kinko
DESCRIPTION
KinkoTRANSCRIPT
-
1
22
3
4
5
6
7
81
92
103
11
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
11.9
18 C16530128
243
-
59 3
1
3
X0, X1, , Xn n1(n 1)
pi( 0), i0, 1, , n p0p1+pn
ipii0, 1, , n
0, 1, , n p0, p1, , pn
p0p1pn1 1
xi
p0 x0p1 x1pn xn 0
p0p1pn
0 x0 1 x1 n xn 0
0 n1
2
1
11.10
11.11
244
-
3 X, Y, Z, px , py , pz
1 px , py , pz 3
(1, 0, 0), (0, 1, 0), (0, 0, 1)
2 1
1 1
2
2
(0,0,1)
(0,1,0)(1,0,0)
px py
pz
1 3
245
-
59 3
2
2
2
1 p0, p1, , pn n 1 (1,
0, 0, , 0), (0, 1, , 0), , (0, 0, , 1)
4
22
1Sperner2
6
6
6 6 36
6
K K K
1 123 1
2 K
123
K 123
1 2
1
246
-
2
3
12 K 1
2
12
1 1
1 2 12 1
n 12
1
1 2
2 1
2 1
1211
2
K 12 K
1
3 2
3 2
3
3 2
247
-
59 3
2 12
2
3 1
23
K
12
1 123
12
1
1
1 1
1
1 1
1
1
2
2
2
2
2
22
2
3
3
33
3
3
3
3
3
248
-
K 12
1 1
12
3
1 3
3
n n
n K
3 (1, 0, 0), (0, 1, 0), (0, 0, 1) 2
n1 (1, 0, 0, , 0), (0, 1, , 0), , (0, 0, , 1)
n 0 1
2 3
1 42
2
2
3
K0n
1 n1 0 n
249
-
59 3
2 K v n1 n1
v v
3 K v n2
v
4 0 n
2K
K n 0
n
n0
0 0 1
n1
n1
01n1 K n1 n1
01n1 n1
1 2
K n
0 n1 K n1
2 0 n1
2
3 n
0 n1 n1 n
1 n 1 n 2
250
-
0 n n
n1
K n1 0 n1
1 0 n
n
0 n1
n1 K n1 0 n1
2 0 n
n 1
1 1 1
0 n n
1
2 0 n
n
0 n
n
1
0 n1
0 n n
0 n n
2
251
-
59 3
0 n
n
2 0 n
n 2
0
n n
0 n
n
2 4
4 2
1
1
1 1
1
1 1
1
1
2
2
2
2
2
22
2
3
3
33
3
3
3
3
0 252
-
0 n
1
4
2 2
0 x1 y
2 z 3
1Brouwer f
m (x0, x1, , xn)
xi 0, i0, 1, , n x0 x1 xn 1 (x0, x1, , xn)
f (x0, x1, , x n) x0x1xn1
1x0 x0 (x0, x1, , xn) 0
2x0 x0 x1 x1 (x0, x1, , xn) 1
3x0 x0x1 x1 x2 x2 (x0, x1, , xn) 2
4 x i xi i (i0, 1, , n)
i (i0, 1, , n)
x i xi x0 x1 xn 1 x0x1 x n 1
i (i0, 1, , n) x i xi
1 (1, 0, 0, , 0) (x0, x1, , x n) x0 1
x0 1 (1, 0, 0, , 0)
253
-
59 3
0
2 (0, 1, 0, , 0) (x0, x1, , x n) x1 1
x0 x0 x11 (0, 1, 0, , 0)
1
3 (0, 0, 1, , 0) (x0, x1, , x n) x2 1
x0 x0x1 x1 x2 1 (0, 0, 1, , 0)
2
4
xn 0 n1 n1
n1 n3 (x0, x1, , 0)
(x0, x1, , x n) xn 0 xn xn
n (x0, x1, , 0) 0 n1
