Учебное пособие 800365
.pdf/ 5
3 / 3 ! #
N @ 3 3 3
3 3 3 5
3 3 #
! ( 2 - # # &
,# ? X = {|x − x0| r; x, r R} J 3
x0. 3 3 τ = {X, R1, }
Y |
|
|
|
! 3 (x0, y0). |
|
! |
|
@ 3 |
|||
? X = |
(x − x0)2 + (y − y0)2 r; (x, y) R2, r R J 3 |
||||
|
|
|
|
|
|
3 τ = {X, R2, } Y |
|||||
O# ? X |
= { |
(x − x0)2 + (y − y0)2 + (z − z0)2 |
r; (x, y, z) R3, |
r R} J 3 ! @ 3 ! 3
(x0, y0, z0). 3 τ = {X, R3, } Y
B# 3 3 3 XY 4F 5
3 #6
+# ? X = {(−∞; a), a R} J 3 3
# 3 3 τ = {X, R, } Y
4H #6
P# 3 Q / 5
3 Y
-# ? X = {[a, b); a, b R} J 3
Q # 3 3 τ = {X, R, } 5
Y
(# 3 Q5
/ 3 Y
.# ? X J 3 3 3 R
# 3 3 τ = {X, R, } Y 4H M #6
,'# 3 τ = {X, R, }/ X J 3 /
Y 4H #6
,,# ? X = {(a, +∞), a R} J 3 3
# 3 3 τ = {X, R, } Y 4H 5
#6
, " 3 R / 5
3 #
,O# " 3 R M # ,B# " 3 R / M 5
3 #
,+# ? 3 R 3 3
/ / M 3
#
* 7 * " ( ',-*. *
G 3 3 3 / 5
3 J 4 3 5
36/ 3 3 @ 3 4 6 5
/ 3 3 3 !
3 / 3 Ax = b
J # ? / 3 5
3 /
@ #
$ 3 &
F E
# 7 - x& x
x E 0 0 ! 7 ,6 x = 0 x = θ:
&6 |
αx = |α|x |
α R, x E: |
O6 |
x1 + x2 x1 + x2 |
x1, x2 E# |
9 / 3 /
#
3 /
3 r(x, y) = x −y 3 3 G 3 3 /
3 3 7
6 r(x, y) = r(y, x) 4 33 6:
6 r(x, y) = 0 x = y 4 6:
6 r(x, y) r(x, z) + r(z, y) 4 6#
* 3 / 3 / 3 / 3
3 7
x, y, z E, α R7
6 r(x, y) = r(x + z, y + z) 4 6:
6 r(αx, αy) = |α|r(x, y) 4 6#
F 3 3 3 5
# F 3 /
f (α, β, x, y) = αx + βy 3 5
@ R/ 3 /
3 @ 3 / ! 5 3 α, β, x, y# * / R 3 3
3 x = |x|/ x J @ #
ρ x0 3 5
(E, · E ) 3
B(x0, ρ) = {x E, x − x0 E ρ}.
'7 ρ x0 3 5
(E, · E ) 3
S(x0, ρ) = {x E, x − x0 E = ρ}.
