 ³Z 1 ´ 21 ρ3 (f, g) = |f (x) − g(x)|2 dx , ∀f, g ∈ C([0, 1]). 0

ApodeÐxte ìti oi metrikèc ρ1 , ρ2 , ρ3 ikanopoioÔn tic sqèseic ρ2 ≤ ρ3 ≤ ρ1 all den eÐnai isodÔnamec metaxÔ touc.

'Askhsh 15 'Estw (X, ρ) metrikìc q¸roc kai A, B ⊆ X mh ken me A sumpagèc, B kleistì kai A ∩ B = ∅. DeÐxte ìti d(A, B) > 0.

'Askhsh 16 'Estw (X, ρ) metrikìc q¸roc kai A, B ⊆ X mh ken me A sumpagèc.

DeÐxte ìti isqÔei d(A, B) = 0 ⇔ A ∩ B 6= ∅. (Parat rhsh: Ja mporoÔsame anti tou A ∩ B 6= ∅ na gr fame A ∩ B = 6 ∅ kai B kleistì, afoÔ B kleistì =⇒ B = B). DeÐxte ìti autì den isqÔei an to A eÐnai kleistì kai ìqi anagkaÐa sumpagèc.

'Askhsh 17 An A anoiktì kai D puknì uposÔnolo tou metrikoÔ q¸rou (X, ρ), deÐxte ìti D ∩ A = A.

'Askhsh 18 'Estw (X, ρ) metrikìc q¸roc kai D ⊆ X mh kenì. DeÐxte ìti D puknì sto X ⇔ D ∩ A 6= ∅, ∀A ⊆ X anoiktì mh kenì. 'Askhsh 19 'Estw (X, ρ) metrikìc q¸roc. An A, B ⊆ X mh ken , poièc apì tic parak tw isìthtec isqÔoun;

(i) A ∪ B = A ∪ B (iii) (A ∩ B)◦ = A◦ ∩ B ◦ (v) A × B = A × B

(ii) A ∩ B = A ∩ B (iv) (A ∪ B)◦ = A◦ ∪ B ◦ (vi) (A × B)◦ = A◦ × B ◦

'Askhsh 20 'Estw ρ1 , ρ2 metrikèc sèna sÔnolo X . DeÐxte ìti ρ1 ∼ ρ2 ⇔ h

tautotik apeikìnish φ : (X, ρ1 ) → (X, ρ2 ) me φ(x) = x, ∀x ∈ X eÐnai omoiomorfismìc.

'Askhsh 21 'Estw (X, ρ) metrikìc q¸roc kai (xn )n akoloujÐa sto X . DeÐxte ρ

ìti an xn → x kai xn → y , tìte x = y (dhl. to ìrio akoloujÐac an up rqei eÐnai monadikì).

'Askhsh 22 'Estw (X, ρ) metrikìc q¸roc kai (xn )n , (yn )n dÔo akoloujÐec sto ρ

X . DeÐxte ìti an xn → xo ∈ X kai yn → yo ∈ X , tìte ρ(xn , yn ) → ρ(xo , yo ) (sÔgklish kat suntetagmènh).

'Askhsh 23 'Estw (X, ρ) metrikìc q¸roc. Tìte x ∈ A ⇔ ∃(xn )n ⊂ A ¸ste ρ

xn → x (dhl. to A eÐnai to sÔnolo twn orÐwn akolouji¸n me ìrouc apì to A).

'Askhsh 24 'Estw (X, ρ) metrikìc q¸roc, xo ∈ X kai ² > 0. DeÐxte ìti S(xo , ²) ⊆ {x ∈ X|ρ(x, xo ) ≤ ²}. D¸ste par deigma metrikoÔ q¸rou ¸ste sthn parap nw sqèsh na èqoume gn sio uposÔnolo. Sunèqeia sunart sewn

'Askhsh 25 D¸ste paradeÐgmata:

(i) Enìc kleistoÔ sunìlou A ⊆ R kai miac suneqoÔc sun rthshc f : R → R ¸ste to f (A) ìqi kleistì sto R. (ii) Enìc anoiktoÔ sunìlou A ⊆ R kai miac suneqoÔc sun rthshc f : R → R ¸ste to f (A) ìqi anoiktì sto R.

'Askhsh 26 'Estw (X, ρ) metrikìc q¸roc, xo ∈ X kai A 6= ∅ uposÔnolo tou X . ApodeÐxte ìti oi parak tw sunart seic eÐnai suneqeic: (i) ρ : X × X → R. (ii) f : X → R, me f (x) = ρ(x, xo ). (iii) g : X → R, me g(x) = d(x, A). EpÐshc deÐxte ìti h g : X → R, me g(x) = d(x, A) eÐnai sun rthsh Lipschitz.

'Askhsh 27 'Estw (X, ρ) metrikìc q¸roc kai A, B ⊆ X mh ken me A ∩ B = ∅.

DeÐxte ìti h sun rthsh

f : X → R me f (x) =

d(x, A) d(x, A) + d(x, B)

eÐnai kal orismènh suneq c sun rthsh me f (x) = 0, an x ∈ A kai f (x) = 1, an x ∈ B . An epiplèon d(A, B) = 0, tìte h f den eÐnai omoiìmorfa suneq c.

'Askhsh 28 'Estw (X, ρ1 ), (Y, ρ2 ) metrikoÐ q¸roi kai f : (X, ρ1 ) → (Y, ρ2 )

sun rthsh. ApodeÐxte ìti ta akìlouja eÐnai isodÔnama: (i) f suneq c (ii) f −1 (G) anoiktì sto X, ∀ G ⊆ Y anoiktì. (iii) f −1 (F) kleistì sto X, ∀ F ⊆ Y kleistì. (iv) f −1 (B) ⊆ f −1 (B), ∀B ⊆ Y . (v) f (A) ⊆ f (A) , ∀A ⊆ X .

'Askhsh 29 'Estw X, Y metrikoÐ q¸roi, f, g : X → Y suneqeÐc sunart seic.

DeÐxte ìti: (i) To F = {x ∈ X : f (x) = g(x)} eÐnai kleistì sto X . (ii) An D ⊆ X puknì kai f |D = g|D , tìte f = g . (iii) To gr fhma thc f , Gf = {(x, f (x)) : x ∈ X} ⊆ X × Y , eÐnai kleistì (sto X × Y ).

'Askhsh 30 'Estw (X, ρ) metrikìc q¸roc, ∅ 6= A, B ⊆ X , a ∈ R kai f, g : X →

R suneqeÐc sunart seic. ApodeÐxte ìti: (i) Ta parak tw sÔnola eÐnai kleist : {x ∈ X|f (x) ≥ a} {x ∈ X|f (x) ≤ a} {x ∈ X|f (x) = a} {x ∈ X|f (x) ≥ g(x)} {x ∈ X|f (x) ≤ g(x)} {x ∈ X|f (x) = g(x)} (ii) Ta parak tw sÔnola eÐnai anoikt :

{x ∈ X|f (x) > a} {x ∈ X|f (x) < a} {x ∈ X|f (x) > g(x)} {x ∈ X|f (x) < g(x)}

'Askhsh 31 'Estw (X, ρ) metrikìc q¸roc, D puknì uposÔnolo tou X kai f |D =

g|D (antistoÐqwc f |D ≤ g|D ). DeÐxte ìti f = g (antistoÐqwc f ≤ g ).

'Askhsh 32 'Estw (X, ρ1 ), (Y, ρ2 ) metrikoÐ q¸roi kai f : (X, ρ1 ) → (Y, ρ2 )

sun rthsh. ApodeÐxte ìti f suneq c sto xo ⇔ ∀(xn )n akoloujÐa sto X me ρ1 ρ2 xn → xo ⇒ f (xn ) → f (xo ).

'Askhsh 33 'Estw (X, ρ1 ), (Y, ρ2 ) metrikoÐ q¸roi, f : (X, ρ1 ) → (Y, ρ2 ) suneq c sun rthsh kai K ⊆ X sumpagèc. DeÐxte ìti f (K) sumpagèc.

'Askhsh 34 'Estw X 6= ∅ sumpag c metrikìc q¸roc kai f : X → R suneq c.

ApodeÐxte ìti h f paÐrnei mègisth kai el qisth tim , dhl. ∃x1 , x2 ∈ X tètoia ¸ste f (x1 ) ≤ f (x) ≤ f (x2 ), ∀x ∈ X .

'Askhsh 35 'Estw X 6= ∅ sumpag c metrikìc q¸roc kai f : X → X sun rthsh me thn idiìthta ρ(f (x), f (y)) < ρ(x, y), ∀x, y ∈ X . DeÐxte ìti up rqei monadikì xo ∈ X tètoio ¸ste f (xo ) = xo . Isosunèqeia, Omoiìmorfh sunèqeia

'Askhsh 36 'Estw X 6= ∅ metrikìc q¸roc kai fn : X → R, n ∈ N akoloujÐa

suneq¸n sunart sewn pou sugklÐnei omoiìmorfa. DeÐxte ìti h (fn ) eÐnai isosuneq c.

'Askhsh 37 (i) 'Estw F = {fn : (0, 1) → R me fn (x) = sunart sewn. Na exetasteÐ h F wc proc thn isosunèqeia.

1 , xn

n ∈ N} oikogèneia

(ii) 'Estw F = {fn : [0, 1) → R me fn (x) = xn , n ∈ N} oikogèneia sunart sewn. Na exetasteÐ h F wc proc thn isosunèqeia. (iii) 'Estw F = {fn : [0, 1] → R me fn (x) = xn , n ∈ N} oikogèneia sunart sewn. Na exetasteÐ h F wc proc thn isosunèqeia.

'Askhsh 38 'Estw F = {fn : [0, 1] → R : f suneq c, |f (x) − f (y)| ≤ M |x − y|} oikogèneia sunart sewn. DeÐxte ìti h F eÐnai isosuneq c.

'Askhsh 39 'Estw (fn )n h akoloujÐa twn sunart sewn fn (x) =

x2 , x ∈ [0, 1], n ∈ N x2 + (1 − nx)2

DeÐxte ìti: (i) H (fn )n eÐnai omoiìmorfa fragmènh. (ii) H (fn )n sugklÐnei kata shmeÐo. (iii) H (fn )n den èqei kami upakoloujÐa pou na sugklinei omoiìmorfa sto di sthma [0, 1]. Up rqei antÐfash me to Je¸rhma Ascoli; Poi eÐnai aut ;

'Askhsh 40 'Estw (X, ρ) sumpag c metrikìc q¸roc, fn : X → R, n ∈ N mia κ.σ.

akoloujÐa suneq¸n sunart sewn me fn → f . Na deiqjeÐ ìti h akoloujÐa fn eÐnai oµ. isosuneq c ⇔ fn → f.

'Askhsh 41 'Estw (X, ρ1 ) sumpag c metrikìc q¸roc kai (Y, ρ2 ) metrikìc q¸roc. κ.σ.

An fn : X → Y , n ∈ N eÐnai mia akoloujÐa isosuneq¸n sunart sewn me fn → f , oµ. na deiqjeÐ ìti fn → f.

'Askhsh 42 'Estw (X, ρ), (Y, ρ) metrikoÐ q¸roi me (Y, ρ) pl rhc, D ⊆ X tètoio ¸ste D = X . An fn : X → Y , n ∈ N eÐnai mia akoloujÐa isosuneq¸n sunart sewn ¸ste h (fn (x))n sugklÐnei ∀x ∈ D, na deiqjeÐ ìti ∃f : X → Y suneq c ¸ste κ.σ. fn → f. EpÐshc, deÐxte ìti h (fn )n sugklÐnei omoiìmorfa sta sumpag uposÔnola oµ. tou X, dhl. ∀K ⊆ X sumpagèc, h fn |K → f |K . Z 1 'Askhsh 43 'Estw f : [0, 1] → R suneq c ¸ste xn f (x)dx = 0, ∀n ∈ N∪{0}. 0

DeÐxte ìti f ≡ 0. AkoloÔjwc, deÐxte ìti an f, g : [0, 1] → R suneqeÐc sunart seic Z 1 Z 1 n tètoiec ¸ste x f (x)dx = xn g(x)dx, ∀n ∈ N, tìte f ≡ g. 0

'Askhsh 44 'Estw X 6= ∅ sÔnolo kai B(X) = {f : X → R : f fragmènh }.