(x0, x1, , 0, xn) n1 (0, x1, x2, , xn)
0 xn1 xn0
n2 n3
(x0, x1, , 0, 0) (x0, x1, , xn1, xn)
xn1 xn0 x n xn x n1 xn1 n
n1 0 n2 1
0 n
0 n n
1 n
(x 0m, 0 , x 1
m, 0 , , x nm, 0), (x 0
m, 1 , x 1m, 1 , , x n
m, 1), , (x 0m, n , x 1
m, n , , x nm, n)
m 0, 1, , n 1
m n 0 n
m
1 x0x2x50 025
254
-
m 1
2m
1 1
1 m
n1 (x0, x1, , xn)
f i0, 1, , n (x 0m, i, x 1
m, i, ,
xnm, i) (x 0
m, i, x 1m, i, , x n
m, i) (x0, x1, , xn) (x0, x1, , xn)
(x 0m, 0, x 1
m, 0, , xnm, 0) 0 x 0
m, 0 x 0m, 0 f
m (x 0m, i, x 1
m, i, , xnm, i) (x0, x1, , xn)
(x 0m, i, x 1
m, i, , x nm, i) (x0, x1, , xn)
x0 x0
(x0m, i, x1
m, i, , xnm, i) (x0, x1, , xn) x 0
m, 0 x0m, 0
(x0x0)
x1 x1, , xn xn x0x1xn x0x1xn 1
(x0, x1, , xn)(x0, x1, , xn) (x0, x1, , xn)
5
1
2n1 X0, X1, , Xn
(p0, p1, , pn) fi (p0, p1, , pn), i0,
1, , n
2 1
255
-
59 3
p0 f0 p1 f1 pn fn 0
fi Xi
(i 0, 1, , n) fi (p0, p1, , pn)0
(p0, p1, , p
n )
(p0, p1, , pn) n1 v(v0, v1, , vn)
vi pi fifi 0
vi pifi 0
(v0, v1, , vn)
( 0, 1, , n)
i (p0, p1, , pn)v0v1vn1
v1
i 0i 0, 1, , n
0 1 n1
( 0, 1, , n) fi
( 0(p0, p
1, , p
n ), 1
(p0, p1, , p
n ), , n(p
0, p
1, , p
n ))(p
0, p
1, , p
n ) (p
0, p
1, , p
n )
i vi pi v
1 i vi (p0, p
1, , p
n )p
i
1 1 pi 0 vi (p0,
p1, , pn )pi fi (p0, p1, , pn )0
i pi 0 1
p0 f0p1 f1p
n fn 0
1 v0 p0, v1 p
1, ,vn p
n fi (p
0, p
1, ,
256
-
pn ) 0
i fi (p0, p
1, , p
n ) 0 p
0 f0p
1 f1p
n fn
0 p1 fi (p0, p
1, , p
n ) 0
pi 0 fi (p0, p1, , pn )0
v
2
3 X
Y A ((xA, yA)), B ((xB, yB)) B
A AB
A B B A
AB
3
3AB
257
-
59 3
B A B A
AB
AB A
1
p ((px, py)) p
pp p AA
A A A
B
p p A B
B p 4
A p B
p
p B 5 A
A p
1 B A
AB
A B B A p p
A A A
4 1
5Bp
258
-
A
A C C p p
A p
A
X Y (x, y) A (xA, yA)B (xB,
yB) xA xB Y A B xA
xB B A xA xB Y
Y
(1, 1) (1, 0)
(1, 1) (1, 0)
(1, 0) (1, 1) X 1
(1, 1) (1x, 2) x
(1, 1) x 0
(1 x, 2) (1, 2) (1, 1)
6
6
6Walrass Existence Theorem and Brouwers Fixed Point Theorem, 8 1962, pp. 59-62.
