(2) hx, yi = hy, xi, ∀x, y ∈ X (*R)

§1 ^.D(

@ 1(^.) ^! X ^)$,l R^!MwmX, ^Cb g= h, i : X × X → R

(x, y) 7→ g(x, y) = hx, yi

[D8Z

(1) hx, xi ≥ 0,^hx, xi = 0 ⇔ x = 0 (^u&R).

(2) hx, yi = hy, xi, ∀x, y ∈ X (^*R).

(3) hλx + µy, zi = λhx, zi + µhy, zi, ∀λ, µ ∈ R, x, y ∈ X (^-IRR)

sg= h, i^>X ^!Y<O, (X, h, i) ^>OmX, hx, yi^>x ^jy^!

O, kxk =phx, xi^>x ^!1,.

%1 ^URⁿ= {(x1,· · · , xn)|xi ∈ R}^>n^meT$,, ^[F!2&, Rⁿ

>R ^!MwmX, ^>n ^{? *mX}. R^{n e
!O} h, i:

hx, yi =

n

X

i=1

xi· yi, ∀ x = (x1,· · · , xn), y= (y₁,· · · , yn) ∈ Rⁿ.

% 2 ^U C⁰[a, b]^>X [a, b] uUE,!7Q!MwmX. ^&_

Oh, i ^D:

hf, gi = Z b

a

f(x) · g(x)dx.

$ 1 (Schwarz ^3) ^! (X, h, i, )^>OmX, ^s

|hx, yi| ≤ kxk · kyk.

"Ftd x ^jy ^IRJ@.

H, x= 0(^My = 0)^#,^dO!IRw h0, yi = h0 · 0, yi = 0h0, yi = 0.

-# SchwarzÆ"&t. ^D!x6= 0, y 6= 0,^s* ∀ t ∈ R,^e hx, xi − 2thx, yi + t²· hx, yi = hx − ty, x − tyi ≥ 0

⇒ ∆ = 4hx, yi²− 4hx, xihy, yi ≤ 0 (^Æ&)

D!v
.

@ 2 (^() ^!X ^>4{QI,^Cb ρ: X × X → R^[D8Z (1) ρ(x, y) ≥ 0^ρ(x, y) = 0 ⇔ x = y;

(2) ρ(x, y) = ρ(y, x);

(3) ρ(x, z) ≤ ρ(x, y) + ρ(y, z). (]Æ"&)

sρ^>X^!Y<)w(^Mgp), (X, ρ)^>)wmX(^MgpmX), ρ(x, y)

>x, y ^xX!gp.

% 3 (^{.C #}) ^! (X, h, i)^>OmX, ^s}

ρ(x, y) = kx − yk, ∀ x, y ∈ X.

Fρ ^&_2 ^!(1), (2), -]Æ"&Xt: ρ²(x, z) = kx − zk² = hx − z, x − zi

= h(x − y) + (y − z), (x − y) + (y − z)i

= hx − y, x − yi + 2hx − y, y − zi + hy − z, y − zi

≤ hx − y, x − yi + 2kx − yk · ky − zk + hy − z, y − zi

= (kx − yk + ky − zk)² = (ρ(x, y) + ρ(y, z))².

`ρ ^>X ^!)w, >dOf!)w.

§2 ^(!8/

_V! (X, ρ) ^>)wmX. ^! x∈ X, r > 0, ^U Br(x) = {y ∈ X | ρ(y, x) < r},

>[x ^>P, r ^>e!i.

@ 1 (^) ^! U ^> X ^!Q, ^C ∀ x ∈ U , ^h ∃ ε > 0, ^% B_ε(x) ⊂ U , ^s U ^>iQ;o&mQX)iQ. CY<QI! Q (^iQ) ⁾

iQ,^sx>Q.

F, X ^>iQ, ^-∅ ^X)Q. ^De x^!iQ>x ^!izl.