OrÐzoume th sun rthsh ρ : B(X) × B(X) → R me ρ(f, g) = sup{|f (x) − g(x)| : x ∈ X}, ∀f, g ∈ B(X). Na deÐxete ìti h ρ eÐnai metrik sto q¸ro B(X) kai ìti o (B(X), ρ) eÐnai pl rhc.

'Askhsh 45 'Estw X 6= ∅ sÔnolo kai C(X) = {f : X → R : f suneq c

kai fragmènh }. OrÐzoume th sun rthsh ρ : C(X) × C(X) → R me ρ(f, g) = sup{|f (x) − g(x)| : x ∈ X}, ∀f, g ∈ C(X). Na deÐxete ìti h ρ eÐnai metrik sto q¸ro C(X) kai ìti o (C(X), ρ) eÐnai pl rhc.

'Askhsh 46 'Estw (X, ρ) sumpag c metrikìc q¸roc kai A ⊆ X. Tìte, ta akìlouja eÐnai isodÔnama: (i) A sumpagèc sto (C([0, 1]), ρ). (ii) A kleistì sto (C([0, 1]), ρ), isosuneqèc kai kat shmeÐo fragmèno.

'Askhsh 47 'Estw B := {f ∈ C([0, 1]) : kf k∞ ≤ 1}, dhl. B = Sek·k∞ ston (C([0, 1]), k · k∞ ). DeÐxte ìti to B den eÐnai sumpagèc ston (C([0, 1]), k · k∞ ).

'Askhsh 48 IsodÔnamh diatÔpwsh tou Jewr matoc Weierstrass: Gia k je f ∈ C([a, b]) kai ² > 0, up rqei p ∈ A ≡ {p ∈ C([a, b]) : p polu¸numo } tètoio ¸ste |f (x) − p(x)| < ², ∀x ∈ [a, b]. 'Askhsh 49 'Estw D = {x1 , x2 , . . .} arijm simo puknì uposÔnolo tou [0, 1] kai fn : [0, 1] → R me

fn (x) = d(x, {x1 , x2 , . . . , xn }), ∀x ∈ [0, 1], ∀n ∈ N. Exet ste thn (fn )n∈N wc proc thn kata shmeÐo kai thn omoiìmorfh sÔgklish.

'Askhsh 50 'Estw X , Y metrikoÐ q¸roi kai f : X → Y mia sun rthsh epÐ. Na

deÐxete ìti: (i) An o X eÐnai sumpag c kai h f suneq c, tìte h f eÐnai omoiìmorfa suneq c. (ii) An o X eÐnai olik fragmènoc kai h f omoiìmorfa suneq c, tìte o Y eÐnai olik fragmènoc.

'Askhsh 51 'Estw f : [0, 1] → R, ∀n ∈ N akoloujÐa Riemann oloklhr¸simwn

sunart sewn, omoiìmorfa Z xfragmènh (dhl. ∃M > 0 : |fn (t)| ≤ M , ∀t ∈ [0, 1],∀n ∈ N). Jètoume Fn (x) = fn (t)dt, x ∈ [0, 1], n ∈ N. ApodeÐxte ìti up rqei up0

akoloujÐa Fkn thc Fn h opoÐa na sugklÐnei omoiìmorfa.

'Askhsh 52 ApodeÐxte ìti k je basik akoloujÐa sena metrikì q¸ro (X, ρ) èqei to polÔ èna shmeÐo suss¸reushc.

'Askhsh 53 DeÐxte ìti den up rqei akoloujÐa (Gn )n anoikt¸n kai pukn¸n uposunìlwn tou R ¸ste

∞ \

n=1

(dhl. to Q den mporeÐ na grafteÐ san Gδ sÔnolo). AntiparadeÐgmata

Antipar deigma 1 'Estw X 6= ∅ metrikìc q¸roc kai ρ1 , ρ2 dÔo isodÔnamec

metrikèc sto X . An o (X, ρ1 ), eÐnai olik fragmènoc, den èpetai ìti o (X, ρ2 ) eÐnai olik fragmènoc.

Antipar deigma 2 'Estw X 6= ∅ metrikìc q¸roc kai ρ1 , ρ2 dÔo isodÔnamec metrikèc sto X . An o (X, ρ1 ) eÐnai pl rhc metrikìc q¸roc, den èpetai ìti o (X, ρ2 ) eÐnai pl rhc.

Antipar deigma 3 DeÐxte, me kat llhla antiparadeÐgmata, ìti:

(i) 'Enac diaqwrÐsimoc metrikìc q¸roc den eÐnai kat n gkhn kai sumpag c. (ii) Den isqÔei p nta ìti an èna sÔnolo K eÐnai kleistì kai fragmèno, tìte eÐnai kai sumpagèc. (iii) 'Enac omoiomorfismìc den eÐnai kat n gkhn isometrÐa. (iv) K je fragmèno sÔnolo den eÐnai p ntote olik fragmèno.

'Askhsh 54 (i) JewroÔme th sun rthsh f : R → [−1, 1] me f (x) =

x . 1 + |x|

DeÐxte ìti h f eÐnai omoiomorfismìc.

(ii) 'Estw (X, k · k) q¸roc me nìrma. JewroÔme th sun rthsh f : X → B(0, 1) ≡ {x ∈ X : kxk < 1} me f (x) = DeÐxte ìti h f eÐnai omoiomorfismìc. [Parat rhsh: To (ii) eÐnai genÐkeush tou (i)].

x . 1 + kxk

LÔsh 'Askhshc 1 EpalhjeÔoume tic idiìthtec thc metrik c: υπoθεση

(a) τ (x, y) = 0 ⇔ φ(ρ(x, y)) = 0 ⇐⇒ ρ(x, y) = 0 ⇔ x = y. (b) τ (x, y) = φ(ρ(x, y)) = φ(ρ(y, x)) = τ (y, x), ∀x, y ∈ X. φ%

(g) τ (x, y) = φ(ρ(x, y)) ≤ φ(ρ(x, z) + ρ(z, y)) τ (x, z) + τ (z, y), ∀x, y, z ∈ X.

υπoθεση

≤

φ(ρ(x, z)) + φ(ρ(z, y)) =

t , φ2 = min(t, 1) 1 + t kai φ3 = ta , 0 < a ≤ 1. Tìte, ρi = φi ◦ ρ, i = 1, 2, 3. 'Ara, gia na deÐxoume ìti oi ρi = φi ◦ ρ, i = 1, 2, 3 eÐnai metrikèc, arkeÐ na epalhjeÔsoume tic upojèseic thc 'Askhshc 1, p.q. gia th φ3 h upoprosjetikìthta apodeiknÔetai wc ex c:

LÔsh 'Askhshc 2 Jètoume φi : [0, ∞) → R me φ1 =

s s (s + t)a ≤ sa + ta ⇐⇒ ( + 1)a ≤ ( )a + 1, t t dhl. thc morf c

(x + 1)a ≤ xa + 1, x ≥ 0,

h opoÐa isqÔei (anisìthta Bernoulli).

LÔsh 'Askhshc 3 EpalhjeÔoume tic idiìthtec thc metrik c: (a) ρ(x, y) = 0 =⇒ f (x) = f (y) kai epeid f gnhsÐwc aÔxousa, èpetai ìti x = y, ∀x, y ∈ R. (b) ρ(x, y) = |f (x) − f (y)| = |f (y) − f (x)| = ρ(y, x), ∀x, y ∈ R. (g) Ja prèpei na deÐxoume thn trigwnik anisìthta, dhl. ∀x, y, z ∈ R

LÔsh 'Askhshc 4 JewroÔme thn apeikìnish f : R → R me f (x) = Gia na deÐxoume ìti h

¯ ¯ ρ : X × X → R me ρ(x, y) = ¯

x . 1 + |x|

x y ¯¯ − ¯, ∀x, y ∈ R 1 + |x| 1 + |y|

11 eÐnai metrik , arkeÐ ('Askhsh 3) na deÐxoume ìti h f eÐnai gnhsÐwc aÔxousa. Autì apodeiknÔetai jewr¸ntac x, y ∈ R me x < y kai akoloÔjwc diakrÐnontac tic peript¸seic x < y < 0, x < 0 < y, 0 ≤ x < y.

LÔsh 'Askhshc 5 Katarq n, h ρ kal eÐnai orismènh, afoÔ f 0 − g0 suneq c wc diafor suneq¸n, ra sup |f 0 (x) − g 0 (x)| < ∞. x∈[α,β]

EpalhjeÔoume tic idiìthtec thc metrik c:

(i) ρ(f, g) = 0 ⇔ sup |f 0 (x) − g 0 (x)| = 0 kai f (β) = g(β) x∈[α,β]

trigwnik anisìthta. 'Ara kai to jroism touc, dhl, h ρ ikanopoieÐ thn trigwnik anisìthta.

LÔsh 'Askhshc 6 (i) H ρ eÐnai metrik ston R([α, β]) giatÐ R([α, β]) upìqwroc tou B([α, β]) = {f : [α, β] → R|f fragmènh }, o opoÐoc, ìpwc xèroume, gÐnetai metrikìc q¸roc me thn ρ. H ρ eÐnai metrik kai ston C([α, β]). H apìdeixh eÐnai an logh me thn perÐptwsh ìpou X = {f : [α, β] → R|f fragmènh }. (ii) H d den eÐnai metrik ston R([α, β]) giatÐ, ∀f, g ∈ R([α, β]) ³Z

d(f, g) = 0 ⇐⇒

|f (x) − g(x)|p dx

´ p1

∗

= 0 <6=> f = g.

(p.q. oi f kai g mporeÐ na diafèroun se èna akrib¸c shmeÐo). H d eÐnai metrik sto C([α, β]), afoÔ ∀f, g, h ∈ C([α, β]) ³Z β ´ p1 p (a) d(f, g) = 0 ⇐⇒ |f (x) − g(x)| dx = 0 ⇐⇒ f = g. α

(b) d(f, g) = d(g, f ), profan¸c.

³Z

´ p1

³Z

´ p1

∗∗

≤

Z [∗ : IsqÔei: 'Estw f ∈ R([α, β]). Tìte, (f ≥ 0, f suneq c) =⇒ [∗∗ : Anisìthta M inkowski]. OmoÐwc kai gia tic τ, σ.

β α

f (x)dx ≥ 0].