259
-
59 3
3
(0,1, ,n)
p (p0, p1, , pn)
zi (p)i(p)pi (p)i0, 1, ,n 2
(p)
p i2
pii(p) i1n
i0n(p)
zi
(2) pi 0 n
i0
n
pi zi i0
n
pii(p)(p)p i
2i0
n
pii(p)i0
n
p i2
pii(p) i1n
i0n p i
2i0
n
pii(p)
i0
n
pii(p)i0
n
0
zi
7 p (p0, p1, , p
n )
i (p) (p) pi (zi(p
)0)
pi 0 i (p)(p) pi i (p
)
pi 0 i (p) 0
i i (p)(p) pi 0 n
7 1
in pipi
i ( 0, 1, , n) 0 i i0
260
-
i(p)
i1
n
(p)i0
n
pi
i (p) i1
n i1n pi 1 (p)1
i (p) pi
p
pi i 1 n1
pn1i0
n1
pi n(p)1i0
n1
i(p)zn(p) 1i0
n1
i(p) (p) 1i0
n1
pi (3)
pii(p) i0
n1 pi) i1n1(1 i(p)) i0
n1(1p i
2 i0n1 pi)
2 i0n1(1
(p) (4)
pn zn
pn zn 1i0
n1
pi 1i0
n1
i(p) (p) 1i0
n1
pi2 (5)
(2) pi 0 n1
i0
n1
pi zi pii(p)i0
n1
(p) p i2
i0
n1
(6)
(5)(6)
i0
n
pi zi pii(p)
i0
n1
1i0
n1
pi 1i0
n1
i(p) (p)p i
2i0
n1
1i0
n1
pi2
(4) i0n pi zi 0 (3)
7
n1m1 pi (i 0,
261
-
59 3
1, , n), qj (j 0, 1, , m) 2
l
l p0 y0lp1 y1
lpn ynlq0 x0
lq1x1lqm xm
l
yil xj
l l i j
k l skl (0 skl 1)
k
p0 x0
kp1 xk1pn xn
kq0 y0kq1 y1
kqm ymk sk
lll
xik i yj
k j l skll
p0 x0kp1 x1
kpn xnkq0 y0
kq1 y1kqm ym
k
skl
l(p0 y0
lp1 y1l pn yn
lq0 x0lq1x1
lqm xml)
0 262
-
kskl1 1
p0k x0kp1k x1
kpnk xnk q0k y0
kq1k y1kqm k ym
k
l( p0 y0
lp1 y1lpn yn
lq0 x0lq1x1
lqmxml )
i k
xik l yi
l XiYi
j k yjk t xjl
YiXi
p0(X0Y0)p1(X1Y1)pn(XnYn)
q0(X0Y0)q1(X1Y1)pm(XmYm)0
n1 nm2 pj , j
n1 , n2, , nm2 nm2
1
8 1
3
263
-
59 3
(1) ab a b a, b
(2)
(3)
(1) (2) n
7
X A X A
1
A
X
1 X
X
X
X
X
1
1
2 3
4
n n
2D
264
-
u xD x u
D D
l xD x l
D D
ul 1
3D
s D D u s u
D sup D
D t D
D l t l D inf D
D D
D D D
D D D
D
DE
(1) DEDEDE
(2)x Dy E x y
D E
1
2
265
-
59 3
2
S 2
AS BS
S A B
A B A
S S s s
21
( s) s S
A A
B
S
2
AS BS
S A B
A B A
S S s s
21
(s ) s S
A A
B
266
-
4ab a b
a, b
a, b 1
a, b
S a,
a, bs supS s W
(sW) b b S s b W
(ss) W 0
0
s S s S
S a, U V1, V2, ,
Vr
t a, b t s
a, ta, (s, s) V1V2 VrW
a, t tS sS
sb s S
s b (s , s) W s t s t S
t s21 b t s
s S bSa, b U
267
-
59 3
(1, 1) (1n1
, 1n1
) (n2, 3, )
(1, 1) (1, 1)
(1n1
, 1n1
)
(1n1
, 1n1
) (1, 1) (1, 1)
5
2 XY (XY ) XY 1
3
2 2 1
n( 3) n 1
3f X Y A X