% 1 ^i>iQ: ^! x ∈ Br(x0), ^s ρ(x, x0) < r,^} ε= r − ρ(x, x0), ^s

y∈ Bε(x)^#,d]Æ"&, ^e

ρ(y, x₀) ≤ ρ(y, x) + ρ(x, x₀) < ε + ρ(x, x₀) = r,

t.y∈ Br(x0,^SB_ε(x) ⊂ Br(x0).

o/kv {y ∈ X|d(y, x0) > r} ^>iQ,^{ Q>}, ^)Q.

-5 1 (1) eH,<iQx\>iQ; ],<iQx>iQ; (2)eH,<Qx>Q; ],<Qx\>Q.

H, (1)^! U₁,· · · , Uk ^>iQ, ∀x ∈

k

T

i=1

U_i,^d&_, ∃ε_i >0,^% B_εi(x) ⊂ U_i, i = 1, · · · , k. ^} ε= min{εi|i = 1, · · · , k}, ^s B_ε(x) ⊂

k

T

i=1

U_i,^>

k

T

i=1

U_i ^>iQ,

iQ!&_tSk[;],<iQx>iQ. (2)^rcQIp1

(A₁∪ · · · ∪ Ak)^c = A^c₁∩ · · · ∩ A^c_h (T

α

Aα)^c =S

α

A^c_α

R(1) ^Sk.

>xlJQ,^AaPH!9, 3G$,y!PH9)Y}!.

@ 2 (^:) ^! {xn}^∞_n=1 ^> X ^%y, ^Cr x₀ ∈ X, ^% ∀ ε > 0,

∃ N = N (ε), n≥ N ^#, xn∈ Bε(x₀),^s{xn} ^+PH x₀,^U>

n→∞lim x_n= x₀.

J (1) lim

n→∞x_n= x0⇔ lim

n→∞ρ(xn, x₀) = 0.

(2)d]Æ"&G (1) ^\Y,^PHCr, ^s=Y.

-5 2 ^QIA ^>Qd A H+v%y!PHr A ^.

H, ! A ^>Q, {xn} ⊂ A, ^ lim

n→∞xn = x₀. ^C x₀ 6∈ A, ^s ∃ ε0 >0,

% B_ε0(x₀) ⊂ A^c, ^ lim

n→∞x_n = x₀ ^]@, ∃ N = N (ε₀) ^% n ≥ N ^# x_n∈ Bε0(x₀),^tj x_n∈ A^J+! ^` x₀ ∈ A.

0x, ^CA H+v%y!PHr A ^,^sx₀6∈ A, ^j r_n= n⁻¹,^C Brn(x₀) ∩ A 6= ∅,^s xn∈ Brn∩ A. ^- xn → x0, ^t)+. ^`,

∃n0 >0^% B_rn0(x0) ⊂ A^c,^SA^{c >iQ}, A^>Q.

§3 ^(!9;

_! (x, ρ) ^>)wmX. ^! {xn}^∞_n=1 ^> X ^%y. ^C ∀ ε > 0, ∃ N = N(ε), n, m≥ N ^#

ρ(xm, xn) < ε

s%y {xn} ^>Cauchy ^y(^MNy).

@ 1 (^9;) ^CX ^Cauchy ^yh>+v%y, ^s(X, ρ) ^><

)wmX.

J (1) ^+v%y> Cauchy^y;

(2) Cauchy^yCe+vy, "XY&+v (^C6).

-5 1 (Rⁿ,k · k) ><)wmX.

H, !{xn} ^>R^{n %y}, 3O5wQ&

x_n= (x¹_n, x²_n,· · · , xⁿ_n),

s

kxⁱ_k− xⁱ_l| ≤





n

X

j=1

(x^j_k− x^j_i)²





1 2

= kxk− xlk

`,^C{xn}^>Cauchy^y,^s{xⁱ_k}^∞_k=1 ^*<i= 1, 2, · · · , n^h>Cauchy^y,

-+v. ^! lim

n→∞xⁱ_k= xⁱ₀,^s

n→∞lim x_n= x₀ = (x¹₀, x²₀,· · · , xⁿ₀).