LÔsh 'Askhshc 7 Katarq n, h d eÐnai kal orismènh afoÔ an x = (xn ), y =

(yn ) ∈ X , tìte, ∞ X d(x, y) = βn · n=1

∞

X |xn − yn | ≤ βn < ∞ (diìti 1 + |xn − yn | n=1

|xn −yn | 1+|xn −yn |

≤ 1). (∗)

Elègqoume mìno thn trigwnik idiìthta thc metrik c: an x = (xn ), y = (yn ), z = (zn ) ∈ X , tìte, ∀k ∈ N, èqoume: t

|xn − yn | τ %ιγωνικη, 1+t % |xn − zn | + |zn − yn | ≤ = 1 + |xn − yn | 1 + |xn − zn | + |zn − yn | |xn − zn | |zn − yn | |xn − zn | |zn − yn | + ≤ + 1 + |xn − zn | + |zn − yn | 1 + |xn − zn | + |zn − yn | 1 + |xn − zn | 1 + |zn − yn | ∞ ∞ ∞ X X X |xn − yn | |xn − zn | |zn − yn | ⇒ d(x, y) = βn · ≤ βn · + βn · 1 + |xn − yn | n=1 1 + |xn − zn | n=1 1 + |zn − yn | n=1 = d(x, z) + d(z, y). ∞ X Gia th di metro, h (∗) mac dÐnei ìti δ(X) := sup{d(x, y)|x, y ∈ X} ≤ βn . n=1

T¸ra, gia xM = (M, M, . . . , M, . . .), M > 0 kai y = (0, 0, . . . , 0, . . .), èqoume: ∞ ∞ X M M X δ(X) ≥ d(xM , y) = βn = βn 1 + M 1 + M n=1 n=1 M kai afoÔ % 1, ìtan M > 0, paÐrnoume 1+M ∞ ∞ X X M δ(X) ≥ sup{d(xM , y) : M > 0} = (sup{ : M > 0}) βn = βn . 1+M n=1 n=1 ∞ X 'Ara, δ(X) = βn . n=1

LÔsh 'Askhshc 8 A 6= ∅, giatÐ 1 ∈ A. Ja deÐxoume ìti an a ∈ A kai 0 < b < a, tìte b ∈ A kai ra to A ja eÐnai anoiktì upodi sthma tou (0, ∞). Pr gmati, apì ton orismì tou A, èqoume ìti ρa metrik kai epeid 0 < b < a, 0 < b b < 1, apì thn skhsh 2, h (ρa ) a eÐnai metrik , dhl. ρb metrik , dhl. b ∈ A. a LÔsh 'Askhshc 9 Sρ ((0, 0), r) = {x ∈ R2 : ρ(x, (0, 0)) < r} = {x ∈ R2 : ∗

max{|x1 |, |x2 |} < r, x = (x1 , x2 )} = (−r, r) × (−r, r) Ã orjog¸nio [∗ Ã |x1 | < r =⇒ −r < x1 < r kai |x2 | < r =⇒ −r < x2 < r].

Sρ1 ((0, 0), r) = {x ∈ R2 : |x1 | + |x2 |} < r} Ã rìmboc Sρ2 ((0, 0), r) = {x ∈ R2 :

|x1 |2 + |x2 |2 } ≤ r} Ã dÐskoc

LÔsh 'Askhshc 10 Gia k je x ∈ X, {x} =

e r) (afoÔ y ∈ S(x,

r>0

e r) ⇐⇒ S(x,

r>0

e r), ∀r > 0. 'Ara, y ∈ S(x, \ e r) ⇐⇒ ρ(x, y) ≤ r, ∀r > 0 ⇐⇒ ρ(x, y) = 0 ⇐⇒ x = y ⇐⇒ {x} = S(x, r>0

e r) eÐnai ky ∈ {x}). 'Ara, to {x} eÐnai tom kleist¸n sunìlwn (afoÔ ta S(x, leist sÔnola ∀r > 0) =⇒ kleistì. T¸ra, an n ∈ N kai x1 , x2 , . . . , xn ∈ X, n [ tìte {x1 , x2 , . . . , xn } = {xi } peperasmènh ènwsh kleist¸n, ra apì prìtash, i=1

kleistì.

πρoτ αση

e ρ), S(x, e ρ) kleistì =⇒ S(x, ρ) ⊆ S(x, e ρ). LÔsh 'Askhshc 11 S(x, ρ) ⊆ S(x, Den isqÔei genik h isìthta, p.q. jewroÔme to diakritì metrikì q¸ro (X, ρδ ) me 2 toul qiston shmeÐa. Tìte, ∀x ∈ X, 10 e 1) = X =⇒ S(x, 1) = {x} = e 1). S(x, 1) = {x}, S(x, {x} X = S(x,

LÔsh 'Askhshc 12 (i) A ⊆ A ⇒ δ(A) ≤ δ(A) (1). Gia to δ(A) ≥ δ(A), èstw ² > 0, x, y ∈ A. Tìte,

S(x, ²) ∩ A 6= ∅ kai S(y, ²) ∩ A 6= ∅. 'Ara, ∃x0 ∈ S(x, ²) ∩ A 6= ∅ kai y 0 ∈ S(y, ²) ∩ A x0 ∈ S(x, ²), x0 ∈ A kai y 0 ∈ S(y, ²), y 0 ∈ A ρ(x, x0 ) < ² kai ρ(y, y 0 ) < ². 'Ara, ρ(x, y) ≤ ρ(x, x0 ) + ρ(x0 , y 0 ) + ρ(y, y 0 ) ≤ ² + ρ(x0 , y 0 ) + ² kai afoÔ x0 , y 0 ∈ A ⇒ ρ(x, y) ≤ 2² + δ(A). PaÐrnontac sup ⇒ δ(A) ≤ 2² + δ(A), ∀² > 0. 'Ara, x,y∈A

δ(A) ≥ δ(A). (2) Apì (1), (2) ⇒ δ(A) = δ(A).

(ii) 'Estw ² > 0. IsqÔei d(x, y) < d(x, A) + ², ∀y ∈ A ⊆ B diìti an d(x, y) ≥ d(x, A) + ², tìte, inf d(x, y) = d(x, A) ≥ d(x, A) + ² > d(x, A), topo. y∈A

d(x, B) =

inf d(x, y) < d(x, y) < d(x, A) + ², dhl. d(x, B) < d(x, A) + ²,

y∈A⊆B

∀² > 0 ⇒ d(x, B) ≤ d(x, A), ∀x ∈ X . (iii) Sth (ii) an B = A, paÐrnoume d(x, A) ≤ d(x, A), ∀x ∈ X. Ja deÐxoume kai to antÐstrofo, dhl. ìti d(x, A) ≥ d(x, A), ∀x ∈ X. 'Estw ² > 0. IsqÔei d(x, y) < d(x, A) + ², ∀y ∈ A diìti an d(x, y) ≥ d(x, A) + ², tìte, inf d(x, y) > d(x, A) + ² > d(x, A), topo. 'Ara, d(x, y) < d(x, A) + ², y∈A

∀² > 0 ⇒ d(x, A) ≥ d(x, A), ∀x ∈ X. (iv) A ⊆ A, B ⊆ B ⇒ d(A, B) ≤ d(A, B). Gia to d(A, B) ≥ d(A, B), èstw ² > 0 kai x ∈ A, y ∈ B. 'Eqoume S(x, ²) ∩ A 6= ∅ kai S(y, ²) ∩ B 6= ∅. Opìtan jewroÔme x0 ∈ S(x, ²) ∩ A ⇒ ρ(x, x0 ) < ², x0 ∈ A kai y 0 ∈ S(y, ²) ∩ B ⇒ ρ(y, y 0 ) < ², y 0 ∈ B. Epomènwc, ρ(x, y 0 ) ≤ ρ(x0 , x) + ρ(x, y) + ρ(y, y 0 ) ≤ ρ(x, y) + 2². PaÐrnontac inf ⇒ d(A, B) ≤ ρ(x, y) + 2² ≤ d(A, B) + 2², ∀² > 0. 'Ara, x0 ∈A,y 0 ∈B

d(A, B) ≥ d(A, B).

LÔsh 'Askhshc 13 (iii) jètoume φB : X → R me φB (x) = d(x, B).

φB suneq c, A sumpagèc =⇒ ('Askhsh 34) h φB |A paÐrnei el qisth tim sto A =⇒ ∃a ∈ A : φB |A (a) = d(a, B) ≤ φB (x) = d(x, B), ∀x ∈ A =⇒ φB |A (a) =

d(a, B) ≤ inf φB (x) = inf d(x, B) = inf inf d(x, y) = d(A, B). x∈A

x∈A

x∈A y∈B

'Ara, d(a, B) ≤ d(A, B). EpÐshc, isqÔei profan¸c ìti d(a, B) ≥ d(A, B). 'Ara, ∃a ∈ A : d(a, B) = d(A, B).

LÔsh 'Askhshc 14 (i) Ja deÐxoume ìti ρ2 ≤ ρ3 : ∀f, g ∈ C([0, Z 1]),

∗

³Z

|f (x)−g(x)|2 dx

´ 12

ρ21 (f, g), (afoÔ ρ21 (f, g) ≥ 0), dhl. ρ3 ≤ ρ1 . Epomènwc ρ2 ≤ ρ3 ≤ ρ1 . Ja deÐxoume ìti ρ1 ρ2 kai ρ1 ρ3 'Estw (fn ) akoloujÐa sunart sewn sto q¸ro C([0, 1]), n ∈ N, me  an x = 0  1 1 − nx an x ∈ (0, n1 ] fn (x) =  0 an x ∈ ( n1 , 1] ρ1

Tìte, ρ1 (fn , 0) = sup{|fn (x)| : x ∈ [0, 1]} = 1 9 0. 'Ara, fn 9 0. 'Omwc,

³Z

ρ3 (fn , 0) = 0

fn2 (x)dx

´ 12

fn2 ≤fn

³Z

≤

fn (x)dx

´ 12

1 1 ) 2 → 0. 2n

ρ2

EpÐshc, ρ2 (fn , 0) ≤ ρ3 (fn , 0), ra kai fn → 0. ρ2

ρ3

ρ1

Epomènwc, fn → 0 kai fn → 0, all , fn 9 0. Sunep¸c, ρ1 ρ2 kai ρ1 ρ3 . Ja deÐxoume ìti ρ2 ρ3 'Estw (fn ) akoloujÐa sunart sewn sto q¸ro C([0, 1]), n ∈ N, me  an x ∈ [0, n12 ]  n 3 2n − n x an x ∈ ( n12 , n22 ] fn (x) =  0 an x ∈ ( n22 , 1] Tìte, ρ3 (fn , 0) =

³Z

1 9 0. ρ3 'Ara, fn 9 0. 'Omwc, ρ2 (fn , 0) =

1 0

fn2 (x)dx

´ 12

³Z ≥ 0

1 n2

fn (x)dx = 0

1 1 ρ2 + → 0, dhl. fn → 0. n 2n

1 n2

fn2 (x)dx

´ 12

Z fn (x)dx +

³Z

1 n2

n2 dx

2 n2 1 n2

fn (x)dx =

´ 12

³ 1 ´ 21 n2 = 2 n

n 1 + 2n = 2 n 2n

Sunep¸c, ρ2 ρ3 .

LÔsh 'Askhshc 15 (=⇒) AfoÔ A sumpagèc, apì thn 'Askhsh 13 (iii), ja

up rqei a ∈ A tètoio ¸ste d(A, B) = d(a, B) ≥ 0. B kleistì =⇒ B = B. Epomènwc, afoÔ / B = {x ∈ X : d(x, B) = 0} =⇒ d(A, B) = A ∩ B 6= ∅ =⇒ A ∩ B = ∅. 'Ara, a ∈ d(a, B) > 0.

LÔsh 'Askhshc 16 (=⇒) 'Estw ìti d(A, B) = 0. A sumpagèc =⇒ A kleistì =⇒ A = A = {x ∈ X : d(x, A) = 0}. 'Ara, x ∈ A ⇔ x ∈ A ⇔ d(x, A) = 0 = d(A, B) = d(A, B) ⇔ A = B ⇔ x ∈ B. 'Ara, x ∈ A, x ∈ B, dhl x ∈ A ∩ B =⇒ A ∩ B = 6 ∅.

(⇐=) 'Estw A ∩ B 6= ∅ =⇒ ∃a ∈ A, a ∈ B = {x ∈ X : d(x, B) = 0} =⇒ d(a, B) = 0 =⇒ d(A, B) = 0. T¸ra, an X = R2 , A := {(x, 0) : x ∈ R}, dhl. A = o xonac twn x,kai B := {(x, x1 ) : x > 0}, tìte, A,B kleist sto X = R2 (qarakthrismìc kleist¸n me akoloujÐec) kai A ∩ B 6= ∅. 'Omwc, d(A, B) = 0, afoÔ 0 ≤ d(A, B) ≤ x→+∞ ρ((x, 0), (x, x1 )) = x1 −→ 0, ìpou ρ = h EukleÐdeia h max metrik .

LÔsh 'Askhshc 17 Apì prìtash, afoÔ D ⊆ X puknì kai A ⊆ X anoiktì,

èpetai ìti D ∩ A puknì sto A, dhl D ∩ A ⊇ A. T¸ra, afoÔ D ∩ A kleistì, èpetai ìti D ∩ A ⊇ A. EpÐshc, D ∩ A ⊆ A =⇒ D ∩ A ⊆ A. Sunep¸c, D ∩ A = A.