Y V f 1(V) X f
2
f 1(V) f V
f 1(V) xX|f(x)V
V f 1(V)
f V X f 1(V)
V Y Y V
V
Y V
YY
2 2
3
268
-
3
2
f(A) Y f(A) f
A
f(A) Y A V
f 1(V)
f 1(V) A A
V V1, V2, , Vk
A f 1(V1) f 1(V2) f 1(Vk)
f(A) V1V2Vk f(A)
4X, Y K Y U XYX
Y 8VxXxK U V X
xV U yK (x, y) Dy
Ey , DyEy U XY DyEy K
K Ey1Ey2Eyk y y1, y2, , yk
Nx Dy1Dy2Dy k Nx X
NxKi1k
(NxEyi)i1k
(Dyi Eyi)U
Nx V NxKU x Nx x
V xV Nx Nx V V Nx
V X
5
2 XY
8 3
269
-
59 3
XY Z XY
Z XY
x X xY Y XY
yY (x, y)
3XY
U1, U2, , Ur xY U1U2Ur
X x xY U1U2Ur
Vx x Vx 4 Vx
VxY U1U2Ur
x X Vx Vxx X X x
Vx XXVx1 Vx2 Vxr
x1, x2, , xr Vx X
Vx1 Vx2 Vxr X XY j1, 2, , r Vxj Y
XY
6
X x1, x2, x3, Fn
xn, xn1, x n2,
xn, xn1, xn2, xn, xn1, xn2,
xn, xn1, xn2,
270
-
F1, F2, F3,
Vn X \ Fn Fn Vn X
V1, V2, V3,
V1 V2 V1V2 F1 F2
X \ (F1F2) Vn F1, F2, F3,
F1, F2, F3,
Vn X
Vn V1, V2, V3, X X
Vn V1 V2 V3
n XVn Fn n
F1, F2, F3, n1 Fn
Fn p p Fn
0 d(xj, p) xj( j n) p xn, xn1,
x n2, xn, xn1, x n2, p
9
n 2 x(x1, x2, , xn)y(y1, y2, , yn)
d(x, y) (x1y1)2+(x2y2)2(xnyn)2
x1, x2, 1 d (xn1, p)1 xn1 1
n1 1p F1 xnn n1 d(xn2, p) 21
xn2(n2 n1) p Fn11
xn3 , xn4 , xnj 1( j1) xnn nj d (xnj 1 , p)
j11
xn1 , xn2 , j p
9 26
271
-
59 3
9 2
5 X Y yf (x) x0 X
2
(1)x x0 f(x) f(x0)
(2) 2 xy d(x, y)
0 d(x0, x ) d( f(x0), f(x ))
0
yf(x) x0 (1)
d(x0, x ) d( f(x0)f(x )) 0
0
d(x0, x) d( f(x0)f(x))
x x
n n1 x 1 x1, x2, , xn
n
d(x0, xn) n
1 d( f(x0)f(xn))
n d(x0, xn)
d( f(x0), f(xn)) f(x) (1) x0
0
yf(x) x0 (2) X x1,
x2, , xn n x0 1
Y f(x1), f(x2), , f(xn) f(x0)
0 N
n N d( f(x0)f(xn))
n f(x) (2)
0
0 272
-
d(x0, xn) d( f(x0)f(xn))
x1, x2, , xn x0
N
n N d(x0, xn)
N n N d( f(x0), f(xn))
n n
d(x0, xn) d( f (x0)f (xn))
(2) f(x)
(1)
X Y f(x) X X
X x
BX(x, )
5 (2)
0 x BX(x0, ) f(x )BY( f(x0), )
0
6f X Y Y V
f 1(V) X f X
5
f V Y x f 1(V)
f 1(V) BX(x, ) f1(V)
f(x) V V
BY( f(x), )Vf xBX(x, )
f(x)BY( f(x), ) x BX(x, )
x f(x)V x f 1(V)
f Y V f 1(V) X
Y V f 1(V) X
273
-
59 3
x X BY( f(x), )
Y f 1(BY ( f(x), )) X
x BX(x, ) f1(BY ( f(x), ))
f (BX(x, )) BY( f(x), )
x BX(x, ) f(x) BY( f(x), )
f