!A ^>X ^Q,^sup{ρ(x, y)|x, y ∈ A} ^>A ^!ye, ^U>diamA. ^ye

eH!QI>eaQI.

D!&q) R¹ X4nq!Y
Q&.

$ 1 ^!(X, ρ) ^>)wmX, sDyT8"W: (1) (X, ρ)><)wmX;

(2) (Cantor) ^Q4nqt:  F₁ ⊃ F₂ ⊃ · · · > Fn ⊃ · · · ^>Yy4m

Q,^ lim

n→+∞diamFn= 0,^sr=Y!% a∈T∞ n=1Fn.

(3)^4nqt: (2)^F_i ^Kye(^e) ^h0^!#eJ:`

.

H, (1)⇒(2): ^an ∈ Fn, ^d Fn ⊃ Fn+1 ⊃ · · · ^w {an, a_n+1,· · · } ⊂ Fn. ^`

, m > n^#

ρ(am, an) ≤ diamFn→ 0 (n → ∞)

- {an} ^> Cauchy ^y, ^!PH> a, ^s a = lim

n→∞a_m ∈ Fn, ∀n ≥ 1, ^S a ∈

∞

T

n=1

F_n. ^C|e b∈

∞

T

n=1

F_n,^s

ρ(a, b) ≤ diamF_n→ 0 (n → ∞)

-a= b.

(2)⇒(3): ^t)F!.

(3)⇒(1). ^! {an} ^> X ^ Cauchy ^y, >xvt)Y<+v%y, ^|S

v3DY<+vySk. ^d Cauchy ^y!&_, ∃ n1 < n₂ < · · · , ^% m, n≥ nk ^#

ρ(am, a_n) < 1 2^k+1.

j X ^! Fk= ¯B₂−k(an_k), k = 1, 2, · · · . x∈ Fn+1 ^#, ρ(x, a_nk) ≤ ρ(x, a_nk+1) + ρ(a_nk+1, a_nk)

≤ 1

2^k+1 + 1 2^k+1 = 1

2^k

t.x∈ Fk. ^SF₁ ⊃ F2 ⊃ · · · ⊃ Fk > F_k+1 ⊃ · · · . ^|Y2, diam F_k≤ 2 · 2^−k → 0, (k → +∞),

>ra∈

∞

T

n=1

F_k. ^-e

0 ≤ ρ(a, an_k) ≤ 2^−k → 0 (k → +∞)

Sy{an_k} ^+vha.

<)wmXeDec!V2bLnq.

@ 2 (^>4A0) ^! A ^> X ^!Q, ^b f : A → A ^C[D8

Z:

(∗)^r, 0 ≤ q < 1, ^% ρ(f (a₁), f (a₂)) ≤ q · ρ(a₁, a₂), ∀ a₁, a₂ ∈ A.

sx>V2b .

$ 2 (^>4A0F$) ^CA><)wmX (X, ρ)^Q, f : A → A

>V2b , ^sr=Y!% a∈ A, ^% f(a) = a (^Æ'%).

H, a₀ ∈ A,^$B#&_ A^%y{an}^D: a_n= f (a_n−1), n= 1, 2, · · · ,

s

ρ(a_n+1, an) = ρ (f (an), f (a_n−1)) ≤ q · ρ(an, a_n−1), ∀ n ≥ 1

-e

ρ(a_n+1, an) ≤ q · ρ(an, a_n−1) ≤ q²ρ(a_n−1, a_n−2) ≤ · · · ≤ qⁿ· ρ(a1, a₀), ∀n ≥ 0 ρ(am, an) ≤ ρ(am, a_m−1) + ρ(a_m−1, a_m−2) + · · · + ρ(a_n+1, an)

≤ (q^m−1+ q^m−2+ · · · + qⁿ) · ρ(a₁, a₀)

≤ qⁿ

1 − q · ρ(a₁, a₀) → 0, (m > n, n → ∞).

t.{an}^> Cauchy^y.^!PH> a,^sa∈ A, ^ ρ(f (a), a) ≤ ρ(f (a), f (an)) + ρ(f (an), f (a) + ρ(an, a)

≤ ρ · ρ(a, an) + qⁿ· ρ(a1, a₀) + ρ(an, a) → 0 (n → ∞)

t.f(a) = a.