LÔsh 'Askhshc 18 (⇒) 'Estw ìti D puknì sto X kai èstw ∅ 6= A ⊂ X anoik-

tì. A 6= ∅ ⇒ ∃x ∈ A kai afou A anoiktì ⇒ ∃r > 0 tètoio ¸ste S(x, r) ⊂ A. (1) D puknì sto X ⇒ D ∩ S(x, r) 6= ∅. (2) Apì (1), (2) ⇒ D ∩ A 6= ∅. (⇐) Profan¸c, afoÔ an x ∈ X, r > 0, h anoikt sfaÐra S(x, r) eÐnai èna par deigma anoiktoÔ sunìlou gia to opoÐo isqÔei D ∩ S(x, r) 6= ∅, dhl. D puknì sto X .

LÔsh 'Askhshc 19 (i) A ⊆ A ∪ B =⇒ A ⊆ A ∪ B kai

B ⊆ A ∪ B =⇒ B ⊆ A ∪ B. =⇒ A ∪ B ⊆ A ∪ B. Gia to A ∪ B ⊇ A ∪ B, A ⊆ A, B ⊆ B =⇒ A ∪ B ⊆ A ∪ B, to opoÐo eÐnai kleistì sÔnolo wc ènwsh kleist¸n. 'Omwc, to A ∪ B eÐnai to mikrìtero kleistì sÔnolo pou perièqei to A ∪ B kai afoÔ A ∪ B ⊆ A ∪ B =⇒ A ∪ B ⊆ A ∪ B ⊆ A ∪ B. 'Ara, A ∪ B = A ∪ B. (ii) Den isqÔei, p.q. an X = R, A = Q, B = R \ Q, tìte, A ∩ B = ∅ = ∅, en¸ A ∩ B = Q ∩ R \ Q = R ∩ (R \ Qo ) = R ∩ R = R. (iii) A ∩ B ⊆ A =⇒ (A ∩ B)o ⊆ Ao

A ∩ B ⊆ B =⇒ (A ∩ B)o ⊆ B o =⇒ (A ∩ B)o ⊆ Ao ∩ B o . Gia to (A ∩ B)o ⊇ Ao ∩ B o , Ao ∩ B o ⊆ A ∩ B kai Ao ∩ B o anoiktì wc tom anoikt¸n. 'Omwc to (A ∩ B)o eÐnai to megalÔtero anoiktì pou perièqetai sto A ∩ B , ra, Ao ∩ B o ⊆ (A ∩ B)o . Epomènwc, (A ∩ B)o = Ao ∩ B o . (iv) Den isqÔei, p.q. an X = R, A = Q,B = R \ Q, tìte, (A ∪ B)o = (Q ∪ (R \ Q)) = Ro = R (afoÔ R anoiktì) kai Ao ∪ B o = Qo ∪ (R \ Q)o = ∅ ∪ (R \ Q) = ∅ ∪ (R \ R) = ∅ ∪ ∅ = ∅. (v) 'Estw (x, y) ∈ A × B. Tìte, up rqei akoloujÐa (xn , yn ) sto A × B me (xn , yn ) → (x, y), dhl. xn ∈ A, yn ∈ B, ∀n ∈ N. Epomènwc, xn → x kai yn → y . 'Ara, x ∈ A kai y ∈ B =⇒ (x, y) ∈ A × B. Sunep¸c, A × B ⊆ A × B. OmoÐwc, deÐqnoume ìti A × B ⊆ A × B. 'Ara, A × B = A × B. (vi) Ao × B o anoiktì (wc ginìmeno anoikt¸n), Ao × B o ⊆ A × B =⇒ Ao × B o ⊆ (A × B)o . Gia to (A × B)o ⊆ Ao × B o , èstw (x, y) ∈ (A × B)o =⇒ ∃r > 0 : S((x, y), r) ⊆ A × B, dhl. S(x, r) × S(y, r) ⊆ A × B. 'Ara, S(x, r) ⊆ A, S(y, r) ⊆ B, dhl. x ∈ Ao , y ∈ B o , dhl. (x, y) ∈ Ao × B o . Epomènwc, (A × B)o ⊆ Ao × B o . Sunep¸c, (A × B)o = Ao × B o .

LÔsh 'Askhshc 20 AfoÔ φ 1-1, epÐ, gia na deÐxoume ìti φ omoiomorfismìc,

arkeÐ na deÐxoume ìti φ, φ−1 suneqeÐc. 'Omwc, φ suneq c ⇔ k je anoiktì sÔnolo sto (X, ρ2 ) eÐnai anoiktì sÔnolo sto (X, ρ1 ) kai φ−1 suneq c ⇔ k je anoiktì sÔnolo sto (X, ρ1 ) eÐnai anoiktì sÔnolo sto (X, ρ2 ). Sunep¸c φ omoiomorfismìc ⇔ (X, ρ1 ) kai (X, ρ2 ) èqoun ta Ðdia anoikt sÔnola ⇔ ρ1 ∼ ρ2 . ρ

LÔsh 'Askhshc 21 xn → x ⇒ ρ(xn , x) → 0 kai xn → y ⇒ ρ(xn , y) → 0.

'Ara, 0 ≤ ρ(x, y) ≤ ρ(xn , x) + ρ(xn , y) → 0 ⇒ ρ(x, y) = 0 ⇒ x = y.

LÔsh 'Askhshc 22 IsqÔei |ρ(x, z) − ρ(y, z)| ≤ ρ(x, y), ∀x, y, z ∈ X (∗). Pr gmati,

ρ(x, z) ≤ ρ(x, y) + ρ(y, z) ⇒ ρ(x, z) − ρ(y, z) ≤ ρ(y, z),

ρ(y, z) ≤ ρ(y, z) + ρ(x, z) ⇒ ρ(y, z) − ρ(x, z) ≤ ρ(y, x). Epomènwc, (∗)

LÔsh 'Askhshc 23 (⇒) AfoÔ x ∈ A èqoume S(x, n1 ) ∩ A 6= ∅, ∀n ∈ N.

Epilègoume xn ∈ S(x, n1 ) ∩ A, gia n ∈ N. OrÐzetai ètsi h akoloujÐa (xn ) sto X me xn ∈ A, ∀n ∈ N kai ρ(xn , x) < n1 , ∀n ∈ N ρ kai ra ρ(xn , x) → 0, dhl. xn → x. ρ

(⇐) 'Estw (xn ) akoloujÐa me xn ∈ A, ∀n ∈ N kai xn → x. 'Estw r > 0. ρ AfoÔ xn → x, ∃n0 ∈ N ¸ste xn ∈ S(x, r), ∀n ≥ n0 . Eidik , xn0 ∈ A ∩ S(x, r). Epomènwc, A ∩ S(x, r) 6= ∅, ∀r > 0. 'Epetai ìti x ∈ A. e o , ²), to opoÐo eÐnai kleistì LÔsh 'Askhshc 24 {x ∈ X|ρ(x, xo ) ≤ ²} = S(x sÔnolo (wc kleist sfaÐra). 'Omwc, to S(xo , ²) eÐnai to mikrìtero dunatì sÔnolo pou perièqei thn S(xo , ²) (apì prìtash). 'Ara,

e o , ²) = {x ∈ X|ρ(x, xo ) ≤ ²}. S(xo , ²) ⊆ S(x AkoloÔjwc, jewroÔme to metrikì q¸ro (X, ρδ ), ìpou ρδ h diakrit metrik kai o X na apoteleÐtai apì dÔo toul qiston shmeÐa. Gia ² = 1,

e o , 1) = X. {x ∈ X|ρδ (x, xo ) ≤ 1} = S(x EpÐshc, S(xo , 1) = {xo } = {xo } (afoÔ {xo } kleistì sÔnolo). 'Ara, (afoÔ o X apoteleÐtai apì dÔo toul qiston shmeÐa),

e o , 1) = X. S(xo , 1) = {xo } ⊂ S(x

LÔsh 'Askhshc 25 (i) JewroÔme th sun rthsh f : R → (−1, 1) me f (x) =

x . kai A = R. Tìte, h f eÐnai suneq c (wc sÔnjesh suneq¸n) kai to A = R 1 + |x| kleistì sto R, en¸ to f (A) = (−1, 1) ìqi anoiktì sto R. (ii) JewroÔme th sun rthsh f : R → R me f (x) = x2 . kai A = (−1, 1). Tìte, h f eÐnai suneq c kai to A = (−1, 1) anoiktì sto R, en¸ to f (A) = [0, 1) ìqi anoiktì sto R.

LÔsh 'Askhshc 26 (i) 'Estw (xn , yn ) akoloujÐa sto X × X kai x, y ∈ X × X πρoτ αση

¸ste (xn , yn ) → (x, y) =⇒ xn → x kai yn → y. Apì 'Askhsh 21, ρ(xn , yn ) → ρ(x, y). Epomènwc, ρ suneq c. ρ

(ii) 'Estw (xn ) akoloujÐa sto X kai x ∈ X ¸ste xn → x, Tìte gia yn = x0 stajerì, n ∈ N πρoτ αση

=⇒ ρ(xn , x0 = yn ) → ρ(x, x0 ), dhl. f (xn ) → f (x0 ), ra f suneq c.

(iii) IsqÔei |d(x, A) − d(y, A)| ≤ ρ(x, y). Pr gmati, ∀x ∈ A, ρ(x, y) ≤ ρ(z, y) + ρ(x, y). Epomènwc, (paÐrnontac inf ) d(x, A) ≤ d(y, A) + ρ(x, y) ⇒ d(x, A) − d(y, A) ≤ ρ(x, y). z∈A

LÔsh 'Askhshc 27 Gia k je x ∈ X èqoume d(x, A) − d(x, B) > 0, giatÐ an d(x, A) − d(x, B) = 0, tìte d(x, A) = d(x, B) = 0 kai afoÔ ta A, B eÐnai kleist (dhl. A = A, B = B ), apì thn 'Askhsh 25, ja eÐqame ìti x ∈ A ∩ B, topo, afoÔ apì upìjesh, A ∩ B = ∅. Sunep¸c, o paronomast c sthn f eÐnai > 0, ra h f eÐnai kal orismènh. EpÐshc, h f eÐnai suneq c wc sÔnjesh suneq¸n, afoÔ apì thn 'Askhsh 26(iii) oi sunart seic d(·, A) kai d(·, B) eÐnai suneqeÐc. T¸ra an a ∈ A, tìte d(a, A) f (a) = = 0, d(a, A) + d(a, B) en¸ an b ∈ B, tìte

f (b) =

d(b, A) = 1. d(b, A) + d(b, B)

LÔsh 'Askhshc 28 (i) =⇒ (ii) 'Estw ìti f suneq c. 'Estw G anoiktì sto X

kai x ∈ f −1 (G). Tìte, afoÔ x ∈ f −1 (G) =⇒ f (x) ∈ G kai afoÔ G anoiktì sto X, èpetai ìti up rqei r > 0 tètoio ¸ste S(f (x), r) ⊆ G AfoÔ f suneq c sto x, ja up rqei δ > 0 tètoio ¸ste f (S(x, δ)) ⊆ S(f (x), r). 'Ara, f (S(x, δ)) ⊆ G, dhl. S(x, δ) ⊆ f −1 (G). Epomènwc, f −1 (G) anoiktì.

(ii) =⇒ (iii) 'Estw F ⊆ Y kleistì. Tìte, Y \ F ⊆ Y anoiktì. Tìte, apì thn upìjesh, to f −1 (Y \ F) anoiktì sto X, dhl. X \ f −1 (F) anoiktì, dhl. f −1 (F) anoiktì sto X. (iii) =⇒ (iv) 'Estw B ⊆ Y. Tìte, B kleistì ston Y kai ra apì thn upìjesh, f −1 (B) kleistì sto X. AfoÔ f −1 (B) ⊆ f −1 (B), èpetai ìti f −1 (B) ⊆ f −1 (B), diìti f −1 (B) to mikrìtero kleistì sÔnolo pou perièqei to f −1 (B).