7f X Y Y G
f 1(G) X f X
Y G X \ f 1(G) f 1(Y \G) f 1(G)
X G Y f
Y \G f 1(Y \G)
G f 1(G)
G f 1(G) Y \G
f 1(Y \G) f
8X, Y, Z fX Y, gY Z X YY
Z X Z f g g
fX Z
V Z g g1(V )
f f 1(g1(V )) f 1(g1(V ))
g V Z Y X
gf V X (gf )1(V )
gf
274
-
10 3
9(1)X, Y n m
XY U X
Y
U (v, w)
(x, y) XYd(x, v)1, d(y, w)2U (7)
1 0, 2 0
(2) X, Y (1) XY U
XY
U (v, w) v V, w W VW U
X V Y W
(1) XY U U (v, w)
(x, y) XYd((x, y) (v, w))U (8)
XY
x(x1, x2, , xn) v(v1, v2, , vn) y(y1, y2, , ym) w(w1, w2, , wm)
d(x, v) (x1v1)2+(x2v2)2(xnvn)2
d(y, w) (y1w1)2+(y2w2)2(ymwm)2
(x, y)(x1, x2, , xn, y1, y2, , ym), (v, w)(v1, v2, , vn, w1, w2, , wm)d((x, y)(v, w))
(x1v1)2+(x2v2)2(xnvn)2(y1w1)2+(y2w2)2(ymwm)2
U (v, w)(8)
1 2 11
2222 d(x, v) 1, d(y, w)2
d((x, y) (v, w)) (x, y) U (7)
1, 2
275
-
59 3
(x, y) XYd(x, v) 1, d(y, w)2U
1, 2 d((x, y) (v, w)) d(x, v) 1
d(y, w)2 (x, y) U (8)
(2)U XY (1)
1 0, 2 0
(x, y) XYd(x, v) 1, d(y, w)2U
V x Xd(x, v) 1, W y Yd(y, w)2
V X W Y vV, wW, VW U
U XY U (v, w) X, Y V,
W vV, wW VW U (v, w)
xd(x, v)1V
yd(y, w)2W
1 0, 2 0
(x, y) XYd(x, v) 1, d(y, w)2VW U
(1) U XY
11
11.1
0 1 1
1 1
276
-
2
1 2
2 3
3
2
3
4 5 , 4
5 n
n 1
2 2 2 x (x1, x2)y (y1, y2)
2
xy (x1y1)2+(x2y2)
2 (xiyi)2
i1
2
23
2 x(x1, x2, x3)y(y1, y2, y3) 2
xy (x1y1)2+(x2y2)
2+(x3y3)2 (xiyi)
2i1
3
2
n 2 x (x1, x2, , xn) y (y1, y2, , yn)
xy (xiyi)2
i1
n
n
(1) x, y n xy 0
277
-
59 3
(2) x, y n xyyx
(3) x, y, z n xy yz xz
(4)xy 0 xyx y
(1), (2), (4) (3) 2
1 3
2
(aibi)2
i1
n
i1
n
ai22
i1
n
ai bii1
n
bi22
ai bii1
n
ai2
i1
n
bi2
i1
n
(aibi)2
i1
n
i1
n
ai22
i1
n
ai bii1
n
bi2
ai22
i1
n
ai2
i1
n
bi2
i1
n
bi2
i1
n
ai2
i1
n
bi2
i1
n2
ai xiyi, bi yizi
(xizi)2
i1
n
(xiyi)2
i1
n
(yizi)2
i1
n
n 3 x x1, x2, , xn, y y1,
y2, , yn, z z1, z2, , zn
xz xy + yz
4 X V
V v V x Xxv V
278
-
0
X
1 (0, 1)
1
0 v 0
212
10 x xv 2
1 (0, 1)
[0, 1]
1
0 1
11.2
5 X
x, y
d(x, y) d
(X, d)
(1) x, y X d(x, y) 02
(2) x, y X d(x, y) d(y, x)
(3) x, y, z X d(x, z) d(x, y)d(y, z)
(4) x y d(x, y) 0 2
2
279
-
59 3
n
1 x
1 x 1 x 0
6(X, d) X x r 0