=YR: f(b) = b,^s

ρ(a, b) = ρ(f (a), f (b)) ≤ q · ρ(a, b)

t.ρ(a, b) = 0,^- a= b.

§4 ^(!DI;

!S ^>)wmX (X, ρ) ^!Q, ^CS ⊂S

α

G_α, ^s {Gα} ^>S ^!Y<6

:. G_α ^h>iQ#, ^ {Gα} ^>i6:; |eeH<m0!6:>eH6 :,^d{Gα} ;m0!6:>6:.

@ 1 (^I;) ^CQI S !Hi6:(eeH6:, ^s S ^>

c}QI.

-5 1 c}QI>eaQ.

H, ! A ^>c}QI, ^ a∈ A, ^`> A⊂

∞

S

n=1

Bn(a), ^> ∃ n1,· · · , nk,^% A⊂S

ni

B_ni(a) = BN(a), N = max{n1,· · · , nk}. ^` A ^>eaQI. ^Dv

A^{c >iQ}. ^($, ^ b ∈ A, ^j A ^]Y% a, ^ 0 < r(a) < 1 2ρ(a, b),

sB_r(a)(a) ∩ B_r(a)(b) = ∅, ^A ⊂ S

a∈A

B_r(a)(a), ^dV!, ^r a₁,· · · , ak ∈ A^% A⊂

k

S

i=1

B_r(ai(ai). ^}r = min

1≤i≤k{r(ai)}, ^sBr(b) ∩ B_r(ai)(ai) = ∅, i = 1, · · · , k. ^- B_r(b) ∩ A = ∅,^S A^{c >iQ}, A^>Q.

$ 1 ^!A ^>R^{n Q}, s[DT8"W: (1) A^>c}QI;

(2) A ^>Tyc}QI, ^S A H%yhe+vy, ^8yPH

rA ^;

(3) A^>eaQ.

H, (1) ⇒ (2). (^0v/). ^!{an} ^>A ^%y, ^3B+vh A^%!

y, ^s ∀a ∈ A, ^ri B_r(a)(a), ^% B_r(a)(a) ^,|De {an} ^eHK.

F, A ⊂ S

a∈A

B_r(a)(a), ^dc}R, ^r a₁,· · · , ak ^% A ⊂ S^k

i=1

B_r(ai)(a_i), ⁵

#, {an} ^|eHKGr A^,^t)+!

(2) ⇒(3). ^Ev A ^ea, (^0v/). ^C ∃ an ∈ A, ^% ρ(a₀, an) → ∞ (n → ∞), ^sF {an} ^B+vy, ^t)+! ^qv A ^>Q, ^c0v/.

sran ∈ A, lim

n→∞an = a 6∈ A. {an} ^!Yyh+vh a6∈ A, ^tjTyc

+!

(3) ⇒ (^0v/). ^! A ^>eaQ, ^r A ^!i6: {Uα}, ^{% }

HeH<m0hB/6: A. ^u27 I₀ ⊃ A, ^[ I₁ ^ 2^{n "5}, ^eY

"5 I₂ ⊂ I1, ^% I₂ ∩ A ^ÆeH< Uα ^6:. ^Zo;, ^Yt27 I₁ ⊃ I2 ⊃ · · · , diam Im→ 0 (m → ∞). ^dQ4nq, ^r=Y!%a∈ ∩Im∩ A.

g`> {Uα} ^> A ^!i6:, ^>r α₀ ^% a ∈ Uα0. ^h) m ^5#e Im ⊂ Uα0. ^tjIm∩ A^ÆeH< Uα ^6:J+!

7+ Rⁿ ea%ye+vy (∵ ^{Dh Mt27}).