(iv) =⇒ (v) 'Estw A ⊆ X. Gia B = f (A), apì thn upìjesh èqoume ìti f −1 (f (A)) ⊆ f −1 (f (A)). 'Omwc, A ⊆ f −1 (f (A)). Epomènwc, A ⊆ f −1 (f (A)), dhl. f (A) ⊆ f (A). (v) =⇒ (i) 'Estw x ∈ X kai r > 0. ArkeÐ na deiqjeÐ ìti x ∈ f −1 (S(f (x), r))◦ , (diìti tìte, ja up rqei δ > 0 tètoio ¸ste S(x, δ) ⊆ f −1 (S(f (x), r)) kai ra, f (S(x, δ)) ∈ S(f (x), r), dhl. f suneq c sto x). (v)

Pr gmati, X\f −1 (S(f (x), r))◦ = f (X \ f (S(f (x), r))) ⊆ f (X \ f −1 (S(f (x), r))) = (∗)

(∗∗)

f (f −1 (Y \ S(f (x), r))) ⊆ Y \ S(f (x), r)) ⊆ Y \ S(f (x), r)), ìpou (∗) : f (f −1 (B)) ⊆ B =⇒ f (f −1 (B)) ⊆ B (∗∗) : Y \ S(f (x), r)) kleistì. T¸ra, afoÔ f (x) ∈ S(f (x), r)) =⇒ f (x) ∈ / Y \ S(f (x), r)) =⇒ f (x) ∈ / f (X \ (S(f (x), r)))◦ ), ra, x ∈ / X \ f −1 (S(f (x), r)))◦ ), epomènwc, x ∈ f −1 (S(f (x), r)))◦ ) =⇒ S(x, δ) ⊆ f −1 (S(f (x), r)) =⇒ f (S(x, δ)) ∈ S(f (x), r).

LÔsh 'Askhshc 29 (i) 'Estw (xn ) akoloujÐa sto F , x ∈ X me xn → x. Ja deÐxoume ìti x ∈ F . Pr gmati, afoÔ xn ∈ F , ∀n ∈ N ⇒ f (xn ) = g(xn ), ∀n ∈ N. T¸ra, afoÔ f kai g suneqeÐc kai xn → x, èpetai ìti f (xn ) → f (x) kai g(xn ) → g(x). Apì th monadikìthta tou orÐou ('Askhsh 20), èpetai ìti f (x) = g(x). (ii) D ⊆ F , afoÔ f |D = g|D ⇒ D ⊆ F . To D eÐnai puknì, apì upìjesh, dhl. D = X . Apì to (i), F kleistì sto X , dhl. F = F . Apì Prìtash, F = X , ra f = g . (iii) 'Estw (xn , f (xn )) akoloujÐa sto X × Y me (xn , f (xn )) ∈ Gf = {(x, f (x)) : x ∈ X} ⊆ X × Y , ∀n ∈ N kai (x, y) ∈ X × Y tètoia ¸ste (xn , f (xn )) → (x, y). Ja deÐxoume ìti (x, y) ∈ Gf . Pr gmati, foÔ (xn , f (xn )) → (x, y) ⇒ xn → x kai f (xn )) → y . 'Omwc, f suneq c ⇒ f (xn ) → f (x). Apì th monadikìthta tou orÐou, f (x) = y ⇒ (x, y) = (x, f (x)) ∈ Gf .

LÔsh 'Askhshc 30 (i) {x ∈ X|f (x) ≥ a} = f −1 ([a, ∞)). f suneq c, [a, ∞), kleistì , ra f −1 ([a, ∞)) kleistì. 'Omoia kai gia ìla ta lla sÔnola.

LÔsh 'Askhshc 31 'Estw F = {x ∈ X : f (x) = g(x)}. Tìte, to F eÐnai

kleistì sto X : Pr gmati, èstw (xn ) mia akoloujÐa sto X me xn ∈ F , ∀n ∈ N tètoia ¸ste xn −→ x gia k poio x ∈ X. AfoÔ xn ∈ F , ∀n ∈ N èpetai ìti f (xn ) = g(xn ), ∀n ∈ N.

f suneq c kai xn −→ x =⇒ f (xn ) −→ f (x). g suneq c kai xn −→ x =⇒ g(xn ) −→ g(x). Apì monadikìthta tou orÐou, f (x) = g(x), dhl. x ∈ F . T¸ra, afoÔ D ⊆ F, kai F kleistì sto X, apì prìtash (to D eÐnai to mikrìtero kleistì pou perièqei to D), èpetai ìti D ⊆ D ⊆ F, 'Omwc D = X, ra F = X, epomènwc, f (x) = g(x), ∀x ∈ X.

LÔsh 'Askhshc 32 (=⇒) 'Estw (xn´) akoloujÐa sto X me xn → a kai ² > 0. ³

f suneq c sto a =⇒ ∃δ > 0 : f S(a, δ) ⊆ S(f (a), ²). xn → a =⇒ ∃n0 ∈ N : xn ∈ S(a, δ), ∀n ≥ n0 . 'Epetai oti f (xn ) ∈ f (S(a, δ)) ⊆ S(f (a), ²), ∀n ≥ n0 . Epomènwc f (x) → f (a).

(⇐=) Upojètoume ìti h f den eÐnai suneq c sto a, opìtan ³ ´ ∃² > 0 : ∀δ > 0, f S(a, δ) * S(f (a), ²). Gia δ = n1 , n ∈ N. ja èqoume ìti

³ 1 ´ ∃² > 0 : ∀δ > 0, f S(a, ) * S(f (a), ²), ∀n ∈ N. n ³ ´ Epomènwc, ∀n ∈ N up rqei xn ∈ S(a, n1 ) : f (xn ) ∈ / f S(a, δ) . gia thn akoloujÐa (xn ) sto X èqoume ρx (xn , a) < n1 kai ra, ρx (xn , a) < n1 → 0, dhl. xn → a, en¸ ρy (f (xn ), f (a)) ≥ ², ∀n ∈ N. Epomènwc, f (xn ) 9 f (a), topo.

LÔsh 'Askhshc 33 'Estw B èna anoiktì k lumma tou f (K). Ja deÐxoume ìti

autì èqei peperasmèno upok lumma. S S B k lumma tou f (K) ⇒ f (K) ⊆ B ≡ {B : B ∈ B} ⇒ K ⊆ S anoiktì {f −1 (B) : B ∈ B}, ìpou f −1 (B) anoiktì ,∀B ∈ B (afoÔ B anoiktì ∀B ∈ B kai f suneq c). Sunep¸c, to {f −1 (B) : B ∈ B} eÐnai upok lumma tou K . 'Omwc o K eÐnai sumpag c, ra to upok lumma autì èqei peperasmèno upok lumma, dhl, ∃n ∈ N, n n [ [ −1 B1 , . . . , Bn ∈ B ¸ste K ⊆ f (Bi ) ⇒ f (K) ⊆ Bi , dhl. {B1 , . . . , Bn } eÐnai i=1

i=1

peperasmèno upok lumma tou f (K). Epomènwc f (K) sumpag c.

LÔsh 'Askhshc 34 AfoÔ f : X → R suneq c kai o X sumpag c, apì thn π%oτ αση

skhsh 33. èqoume ìti to eÐnai sumpagèc sto R =⇒ kleistì kai fragmèno. AfoÔ f (X) 6= ∅ kai fragmèno ⇒ ∃ sup f (x) := s ∈ R kaj¸c kai to inf f (x) := i ∈ R. Ta s, i ∈ f (X) (wc ìria akolouji¸n me ìrouc apto f (X)) kai epeid f (X) = f (X) (dhl. f (X) kleistì), èpetai ìti s, i ∈ f (X). 'Ara, up rqoun x1 , x2 ∈ X tètoia ¸ste i = f (x1 ) kai s = f (x2 ). Tìte, ∀x ∈ X , f (x1 ) ≤ f (x) ≤ f (x2 ).

LÔsh 'Askhshc 35 'Estw h : X → R me h(x) = ρ(x, f (x)). H h eÐnai suneq c,

to X sumpagèc, ra paÐrnei el qisth (kai mègisth) tim ('Askhsh 34), dhl. ∃x0 ∈ X ¸ste h(x0 ) ≤ h(x), ∀x ∈ X. EpÐshc, h(x) ≥ 0. An

h(x0 ) > 0 =⇒ h(x0 ) = ρ(x0 , f (x0 ))

υπoθ²ση

ρ(f (x0 ), f (f (x0 ))) = h(f (x0 )),

dhl. h(x0 ) > h(f (x0 )), topo, afoÔ to x0 eÐnai h el qisth tim thc h. 'Ara, h(x0 ) = 0 =⇒ ρ(x0 , f (x0 )) = 0 =⇒ x0 = f (x0 ). Monadikìthta 'Estw x0 , ∈ X me f (x0 ) = x0 kai f (y0 ) = y0 , x0 6= y0 . Tìte, ρ(x0 , y0 ) > ρ(f (x0 ), f (y0 )) = ρ(x0 , y0 ), topo.

LÔsh 'Askhshc 36 'Estw f : X −→ R h sun rthsh sthn opoÐa sugklÐnei

omoiìmorfa h akoloujÐa (fn ). Apì prìtash, afoÔ h (fn ) eÐnai akoloujÐa suneq¸n oµ. sunart sewn me fn −→ f, èpetai ìti h f eÐnai suneq c. T¸ra, ja epalhjeÔsoume thn isosunèqeia thc (fn ) : 'Estw loipìn x ∈ X kai ² > 0. f suneq c sto x ∈ X =⇒ ∃δ1 > 0 : |f (y) − f (x)| < 3² , ∀y ∈ S(x, δ) (∗) Lìgw thc omoiìmorfhc sÔgklishc,

² ∃n0 ∈ N : |fn (y) − f (x)| < , ∀n ≥ n0 , ∀y ∈ X(∗∗). 3 Tìte, ∀n ≥ n0 kai ∀y ∈ S(x, δ1 ) ja isqÔoun oi (∗), (∗∗) :

|fn (y)−fn (x)| = |fn (y)−f (y)+f (y)−fn (x)| ≤ |fn (y)−f (y)|+|f (y)−f (x)|+|f (x)−fn (x)| ² ² ² < + + = ². 3 3 3 Dhl. isqÔei h isosunèqeia gia ìlec tic sunart seic fn , n ≥ n0 . Gia ìlec tic upìloipec sunart seic fi , 1 ≤ i ≤ n0 − 1, èqoume ìti to sÔnolo {f1 , f2 , · · · , fn0 −1 } eÐnai èna peperasmèno uposÔnolo suneq¸n sunart sewn =⇒ isosuneqèc kai ra ∃δ2 > 0 : |fn (y) − fn (x)| < ², ∀y ∈ S(x, δ2 ), ∀1 ≤ n ≤ n0 − 1. PaÐrnontac wc δ := min{δ1 , δ2 } èqoume ìti: |fn (y) − fn (x)| < ², ∀y ∈ S(x, δ), ∀n ∈ N, dhl. (fn ) isosuneq c.

LÔsh 'Askhshc 37 (i) 'Estw x0 ∈ (0, 1) An h F = {fn : (0, 1) → R me

fn (x) = x1n , n ∈ N} tan isosuneq c sto x0 , tìte gia ² = 1 ja up rqe δ > 0 tètoio ¸ste |fn (x) − fn (x0 )| < ² = 1, ∀n ∈ N, ∀x ∈ (x0 − δ, x0 + δ),

1 1 − | < 1. 'Omwc, xn xn0 1 xn − xn |x − x0 | · |xn−1 + xn−2 + . . . + x0n−1 | 1 1 | n − n| = | n n0 | = = |x−x0 |·| + n n x x0 x x0 x x0 x · xn0 1 1 ... + n | ≥ n · |x − x0 |, ∀n ∈ N, dhl. > n, ∀n ∈ N, topo. x · x0 |x − x0 | dhl. |

(ii) 'Estw x ∈ [0, 1). Epilègoume a ¸ste x < a < 1 kai jewroÔme thn akoloujÐa gn := fn |[0,a] , n ∈ N. Tìte, (gn ) eÐnai mia akoloujÐa sto q¸ro C([0, a]) (afoÔ gn = fn |[0,a] = xn , x ∈ [0, a]) me tic idiìthtec • fjÐnousa • gn −→ 0 ≡ h ∈ C([0, a]) kat shmeÐo • To pedÐo orismoÔ thc gn eÐnai sumpagèc oµ.