BX (x, r) X x r
BX (x, r) xXd(x , x) r
x r X
3
7(X, d) X V
V v V BX (v, ), V 0
4
10X d x0 X
r 0 r BX (x0, r) X
x BX (x0, r) 0 BX (x, ) BX (x0, r)
r d(x, x0) d(x, x0) rx BX (x0, r)
0 xBX (x, )
d(x , x0) d(x , x)d(x, x0)d(x, x0)r
xBX(x0, r) BX(x, ) BX(x0, r) BX(x, )
BX(x0, r) BX(x0, r)
1
280
-
[0, 1]
y f(x), yg(x)(y , x [0, 1]) 2
d(f, g)
[0, 1]max f(x)g(x)
[0, 1] f(x) g(x)
[0, 1]
f(x)g(x) f(x) g(x) f(x) g(x)
d(f, g) 0
[0, 1] f(x)g(x) d(f, g)0
f(x) g(x) d(f, g)0 3 y f(x), yg(x), yh(x)
d(f, h)[0, 1]
max f(x)h(x) [0, 1]
max f(x)g(x)g(x) h(x)
[0, 1]
max f(x)g(x) [0, 1]
max g(x)h(x) d(f, g)d(g, h)
[0, 1]
11.3
8 X
X
(1) X
(2)
(3)
X X (X,
) X
281
-
59 3
11X X
(1) X
X X x
X
(2) X A U A
U x U x
A V BX (x, ) V
V U BX (x, ) U U
(3) V1, V2, , Vk X
V V1V2 Vk x j x Vj
V1, V2, , Vk j
BX (x, j) Vj 1, 2, , k 1, 2, , k
k j 0
x Vj j
BX (x, ) BX (x, j) Vj BX (x, ) V V
(3)
1n (n1
, 1n1
)
n 1, 2, , n ( n1
, 1n1
)
[0, 1]
0 282
-
11.4
9X X
F X \ FF X
X X X
X X
1 F[0, 1]
\ F( , 0)(1, ) ,
v x
xv \ F 0
F
1 [1, 1]1 1
(1, 1)1 1 (1, 1]1 1
X [2, 1] (1, 1]
1 (3) 1
[2, 1] [2, 1]
X
X
12
(1) X
(2)
283
-
59 3
(3)
(2)
(3)
(2) 1
n [ n1
, 1n1
] n3, 4, ,
0 1 n [ n1
, 1n1
]
n
(0, 1)
11.5
10 X
x y X x Uy V UVU
V U V
13
XdxyX
21
d(x, y) x, y BX (x, ) BX (y, )
x, y
BX (x, )BX (y, )
d(x, z), d(y, z) zX z BX (x, ) BX (y, )
d(x, y) d(x, z) + d(y, z)
221
d(x, y) BX (x, )BX (y, )
X
284
-
11.6
X A X A V
VA A A A
VA
V
A A
2 V1, V2 V1A, V2A V1 A
V2 A
2 (V1 A)(V2 A) A V1
V2 (V1V2)A
VA V
A
A
2 V1, V2 V1A, V2A (V1A) (V2A)
A V1 V2 (V1V2)A
14X X A
A
A X A V1V2
Vr V1,V2, ,Vr
A B X V VA
A A V1V2 Vr
A V1A, V2A, , VrA
15A X A
285
-
59 3
A X X \ A
X A
X \ A X X
A
14
A
16X K X
x X \ K x V, KW, VW X
V, W
X y K x Vx, y, y Wx, y,
Vx, y Wx, y Vx, y, Wx, y K
K Wx, y1 Wx, y2 Wx, yr y1,y2,
, yrK y K Wx, y K
K K
VVx, y1 Vx, y2 Vx, yr, WWx, y1 Wx, y2 Wx, yr
V, W x V, KW VW Vx, y1
Vx, y2 Vx, yr Vx, yi(i1, 2, , r) Wx, y1 , Wx, y2, ,
Wx, yr WWx, y1 Wx, y2 Wx, yr
17
K X
16xX \ KxVx, VxKVx
X \ K x Vx
Vx K K Vx
X \ K X \ K K