@ 2 (^Æ*&6)

§5 &<A0

L^YDuUE,!&_: f : R → R ^r x₀ ^uU){, ∀ ε > 0, ∃ δ > 0

% |x − x0| < δ^# |f (x) − f (x0)| < ε. c)wmX!kWkD;A:

@ 1 (&<A0) ^!f : X → Y ^>)wmX (X, ρ₁), (Y, ρ₂)^xX!b ,

! x₀ ∈ X. ^C ∀ ε > 0, ∃ δ > 0 ^% f(B_δ^X(x0)) ⊂ B_ε^Y(f (x0)), ^s f ^rx₀ ^

uU. ^C f ^uU, ^sf ^>uUb .

-5 1 (&<A0 ) ^! f : X → Y >)wmXxX!b , x₀ ∈ X.

s

(1) f ^rx₀ ^uU⇔ ^*]+vhx₀ ^!%yxn,^he lim

n→∞f(xn) = f (x₀);

(2) f ^>uUb ⇔^* ∀^iQ V ⊂ Y , f⁻¹(V ) ^>X ^iQ; (3) f ^>uUb ⇔^* ∀^Q V ⊂ Y , f⁻¹(V ) ^>X ^Q.

H, (1) “⇒” ^!f ^rx₀ ^uU,^s ∀ε > 0, ∃ δ > 0^% f(B_δ^X(x₀) ⊂ B_ε^Y(f (x₀)).

`>x_n→ x0,^>n^5#x_n∈ B_δ^X(x₀), ^- f(xn) ∈ B_ε^Y(f (x₀)), ^Sf(xn) → f(x₀).

“⇐” (^0v/) ^C f ^r x₀ ^uU, ^s ∃ ε0 > 0 ^{% *} δ = 1

n, ∃ xn ∈ B^X1

n

(x₀), ^-f(xn) 6∈ B_ε^Y₀(f (x₀)). ^Fx_n→ x0,^f(xn) 6→ f (x₀),^+! (2), (3): ^~C6.

J (1) ^CA ^> X ^xQ, f : A → Y ^>b , ^s X ^!)wH~h A,

-A ^X>)wmX (^)wmX), ^#k[&_ f ^!uUR, ^eo/!l

J.

(2) ^! f : A → Y ^uU, ^C ∀ ε > 0, ∃ δ > 0 ^% ρ₁(a₁, a₂) < δ ^# ρ₂(f (a₁), f (a₂)) < ε, ^s f ^>Y}uUb .

$ 1 (&<A0D;) ^! f : X → Y ^>uUb ,^s

(1) f X ^c}QIb> Y ^c}QI;

(2) f ^rc}QIY}uU.

H, (1)^! A^>X ^c}QI, ^f(A)^!i6: {Vα}, ^s{f⁻¹(Vα)}^>

A ^!i6:,^- ∃ α1,· · · , αk ^% A⊂

k

S

i=1

f⁻¹(Vαi). ^t. f(A) ⊂

k

S

i=1

Vαi. (2)^! A ^>c}QI.^C f ^r A ^Æ)Y}uU!, ^s ∃ ε0 >0, ^{% *} δ= 1

n, ∃ am, b_n∈ A,^%

ρ₁(an, b_n) < 1

n, ρ₂(f (an), f (bn)) > ε₀.

d§4 ^&q1, {a_n}^G{bn}^5r+vy,Æ3!3")+v!, a_n→ a₀, bn→ b0,^s

ρ₁(a₀, b₀) ≤ ρ₁(a₀, a_n) + ρ₁(an, b_n) + ρ₁(bn, b₀)

< 1

n+ ρ₁(a₀, a_n) + ρ₁(bm, b₀) → 0

Sa₀ = b₀. ^

ε₀< ρ₂(f (an), f (bn)) ≤ ρ₂(f (an), f (a₀)) + ρ₂(f (b₀), f (bn)) → 0.

t)+!

7 uUb (^uUE,)f : X → R ^{rc}QIk[z}, ^

Nz.

@ 2 (^Æ*&6) ^! G^> X ^!Q, ^C= x₁, x₂ ∈ G,^hruU

b (^uUI) σ : I = [0, 1] → X ^% σ(0) = x1, σ(1) = x2, σ(I) ⊂ G, ^sG

u9.