'Ara apì to Je¸rhma Dini, h gn −→ 0 ≡ h. Epomènwc, apì thn 'Askhsh 36, h (gn ) eÐnai isosuneq c sto x. T¸ra, afoÔ x < a, èpetai ìti kai h (fn ) eÐnai isosuneq c sto x.

(iii) Gia x ∈ [0, 1), fn (x) = 0 → 0 = f (0). Gia x = 1,

fn (1) = 1 → 1 = f (1), ìpou

½ f (x) =

0 an 0 ≤ x < 1 1 an x = 1

Dhl. fn → f kat shmeÐo. An h (fn ) tan isosuneq c, apì thn 'Askhsh 36, ja oµ. oµ. èprepe fn → f. 'Omwc, kfn − f k∞ = 1 → 1 6= 0, dhl. fn 9 f. 'Ara, (fn ) ìqi isosuneq c

LÔsh 'Askhshc 38 'Estw x0 ∈ [0, 1] kai ² > 0. Gia δ = [0, 1] : |x − x0 | < δ =

² M

kai f ∈ F

|f (x) − f (x0 )| ≤ M · |x − x0 | < M ·

² M

èqoume ìti ∀x ∈

² = ². M

'Ara h oikogèneia F = {f : [0, 1] → R : f suneq c, |f (x) − f (y)| ≤ M |x − y|} eÐnai isosuneq c.

LÔsh 'Askhshc 39 (i) IsqÔei |fn (x)| =

x2 ≤ 1, ∀x ∈ [0, 1], ∀n ∈ N. x2 + (1 − nx)2

'Ara, h fn (x) eÐnai omoiìmorfa fragmènh.

(ii) Gia x = 0, fn (0) = 0 → 0 ≡ f. Gia x 6= 0, ,

x2 fn (x) = 2 → 0 ≡ f. x + (1 − nx)2

'Ara, fn → f ≡ 0 kat shmeÐo.

(iii) 'Estw (nk )k upakoloujÐa sto N. Tìte kfnk − 0k∞ = sup |fnk (x) − 0| ≥ fnk ( x∈[0,1]

( n1 )2 1 )= 1 2 k nk ( nk ) + (1 −

nk 2 ) nk

= 1.

oµ

'Ara, fnk 9 f ≡ 0. Tèloc, o ([0, 1], | · |) eÐnai sumpag c =⇒ diaqwrÐsimoc, h (fn ) eÐnai kat shmeÐo fragmènh all ìqi isosuneq c. Epomènwc, den efarmìzetai to Je¸rhma Ascoli.

LÔsh 'Askhshc 40 (=⇒) 'Estw ² > 0.

[

S(x, δx ) kai X sumpag c, èpetai ìti ∃n ∈ N kai x1 , x2 , . . . , xn

x∈X

X⊆

n [

S(x, δxi ).

i=1

AfoÔ fn → f kata shmeÐo, èpetai ìti

fn (xi ) → f (x), ∀i = 1, 2, . . . , k. Jètontac n0 = max{n1 , n2 , . . . , nk }, èqoume ìti

² ∃n0 ∈ N : |fn (xi ) − f (xi )| < , ∀n ≥ n0 . 3

25 Epomènwc, sunajroÐzontac ta pÐo p nw, èqoume ìti

∀n ≥ n0 , ∀x ∈ X∃i ∈ {1, 2, . . . , k} : x ∈ S(x, δxi ) kai ra

|fn (x) − f (x)| ≤ |fn (x) − fn (xi )| + |fn (x) − f (xi )| + |f (xi ) − f (x)| <

² ² ² + + = ², ∀n ≥ n0 , 3 3 3

oµ.

dhl. fn → f. oµ.

(⇐=) 'Estw f t.w. fn → f. fn suneq c, ∀n ∈ N =⇒ f suneq c . 'Estw x ∈ X, ² > 0.

f suneq c sto x ∈ X =⇒ ∃δ1 > 0 : |f (y) − f (x)| < 3² , ∀y ∈ S(x, δ1 ). (∗) oµ.

fn → f =⇒ ∃n0 ∈ N : |fn (y) − f (y)| < 3² , ∀y ∈ X, ∀n ≥ n0 . (∗∗) Gia na isqÔoun tautoqrìnwc oi sqèseic (∗), (∗∗) paÐrnoume ∀y ∈ X, ∀n ≥ n0 :

|fn (y) − fn (y)| ≤ |fn (y) − f (y)| + |f (y) − f (x)| + |f (x) − fn (x)| <

² ² ² + + = ².(1) 3 3 3

Gia to sÔnolo {f1 , f2 , . . . , fn0 −1 } èqoume ìti, afoÔ autì eÐnai peperasmèno sÔnolo suneq¸n sunart sewn, eÐnai isosuneqèc, ra

∃δ2 > 0 : |fn (y) − fn (x)| < ², ∀y ∈ S(x, δ2 ).(2) PaÐrnontac loipìn δ := min{δ1 , δ2 }, èqoume apì tic sqèseic (1), (2) ìti:

|fn (y) − fn (x)| < ², ∀y ∈ S(x, δ)∀n ≥ n0 , dhl. (fn ) isosuneq c.

LÔsh 'Askhshc 41 'Opwc thn prohgoÔmenh 'Askhsh. LÔsh 'Askhshc 42

• ArkeÐ na deÐxoume ìti ∀x ∈ X, h akoloujÐa (fn (x))n eÐnai basik akoloujÐa (sto R), diìti afoÔ o (R, | · |) eÐnai pl rhc, aut ja sugklÐnei. 'Estw x ∈ X, ² > 0. (fn )n isosuneq c sto x =⇒ ∃δx > 0 : |fn (x) − fn (y)| < 3² , ∀y ∈ S(x, δx ),

∀n ∈ N. D = X =⇒ D ∩ S(x, δx ) 6= ∅ =⇒ ∃y0 ∈ D ∩ S(x, δx ), dhl. y0 ∈ D. Tìte, apì upìjesh, h (fn (y0 ))n sugklÐnei, dhl. eÐnai basik =⇒ ∃n0 ∈ N : |fn (y0 ) − fm (y0 )| < 3² , ∀n, m ≥ n0 . 'Ara, ∀n, m ≥ n0 ,

• Ja deÐxoume ìti h (fn )n sugklÐnei omoiìmorfa sta sumpag uposÔnola tou X. 'Estw K ⊆ X sumpagèc kai ² > 0. Gia k je x ∈ K, h (fn (x))n eÐnai isosuneq c =⇒ ∃δx > 0 : |fn (x) − fn (y)| < 3² , ∀y ∈ S(x, δx ), ∀n ∈ N. Kaj¸c n → ∞, lìgw sunèqeiac thc | · |, |f (x) − f (y)| < 3² , ∀y ∈ S(x, δx ). T¸ra, afoÔ K ⊆

[

S(xi , δxi ) kai K sumpagèc, up rqei k ∈ N kai x1 , x2 , . . . , xk

x∈K

¸ste K ⊆

k [

S(xi , δxi ).

i=1

AfoÔ fn (xi ) → f (xi ), ∀1 ≤ i ≤ k, jètontac n0 := max{n1 , n2 , . . . , nk }, èqoume ìti |fn (xi ) − f (xi )| < 3² , ∀n ≥ n0 , ∀1 ≤ i ≤ k. 'Eqoume loipìn ìti ∀n ≥ n0 kai x ∈ K, ∃i ∈ {1, 2, . . . , k} ¸ste x ∈ S(xi , δxi ) kai epomènwc

|fn (x)−f (x)| ≤ |fn (x)−fn (xi )|+|fn (xi )−f (xi )|+|f (xi )−f (x)| < 3² + 3² + 3² = ², dhl. oµ. |fn (x) − f (x)| ≤ ², ∀n ≥ n0 kai x ∈ K =⇒ fn |K → f |K . Z 1 LÔsh 'Askhshc 43 AfoÔ xn f (x)dx = 0, ∀n ∈ N ∪ {0} kai epeid to olok0 Z 1 l rwma eÐnai grammikì, èpetai ìti p(x)f (x)dx = 0 (∗), gia k je polu¸numo 0

p. 'Estw ² > 0. Apì to Je¸rhma W eierstrass (P([0, 1]) = C([0, 1])), afoÔ f ∈ C([0, 1]), up rqei polu¸numo p tètoio ¸ste |f (x) − p(x)| < ², ∀x ∈ [0, 1] (∗∗). 'Eqoume loipìn: Z 1 Z 1 Z 1 (∗) 2 2 0 ≤ | f (x)dx| = | (f (x) − p(x)f (x))dx| ≤ |f 2 (x) − p(x)f (x)|dx = 0 0 0 Z 1 Z 1 (∗∗) |f 2 (x)||f (x) − p(x)|dx ≤ ² f (x)dx. 0 0 Z 1 'Ara, telik , f 2 (x)dx = 0. 'Omwc, afoÔ f 2 suneq c ⇒ f 2 ≡ 0 ⇒ f ≡ 0. 0

Gia to deÔtero er¸thma, efarmìzoume to prohgoÔmeno gia th sun rthsh f − g .

LÔsh 'Askhshc 44 'Estw (fn ) mia basik akoloujÐa sto q¸ro (B(X), ρ). AfoÔ |fn (x) − fm (x)| ≤ sup{|fn (x) − fm (x)| : x ∈ X} = ρ(fn , fm ), ∀x ∈ X, ∀n, m ∈ N, èpetai ìti h (fn (x)) eÐnai basik sto R, ∀x ∈ X. O R eÐnai pl rhc =⇒ ∃ lim fn (x) sto R, ∀x ∈ X. n

OrÐzoume loipìn

f : X → R me f (a) = lim fn (x), ∀x ∈ X. n

ArkeÐ na deÐxoume ìti f ∈ B(X) kai fn → f. 'Estw ² > 0. AfoÔ (fn ) basik , ∃n0 ∈ N : ρ(fn , fm ) < 2² , ∀n, m ≥ n0 . 'Ara, ² |fn (x) − fm (x)| < , ∀n, m ≥ n0 , ∀x ∈ X. 2 Kaj¸c m → ∞, apì th sunèqeia thc sun rthshc | · |, èqoume ìti

² |fn (x) − f (x)| ≤ , ∀n ≥ n0 , ∀x ∈ X. 2 PaÐrnontac sup, èpetai ìti ρ(fn , f ) ≤ x∈X

² 2

< ², ∀n ≥ n0 . Epomènwc, fn → f.

EpÐshc, gia n = n0 , h fn0 − fn eÐnai fragmènh, dhl. fn0 − fn ∈ B(X). AfoÔ f = fn0 − (fn0 − f ) diafor dÔo fragmènwn, èpetai ìti f ∈ B(X).

LÔsh 'Askhshc 45 IsqÔei ìti B(X)) ⊆ C(X) kai metrik tou C(X) eÐnai o

periorismìc thc metrik c tou B(X). Epomènwc, o C(X) eÐnai upìqwroc tou B(X). 'Omwc, afoÔ o (B(X), k · k∞ ≡ ρ) eÐnai pl rhc ('Askhsh 44), gia na deÐxoume ìti o (C(X), ρ|C(X) ) eÐnai pl rhc, arkeÐ na deÐxoume ìti autìc eÐnai kleistìc upìqwroc tou (B(X), ρ). ρ 'Estw loipìn (fn )n mia akoloujÐa ston (C(X), ρ|C(X) ) kai f ∈ B(X) ¸ste fn → f. Ja deÐxoume ìti f ∈ (C(X), ρ|C(X) ). 'Estw x0 ∈ X kai ² > 0.