286
-
11X d X A
x, y A d(x, y) K K
A diamA
diamA A x, y d(x, y)
7K n K
K n
17
K m Bm n
m Bm xn x m x x
x
Bmm n K K
Bm1 Bm1 Bmk m1, m2, , mk Bm
M m1, m2, , mk K BM K
K K
C L
C (x1, x2, , xn)nL xj L.j1, 2, , n
4 [L, L]
C [L, L] n 5 C
K C
15K
287
-
59 3
11.7
12 X x1, x2, X p
p U N
j NN xj U
p p U
p
1
18 X 1
(xjj ) 1, 2, 3,
(xj) (xj) p q 2 X
p U, q V , UV U, V p, q
j N1 j N2 xj U, xj V
N1, N2 xj UVUV
1
1 1, 2, 1, 2,
1, 2, 1 2 1, 2, 3, 4, 5,
19X F X (xjj ) F (xj
j ) X p p F
p F F X \ F
F X \ F (xjj ) p
j N xj X \ F N
xj F p F
288
-
20f X 1
f
X
3 f(X) m
(m, m) m 1
f(X)
I1, I2, , IkM (M,
M) I1, I2, , Ik f(X)(M, M)
f
21f X
xX f(u) f(x) f(v)Xu, v
minf f(x)xX , Msup f(x)xX minf f(x)xX
xX m f(x) Msup f(x)xX
xX M f(x) mM
f(x)
X x f(x) xX
f(x) M xX g(x)Mf(x)1
g(x)Mf(x)1
y f(x) zMy1
M f(x)
M f(x) M M f(x) M M
20 f(v)M v
X x X f(x) m x X
h(x) f(x)m1
289
-
59 3
m f(x) m f(x) m
m f(x) m m f(u)m u
X m M f(u) f(x) f(v)
11.8
13X x X N x X
x U U N U N x
22X X V V
X
V V
V
V v Uv, Uv V Uv
V Uv V v
v Uv Uv Uv V Uv
V
14
(1) X A A
XXA
(2) X A A X
X A A0
A
(1)A A
(2) F A F F
290
-
A
(1) A A0 A
(2) U A U A0 A0 U
23X A A x1, x2, x3, X
p p A
F A j xj F
p F 19
p
p A
1 (0, 1)1, 21
, 31
, 0
0 (0, 1) [0, 1]
11.9
15X d
X x1, x2, x3,
j N, k N j, k d(xj, xk)
N
X X
X
(0, 1) 21
, 31
,
n(n1) 1 n
n1
n11
-
59 3
24X X
X
x1, x2, x3, X X p
xn1, xn2, xn3 p
x1, x2, x3, 0 m Nn N
m, n d(xm, xn) 21N2
1
p N
nj N nj d(xnj, p) 21
n N
d(xn, p) d(xn, xnj)d(xnj, p)212
1
x1, x2, x3, p
X
6
160 X
X X
X X Bi (i1, 2, , k)
X XB1B2 Bk
X A A(B1 A) (B2 A) (Bk
A) Bi A A
25X X
0 292
-
X
X 3 X
X 3
X m k, n k, mn d(xm, xn)
k1 x1, x2, , xk1 xm
BX (xm, ) 23X
mk BX (xm, ) X
BX (xm, ) xk m k
d(xm, xk) k
x1, x2,
X X
26X A A
diamAdiam
A diam A diam x, y
d(x, x), d(y, y) A x, y
x d(x, x)
A x BX (x, ) A
x
d(x, y) d(x, x)d(x, y)d(y, y)diam A2
0 d(x, y) diamA2
d(x, y) diam A d(x, y)diam A0
21d(x, y)diam A d(x, y) diam A2
diam diam A diam A diam diam A diam
293
-
59 3
27
X X
X
X x1, x2, x3, X
X
0 X 26 X
X
X X
0
X