-5 2 R¹ u9QI>X.

H, !G⊂ R^{1 u9}, a, b ∈ G. ^Av [a, b] ⊂ G. ^($, ^d&_,

ruUb f : [0, 1] → R,^% f(0) = a, f (1) = b, f ([0, 1]) ⊂ G. f ^>Ymu

UE,, ^dbz&q, [a, b] ⊂ f ([0, 1]) ⊂ G.

$ 2 (&<A0D&6;) ^{uUb u9!QIb>u9}

QI.

H, ! G⊂ X ^u9, f : X → Y ^uU. ^ y₁, y₂ ∈ f (G), ^sr x₁, x₂ ∈ G ^% f(x1) = y1, f (x2) = y2. ^d&_, ^ruUb σ : [0, 1] → X

% σ(0) = x₁, σ(1) = x₂, σ([0, 1]) ⊂ G. ^7Ib f ◦ σ : [0, 1] → Y ^uU, ^ f◦ σ(0) = f (x1) = y₁, f ◦ σ(1) = f (x₂) = y₂, f ◦ σ([0, 1]) ⊂ f (G). f ◦ σ^f) f(G)

u^y₁, y₂ ^!.

7+ !f : X → R^>uUE,, G ⊂ X ^u9.

(1) ^Cr x₁, x2 ∈ G, ^% f(x1) · f (x2) < 0, ^sr x₀ ∈ G ^% f(x₀) = 0;

(2) ^*h8Z f(x₁) ≤ y ≤ f (x₂) ^!] y, ^Y&r x ∈ G ^% y= f (x).

b f : R² → R^>.mE,. R^{2 !%c} (x, y)^'. ^!(x₀, y₀) ∈ R²,^C∃ A ∈ R, ^{% =} ε >0, ∃ δ > 0^%

0 < k(x, y) − (x0, y₀)k =p(x − x0)²+ (y − y0)² < δ

#,

|f (x, y) − A| < ε,

ff ^r (x₀, y₀) ^ePH ( ^PH), ^U>

(x,y)→(xlim0,y0)f(x, y) = A

M

x→x0lim

y→y0

f(x, y) = A.

C*hY<?&! y,^PH lim

x→x0

f(x, y) = ϕ(y) ^r, sk[&_PH

y→ylim0

x→xlim0

f(x, y) = lim

y→y0

ϕ(y).

o/#&_ lim

x→x0

y→ylim0

f(x, y),3>nPH.

% 1 (1) lim

(x,y)→(0,0)

x²y² x²+ y² = 0



∵ x²y² x²+ y² ≤ 1

2xy ≤ 1

4px²+ y²

 . (2) xy

x²+ y^{2 r} (0, 0) ^BPH (^5j y= x^G y= x².) (3) f (x, y) = x · sin1

y. ^d|f (x, y)| ≤ x ^w lim

(x,y)→(0,0)f(x, y) = 0, ^ lim

y→0f(x, y)

Ær. (4) lim

y→0lim

x→0

xy

x²+ y² = lim

x→0lim

y→0

xy

x²+ y² = 0,^d (2), ^PHÆr.

$ 3 ^C lim

(x,y)→(a,b)f(x, y) = A,^ ∀ y 6= b, lim

x→af(x, y) = ϕ(y) ^r,^s

y→blimlim

x→af(x, y) = lim

y→bϕ(y) = A

C∀ x 6= a, lim

y→bf(x, y)^Xr,^s

x→alimlim

y→bf(x, y) = A = lim

y→blim

x→af(x, y).

H, [A ^eH>s. ^dV!, ∀ ε > 0, ∃ δ > 0,  0 <p(x − a)²+ (y − b)² < δ

#|f (x, y) − A| < ε

2. ^?&y,^}x→ a, 

 lim_x→af(x, y) − A   ≤

ε

2 < ε, ∀ 0 < |y − b| < δ 2.

t.

y→blim lim

x→af(x, y) = A.