² ρ fn → f =⇒ ∃n0 ∈ N : |fn (x) − f (x)| ≤ ρ(fn , f ) < , ∀n ≥ n0 , ∀x ∈ X. 3 fn0 suneq c sto x0 =⇒ ∃δ > 0 : |f (x) − fn0 (x0 )| < 3² , ∀x ∈ S(x0 , δ). 'Ara, ∀x ∈ S(x0 , δ),

² ² ² |f (x)−f (x0 )| < |f (x)−fn0 (x)|+|fn0 (x)+fn0 (x0 )|+|fn0 (x0 )−f (x0 )| < + + = ². 3 3 3 Epomènwc, f suneq c sto (tuqìn) x0 ∈ X, dhl. f ∈ (C(X), ρ|C(X) ).

LÔsh 'Askhshc 46 ((i) =⇒ (ii)) A sumpagèc sto (C([0, 1]), ρ) =⇒ kleistì

kai fragmèno.

A isosuneqèc: 'Estw x ∈ X kai ² > 0. AfoÔ to A eÐnai sumpagèc, tìte ja eÐnai kai olik fragmèno (apì prìtash) kai ra ja up rqei n ∈ N kai f1 , f2 , . . . , fn ∈ A ¸ste A⊆

³ ²´ S fi , . 3 i=1 n [

To sÔnolo {f1 , f2 , . . . , fn }, wc peperasmèno uposÔnolo tou C(X) eÐnai isosuneqèc sto X (diìti epalhjeÔetai h isosunèqeia gia δ := min δ(fi , ²)), epomènwc up rqei 1≤i≤n

δ > 0 ( koinì gia ìlec tic f ) tètoio ¸ste

² |fi (y) − fi (x)| < , ∀y ∈ S(x, δ) kai ∀i = 1, 2, . . . , n (∗). 3 Tìte, ∀y ∈ S(x, δ) kai ∀f ∈ A èqoume |f (y) − f (x)| < ² kai ra A isosuneqèc sto (∗)

x, ∀x ∈ X, afoÔ gia y ∈ S(x, δ), f ∈ A =⇒ ∃i ∈ {1, 2, . . . , n} : f ∈ S(fi , 3² ) ≡ Sρ (fi , 3² ) kai ra, |f (y) − f (x)| ≤ |f (y) − fi (y)| + |fi (y) − fi (x)| + |fi (x) − f (x)| <

² ² ² + + = ². 3 3 3

A kat shmeÐo fragmèno: Ja deÐxoume ìti to sÔnolo {f (x) : f ∈ A} ⊆ R eÐnai fragmèno, ∀x ∈ X.

29 Autì èpetai apì th sunèqeia thc sun rthshc ϕ : A → R me ϕ(f ) = ϕ(x). Pr gρA mati h ϕ eÐnai suneq c: 'Estw (fn ) mia akoloujÐa sto A kai f ∈ A me fn → f. øµ. κ.σ. Tìte, fn → f, epomènwc, fn → f kai eidik fn (x) → f (x), dhl. ϕ(fn ) → ϕ(f ). AfoÔ to A eÐnai sumpagèc kai h ϕ eÐnai suneq c, èpetai oti to ϕ(A) eÐnai fragmèno, dhl. {ϕ(f ) : f ∈ A} = {f (x) : f ∈ A} eÐnai fragmèno. Epomènwc, A kat shmeÐo fragmèno.

((ii) =⇒ (i))'Estw (fn ) mia akoloujÐa sto A. Tìte, h (fn ) eÐnai isosuneq c kai kat shmeÐo fragmènh. O X eÐnai diaqwrÐsimoc wc sumpag c. Epomènwc, apì to Je¸rhma Ascoli, up rqei upakoloujÐa (fkn ) thc (fn ) h opoÐa sugklÐnei kat shmeÐo se mia suneq sun rthsh f kai h sÔgklish eÐnai omoiìmorfh sta sumpag ρ uposÔnola. 'Ara, fkn → f omoiìmorfa (afoÔ X sumpag c), dhl. fkn → f. AfoÔ A kleistì, èqoume ìti f ∈ A. Epomènwc, A akoloujiak sumpagèc kai ra sumpagèc (Je¸rhma).

LÔsh 'Askhshc 47 1oc trìpoc:

Upojètoume ìti to B eÐnai sumpagèc. Tìte, apì thn prohgoÔmenh 'Askhsh (46), èqoume oti to B eÐnai isosuneqèc. Epomènwc, h akoloujÐa (fn ) ⊆ (C([0, 1]), k · k∞ ) me fn (x) = xn , ∀n ∈ N, x ∈ [0, 1] wc uposÔnolo tou B eÐnai isosuneq c, topo, afoÔ eÐdame ìti h akoloujÐa aut den eÐnai isosuneq c.

2oc trìpoc: 'Estw (fn ) ⊆ (C([0, 1]), k · k∞ ) me fn (x) = xn , ∀n ∈ N, x ∈ [0, 1] akoloujÐa. Upojètoume ìti to B eÐnai sumpagèc, ra akoloujiak sumpagèc. Epomènwc, h akoloujÐa (fn ) sto B èqei sugklÐnousa upakoloujÐa (fkn ) se mia f ∈ B, dhl. ρ oµ. κ.σ. fkn → f. 'Ara, fkn → f (Je¸rhma Dini) kai epomènwc fkn −→ f. All , h (fn ) sugklÐnei kat shmeÐo sth ϕ me   0 an 0 ≤ x < 1 ϕ(x) =  1 an x = 1 κ.σ.

Epomènwc, fkn −→ ϕ. 'Ara, f ≡ ϕ, topì, diìti f ∈ C([0, 1]) en¸ h ϕ den eÐnai suneq c (sto 1).

LÔsh 'Askhshc 48 'Estw f ∈ C([a, b]) kai ² > 0. Apì upìjesh, afoÔ A =

C([a, b]) =⇒ A ∩ Sρ (f, ²) 6= ∅. 'Estw loipìn p ∈ A ∩ Sρ (f, ²). Tìte, afoÔ p ∈ A, kai p ∈ Sρ (f, ²), èqoume ρ(p, f ) < ² =⇒ |p(x) − f (x)| < ², ∀x ∈ [a, b]. Antistrìfwc, upojètoume ìti isqÔei h sqèsh

∀f ∈ C([a, b]) kai ² > 0, ∃p ∈ A ≡ {p ∈ C([a, b]) : p polu¸numo } : |f (x)−p(x)| < ², ∀x ∈ [a, b].

30 PaÐrnoume loipìn èna p ∈ A ≡ {p ∈ C([a, b]) : p polu¸numo } tètoio ¸ste |p(x) − ² f (x)| < 2² , ∀x ∈ [a, b]. Tìte, ρ(f, p) := sup |p(x) − f (x)| ≤ < ², ra, p ∈ 2 x∈[a,b] Sρ (f, ²). Epomèwc, A ∩ Sρ (f, ²) 6= ∅, dhl. A = C([a, b]).

LÔsh 'Askhshc 49 'Estw x ∈ [0, 1] kai ² > 0. Jètoume Fn := {x1 , x2 , . . . , xn }, ∀n ∈ N. Tìte,

F1 ⊆ F2 ⊆ . . . ⊆ Fn ⊆ Fn+1 ⊆ . . . . 'Ara, fn (x) = d(x0 , Fn ) = inf ρ(x, y) =⇒ f1 (x) ≥ f2 (x) ≥ . . . ≥ fn (x) ≥ y∈Fn

fn+1 (x0 ) ≥ . . . , dhl. (fn (x))n & . D puknì sto [0, 1] =⇒ D = [0, 1] =⇒ S(x0 , ²) ∩ D 6= ∅ =⇒ ∃n0 ∈ N : xn0 ∈ S(x, ²), dhl. ρ(x, xn0 ) < ², dhl. |x − xn0 | < ². Tìte, fn0 (x) = d(x, {x1 , x2 , . . . , xn0 }) = inf ρ(x, y) ≤ ρ(x, xn0 ) < ² y∈Fn0

Epomènwc, ∀n ≥ n0 , 0 ≤ fn (x) ≤ fn0 (x) < ², afoÔ (fn (x))n &=⇒ fn (x) → 0, ∀x ∈ [0, 1], dhl. fn → 0 kata shmeÐo. AfoÔ fn suneq c, (fn )n fjÐnousa, [0, 1] oµ. sumpagèc kai fn → 0 kata shmeÐo, apì to Je¸rhma Dini èpetai ìti fn → 0.

LÔsh 'Askhshc 50 (i) 'Estw ² > 0. Ja epalhjeÔsoume ton orismì thc omoiì-

morfhc sunèqeiac, dhl. ja deÐxoume ìti ∃δ > 0 tètoio ¸ste an x, y ∈ X me ρX (x, y) < δ, tìte ρY (f (x), f (y)) < ². f suneq c sto x ∈ X =⇒ ∃δ ≡ δx > 0 ¸ste

f (S(x,

δx ² )) ⊆ S(f (x), ).(∗) 2 2

'Omwc,

n [

S(x,

i=1

δx ), 2

ra, to sÔnolo {S(x, δ2x ) : x ∈ X} apoteleÐ anoikt k luyh tou sumpagoÔc X , ra ∃n ∈ N kai x1 , x2 , . . . , xn ∈ X ¸ste

n [ i=1

Jètoume δ := min{ y qnoume!).

δx1 , . . . , δx2n }. 2

S(xi ,

δxi ). 2

(To δ autì eÐnai thc omoiìmorfhc sunèqeiac pou

'Estw loipìn x, y ∈ X me ρX (x, y) < δ. Tìte ∃i ∈ {1, 2, . . . , n} ¸ste x ∈

31 δ

S(xi , 2xi ) ⊆ S(xi , δxi ). EpÐshc, ρX (y, xi ) ≤ ρX (y, x) + ρX (x, xi ) < δ +

δxi 2

≤ δxi =⇒ x ∈ S(xi , δxi ).

Apì thn (∗) gia x = xi , paÐrnoume f (x), f (y) ∈ S(f (xi ), 2² ) =⇒ ρY (f (x), f (y)) ≤ ρY (f (x), f (xi )) + ρY (f (xi ), f (y)) < 2² + 2² = ².

(ii) 'Estw ² > 0 kai x ∈ X. f omoiìmorfa suneq c sto x ⇒ ∃δ > 0 ¸ste f (S(x, δ)) ⊆ S(f (x), ²).(1) X eÐnai olik fragmènoc ⇒ ∃x1 , x2 , . . . , xn ∈ X ¸ste X⊆

n [

S(xi , δ).(2)

i=1 (2)

f epÐ ⇒ Y = f (X) ⊆ f Y eÐnai olik fragmènoc.

n ³[

n n ´ [ (1) [ S(xi , δ) = f (S(xi , δ)) ⊆ S(f (xi ), ²), dhl. o

i=1

LÔsh 'Askhshc 51 Gia na deÐxoume ìti up rqei upakoloujÐa (Fkn ) thc (Fn ) h opoÐa na sugklÐnei omoiìmorfa, arkeÐ na epalhjeÔsoume tic upojèseic sto Je¸rhma Ascoli :

• ([0, 1], | · |) eÐnai sumpag c =⇒ diaqwrÐsimoc • H (Fn ) eÐnai kat shmeÐo ¯Z xfragmènh: ¯ Z ¯ ¯ ¯ ∀x ∈ [0, 1], |Fn (x)| = ¯ fn (t) dt¯¯ ≤ 0

|fn (t)|dt ≤ x · M ≤ M, ∀n ∈ N. 0

• H (Fn ) eÐnai isosuneq c: 'Estw x ∈ [0, 1]. Tìte ∀y ∈ [0, 1] me |y − x| < M² ≡ δ èqoume: ¯ ¯Z y Z x ¯ ¯ fn (t) dt¯¯ = |Fn (x) − Fn (y)| = ¯¯ fn (t) dt − 0 0 ¯ Z y  ¯Z y ¯ ¯  ¯  |fn (t)| dt ≤ M |y − x| < ² ,an y > x fn (t) dt¯¯ ≤    ¯ x x = ¯ Z y ¯Z x  ¯ ¯    |fn (t)| dt ≤ M |x − y| < ² ,an y < x fn (t) dt¯¯ ≤  ¯¯ y

'Ara, plhroÔntai oi proôpojèseic tou Jewr matoc Ascoli =⇒ up rqei upakoloujÐa (Fkn ) thc (Fn ) h opoÐa na sugklÐnei omoiìmorfa sta sumpag uposÔnola tou [0, 1]. 'Omwc [0, 1] sumpagèc uposÔnolo tou [0, 1] kai ra h (Fkn ) sugklÐnei omoiìmorfa sto [0, 1].