F1 F2 F3 Fn 2n
1
X F1 ,
F2 , F3,
X 21 F1 X
F1 0
F1
2214
1 F2 n
n xn Fn mn xm, xn
xm, xn Fn diamFn2n1 d(xm, xn)2n
1
x1, x2, x3, X p
294
-
Fn m n xm Fn n
pFn
X
V pV BX (p, ) X p
BX (p, )V V
n 2n1
p Fn
diamFn FnBX (p, ) V
Fn x1, x2, x3,
X
8X d 3
(1)X
(2)X
(3)X
6 242527 (1) (2) (2) (3) (3) (1)
(1), (2), (3)
11.10
28(X, d) d
X X
1
X 1 X
x x 2x B(x, 2x) 1
295
-
59 3
x x x
B(x, x) X x B(x, x) X
X
B(x1, 1)B(x2, 2) B(xr, r)X, i xii1, 2, , r
x1, x2, , xr 1, 2, , r
A X u
u 1 r i B(xi, i) A
v
d(v, xi) d(v, u)d(u, xi)i 2i
AB(xi, 2i) B(xi, 2i) 1
A
X X
1
X, Y dX, dY fX Y X
Y f
0 0 dX (x, x) X x, x
dY (f(x), f(x)) x, x
9X, Y X X
Y
dX, dY X, Y fX Y X
Y yY
VyxXdY (f(x), y) 21
f(x) y 21 x y 2
1
296
-
Y BY (y, 21) Vyf
1(BY (y, 21)) BY (y, 2
1)
f y Vy X
xX xVf (x)
Vy VyyY X
28 X
Vy 0 x, x dX (x, x)
x, xx x 2 dX (x, x)
xVy xVy y Y
dY (f(x), y) 21, dY (f(x), y) 2
1
dY ( f(x), f(x)) dY(f(x), y)dY(f(x), y)
fX Y
11.11
17 X X f(x) X
x x idX A
X A A X
iA X 1 Y
X Y
18XY X
Y h
(1) h Y X h1
(2)h h1
f(x) x 1
297
-
59 3
Y y Y x
y Y x 1
f(x) x 1 X Y 1 1
h Y
h f(x) x
h
Y 1 x
X Y X Y
X Y h h1
X h X, Y h1
X h1hidX
h1 h Y hh1idY
19xyz
R
(1)xRx
(2)xRy yRx
(3)xRy yRz xRz
29
X X X
X X f X Y
f 1 Y X X Y
Y X
X Y Y Z fX Y gY
Z f, g f 1, g1
298
-
gf (gf)1f1g1
gf X Z X Z
2
2 2
1t (t1) 2
t
r(0)
( r)
r
r
31 X Y(1, 1) X
Y
f(x)1 xx
, xX
X Y f(0)0, x
f(x) 1, x f(x) 1
y1 xx
x x 0 y 0 x1yy
x 0 y 0
299
-
59 3
x1yy
x1 y
y
f
f 1(y)1 y y
f1(0)0, y 1 f1(y) , y 1
f1(y)
300
-
The Doshisha University Economic Review Vol.59 No.3
Abstract
Yasuhito TANAKA, Introduction to General Equilibrium Theory with a Proof of
Brouwer's Fixed Point Theorem
I present an introductory explanation of the proof of the existence of
general equilibrum mainly in an exchange economy. Present elementary proofs
of Sperner's lemma and Brouwer's fixed point theorem are also indicated. The
appendices provide expositions of related mathematical concepts.
301
1223456781921031111.111.211.311.411.511.611.711.811.911.1011.11
Abstract