LÔsh 'Askhshc 52 'Estw (xn ) basik akoloujÐa sto metrikì q¸ro (X, ρ) kai

α, β dÔo shmeÐa suss¸reushc. 'Estw ² > 0. (xn ) basik =⇒ ∃n0 ∈ N tètoio ¸ste ρ(xn , xm ) < ², ∀n ∈ N.

α shmeÐo suss¸reushc =⇒ ∃n1 ≥ n0 tètoio ¸ste ρ(xn1 , α) <

² 2

β shmeÐo suss¸reushc =⇒ ∃n2 ≥ n0 tètoio ¸ste ρ(xn1 , β) <

² 2

ρ(α, β) ≤ ρ(α, xn1 ) + ρ(xn1 , xn2 ) + ρ(xn2 , β) < Gia ² :=

² 2

+²+

² 2

= 2². (∗)

ρ(α, β) , paÐrnoume ρ(α, β) < ρ(α, β), topo. 2

(diaforetik , ja mporoÔsame na poÔme ìti (∗) =⇒ ρ(α, β) −→ 0 =⇒ α = β).

LÔsh 'Askhshc 53 'Estw ìti up rqei akoloujÐa (Gn )n anoikt¸n kai pukn¸n uposunìlwn tou R ¸ste

∞ \

n=1

AfoÔ Gn ⊆ Q puknì kai Q puknì sto R, èpetai ìti Gn puknì kai sto R. Q arijm simo, opìtan jewroÔme mia arÐjmhs tou, èstw Q = {q1 , q2 , . . .} kai jètoume Vm = R \ {qm }, m ∈ N. ('Epetai ìti Vm anoiktì sto R). Tìte,

Vm = R \ {qm } = R \ {qm }o = R, dhl. Vm puknì sto R. T¸ra, ∞ ³\ n=1

∞ ∞ ∞ ´ ³\ ´ ³\ ´ ³ ´ ³ ´ [ Gn ∩ Vm = Q ∩ Vm = Q ∩ R \ {qm } = Q ∩ Q \ R = ∅. m=1

m=1

'Ara, h tom thc arijm simhc oikogèneiac {Gn : n ∈ N} ∪ {Vm : m ∈ N} eÐnai to kenì, ra ìqi puknì sto R, topo, apì to Je¸rhma Baire, afoÔ R pl rhc. AntiparadeÐgmata

Antipar deigma 4 'Estw X 6= ∅ metrikìc q¸roc kai ρ1 , ρ2 dÔo isodÔnamec

metrikèc sto X . An o (X, ρ1 ), eÐnai olik fragmènoc, den èpetai ìti o (X, ρ2 ) eÐnai olik fragmènoc.

33 Pr gmati, gia X = R,

ρ1 : R × R → R me ρ1 (x, y) = |x − y|, ∀x, y ∈ R kai

¯ ¯ ¯ x y ¯¯ ¯ , ∀x, y ∈ R. ρ2 : R × R → R me ρ2 (x, y) = ¯ − 1 + |x| 1 + |y| ¯ èqoume ìti: • ρ2 metrik ('Askhsh 4) • ρ1 ∼ ρ2 (afoÔ kajorÐzoun ta Ðdia anoikt sÔnola) • (R, ρ2 ) olik fragmènoc: ArkeÐ na deÐxoume oti o R gr fetai wc peperasmènh ènwsh sunìlwn me di metro ≤ ². 'Estw ² > 0. Jètoume f : R → [−1, 1] me f (x) =

x , ∀x ∈ R 1 + |x|

f 1-1, epÐ me suneq antÐstrofo kai ρ2 (x, y) = |f (x) − f (y)|. JewroÔme mia diamèrish tou diast matoc [−1, 1]: −1 = x0 < x1 < . . . < xn = 1, |xi − xi−1 | < ². Tìte,

R⊆

n [

f −1 ([xi−1 , xi ]).

i=1

Pr gmati, èstw x ∈ R. Tìte, f (x) ∈ [−1, 1] =⇒ x ∈ f −1 ([xi−1 , xi ]), gia n [ k poio i ∈ {1, 2, . . . , n} ⇐⇒ x ∈ f −1 ([xi−1 , xi ]). i=1 −1

T¸ra, an α, β ∈ f ([xi−1 , xi ]) =⇒ f (α), f (β) ∈ [xi−1 , xi ] =⇒ ρ2 (α, β) = |f (α) − f (β)| < ² =⇒ δ(f −1 ([xi−1 , xi ])) ≤ ².

• (R, ρ1 ) ìqi olik fragmènoc afoÔ δ(R) = ∞ (wc proc thn ρ1 ). Parat rhsh: Shmei¸ste oti h sun rthsh f k nei to di sthma [−1, 1] omoiomorfikì me to R !

Antipar deigma 5 'Estw X 6= ∅ metrikìc q¸roc kai ρ1 , ρ2 dÔo isodÔnamec

metrikèc sto X . An o (X, ρ1 ) eÐnai pl rhc metrikìc q¸roc, den èpetai ìti o (X, ρ2 ) eÐnai pl rhc. Pr gmati, gia X = R,

ρ1 : R × R → R me ρ1 (x, y) = |x − y|, ∀x, y ∈ R

kai

¯ ¯ ρ2 : R × R → R me ρ2 (x, y) = ¯

y ¯¯ x − ¯, ∀x, y ∈ R. 1 + |x| 1 + |y|

èqoume ìti:

• ρ2 metrik ('Askhsh 4) • ρ1 ∼ ρ2 (afoÔ kajorÐzoun ta Ðdia anoikt sÔnola) • (R, ρ2 ) den eÐnai pl rhc afoÔ h akoloujÐa (xn ) me xn = n, ∀n ∈ N ston (R, ρ2 ) eÐnai basik (wc proc thn ρ2 ), diìti gia ² > 0 kai n0 ∈ N, ρ2 (xn , xm ) = | an 1 −

n m − | < ², ∀m > n ≥ n0 , 1+n 1+m

² n ² ² m ² < < 1 + ,1 − < <1+ 2 1+n 2 2 1+m 2

kai an h (xn ) sunèkline wc proc thn ρ1 , afoÔ ρ1 ∼ ρ2 , ja sunèkline kai wc proc thn ∼ ρ2 , topo.

Antipar deigma 6 DeÐxte, me kat llhla antiparadeÐgmata, ìti:

(i) 'Enac diaqwrÐsimoc metrikìc q¸roc den eÐnai kat n gkhn kai sumpag c. Pr gmati, an X = R, ρ1 : R × R → R me ρ1 (x, y) = |x − y|, ∀x, y ∈ R JewroÔme mia arÐjmhsh tou Q : {q1 , q2 , . . . , qn , . . .}. EpÐshc, xèroume ìti Q = R. Br kame loipìn èna arijm simo puknì uposÔnolo tou R. 'Ara, o (R, ρ1 ) eÐnai diaqwrÐsimoc [ O (R, ρ1 ) ìmwc den eÐnai sumpag c, afoÔ an tan, tìte h anoikt k luyh (−n, n) n∈N

tou R, ja eÐqe peperasmènh upok luyh, dhl. ja up rqe k ∈ N kai {n1 , n2 , . . . , nk } ∈ N ¸ste n [ R = (−ni , ni ) = (−n, n), i=1

topo.

(ii) Den isqÔei p nta ìti an èna sÔnolo K eÐnai kleistì kai fragmèno, tìte eÐnai kai sumpagèc. DÐnoume dÔo antiparadeÐgmata:

• O q¸roc (R, ρ2 ), ìpou ρ2 : R × R → R me ρ2 (x, y) =

ρ1 (x, y) , ∀x, y ∈ R, 1 + ρ1 (x, y)

eÐnai fragmènoc (afou ρ2 (x, y) ≤ 1) kai kleistìc sto R. 'Omwc, ρ1 ∼ ρ2 kai ra, afoÔ (R, ρ1 ) ìqi sumpag c, oÔte kai o (R, ρ2 ).

• O q¸roc (X, ρδ ), ìpou ρδ h diakrit metrik kai X peiro sÔnolo eÐnai kleistìc kai fragmènoc all ìqi sumpag c, diìti isqÔei: (X, ρδ ) sumpag c ⇐⇒ X peperasmèno sÔnolo (iii) 'Enac omoiomorfismìc den eÐnai katan gkhn isometrÐa. Pr gmati, an X = { n1 : n ∈ N∗ }, tìte (X, ρ1 ) ⊂ (R, ρ1 ), me ρ1 ìpwc sto (i). JewroÔme th sun rthsh f : (N, ρ1 |N ) → (X, ρ1 ) me f (x) = n1 , ∀x ∈ N H f eÐnai 1-1 kai epÐ me suneq antÐstrofo ( ra f omoiomorfismìc). O (N, ρ1 |N ) eÐnai pl rhc en¸ o (R, ρ1 ) ìqi. 'Ara f ìqi isometrÐa.

(iv) K je fragmèno sÔnolo den eÐnai p ntote olik fragmèno. Pr gmati, o q¸roc (R, ρ), ìpou ρ = min(ρ1 , 1), me ρ1 ìpwc sto (i) eÐnai fragmènoc, all ìqi olik fragmènoc: Gia x ∈ R, 0 < ² < 1 isqÔei Sρ (x, ²) = Sρ1 (x, ²) = (xi − ², xi + ²). Tìte den up rqei n ∈ N kai {x1 , x2 , . . . , xn } ∈ R ¸ste R=

n [ i=1

afoÔ

n [

Sρ (xi , ²) =

n [

(xi − ², xi + ²),

i=1

(xi − ², xi + ²) fragmèno wc proc th ρ1 kai ra eÐnai 6= R.

i=1

LÔsh 'Askhshc 54 (i)f 1-1 kai epÐ: ArkeÐ na broÔme thn antÐstrofh:

Gia x > 0, èqoume ìti |x| = x, ra x y f (x) = := y =⇒ y(1 + x) = x =⇒ x(1 − y) = y =⇒ x = . 1+x 1−y Gia x < 0, èqoume ìti |x| = −x, ra x y f (x) = := y =⇒ y(1 − x) = x =⇒ x(1 + y) = y =⇒ x = . 1−x 1+y

y eÐnai h antÐstrofh thc f. 1 − |y| ¶ µ y y 1−|y| (f ◦ g) (y) = f = = y =⇒ g ◦ f = id|[−1,1] (qrhsimopoi¸ntac y 1 − |y| 1 + 1−|y| th summetrÐa thc | · | ). OmoÐwc, (g ◦ f )(x) = x =⇒ g ◦ f = id|R . H g(y) =

Epomènwc, g = f −1 . Oi sunart seic f kai g eÐnai suneqeÐc wc sÔnjesh twn suneq¸n sunart sewn

k · k : x 7→ kxk − : x 7→ −x 1 h : x 7→ . x 'Ara, f omoiomorfismìc.

(ii)Ergazìmenoi ìpwc sto (i), brÐskoume thn antÐstrofh thc f, h opoÐa eÐnai h g(y) =

y . 1 − kyk

Oi sunart seic f kai g eÐnai suneqeÐc wc sÔnjesh twn suneq¸n sunart sewn

k · k : x 7→ kxk − : x 7→ −x 1 h : x 7→ . x 'Ara, f omoiomorfismìc.