" 3 @ 3 5
3 # ? G 3 3 3
G 3 3 3 7 #
? G 3 x1, x2, ..., xn, ... E 3 @
G 3 a E/ r(a, xn) → 0 n → ∞#
? x1, x2, ..., xn, ... E 5
* / 3 / m, n → ∞ lim r(xm, xn) = 0#
@ 3 4 G 6# 1 3 / 3 5
3 G 3 5 / & /
! J " A #
? 3 3 3 5
3 3 (x, y) J # 1
/ #
? 3/ 3 3 x = (x, x)
O6 4 3 5
6# ) / 3
* 5A |(z, y)| x y
x + y 2 = (x + y, x + y) = (x, x) + 2(x, y) + (y, y)
x 2 + 2 x y + y 2 = ( x + y )2,
#
? 3 3 3 3
! #
? E J n53 3 e1, e2, ..., en
x E [x]e = (x1, x2, ..., xn)T G 3 # H
p, 1 p ∞/ 3
x p = |
|
|
1 |
|xn|p |
|
|
, |
|
|
|
|
|
|
1 p < ∞, |
|
(1) |
||||||||||
|
|
n |
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||
|
|
|
|
|
|
| |
|
1 |
| | |
|
|
2 |
| |
|
|
| |
n |
| |
|
|
|
|
∞ |
|
|
|
|
|
|
|
|
|
|
|
x |
|
, ..., |
|
), |
|
p = |
. |
|
|
|||||||||
|
|
max( x |
|
, |
|
|
x |
|
|
|
|
|
|
|||||||||||||
? E = C[a, b] |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
[a, b] |
|
|
|
J |
|
5 |
|||||||||||||||||||||||
! # H p, 1 p ∞/ |
|
f C[a, b] 3 |
|
|
|
|||||||||||||||||||||
|
|
|
|
b |
|
f (t) p dt |
p |
, |
|
|
|
1 p < , |
|
|
|
|||||||||||
f p = |
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
(2) |
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||
|
|
|
a |
|
| |
|
|
| |
|
|
|
|
|
|
|
|
|
|
∞ |
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
| |
|
|
| |
|
|
|
|
|
|
|
|
= |
∞ |
|
|
|
|||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
sup( f (t) , t |
|
|
[a, b]), |
|
p . |
|
|
|
||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
E0 · 1 · 2
E - & 0
0 < c1 < c2 < ∞ &
c1 x 2 x 1 c2 x 2
0 x E# & 5 6&
0 & 0
#
- x
- &
5 H 6 0
x#
F G 3 3 T(U#
H 3 3/ 3 / 3 4,6/ G 5 3 3 / c1, c2 3 5
c2/c1 → ∞ n → ∞#
F / 3 / 3 4&6/ G 3
C[a, b]/ · ∞ G 3 4 5 36# ? G 3/ 3 · p, p < ∞, @ ! / @ 5
! # F 3 / [0, 1] 5
xn(t) = tn, n = 1, 2, .../ @ ! /
, , ' [0, 1]#
? A J / @ 3 5
(E, · E ) 3 (F, · F )# F 3 3/ 3/
3 ! 5
3 ! 7 A(αx + βy) = αAx + βAy#
F A E F
A = sup Ax F .
x E, x E=1
, - & #
#
H x E, x = θ
|
A |
x |
F |
A |
|
, |
|
A |
x |
|
= |
Ax F |
, |
|
x E |
|
|
x E F |
|
||||||||
|
|
|
|
|
x E |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
Ax F A x E , |
(3) |
|||||||||
x E# |
|
|
|
|
|
|
|
|
|
|
|
||
|
) LB(E, F ) & |
||||||||||||
0 E F & |
& |
#
I E & 0
& 0 E F #
, A : E → F, B : F → G ! &
BA&
&
BA B A . |
(4) |
? 3 A 3 #
,6 1 A = 0/ 4O6 / A = O#
&6 |
αA = |
sup |
αAx F = |
sup |α|Ax F = |α|A . |
||
|
|
x E, x E=1 |
|
x E, x E=1 |
||
O6 |
? 3 O6 B#,/ 3 3 A + B = |
|||||
= sup Ax + Bx F |
sup |
( Ax F + Bx F ) |
||||
x E, x E=1 |
|
|
x E, x E=1 |
|
||
|
|
sup |
Ax F + |
sup |
Bx F = A + B . |
|
|
x E, x E=1 |
|
x E, x E=1 |
H / @ n53 5
m53 3 3 mn/ 5
/ 3 G / / #
E 3 3 3 5
# 4O6
BA = sup BAx G |
sup B Ax F = B A . |
x E, x E=1 |
x E, x E=1 |
H 3 #
1 F = R/ f LB(E, R)
G 3 E # ? G 3 f LB(E, R)
3 3 7/ 3 3 5
3 E/ 3 3 LB(E, R) 3
3 E #
* 3 3 4 3 6/ E 3
3 / 5
G 3 {xn} E x E /
f E limn→∞ f (xn) = f (x)#
% E 3 / 5
E # ?
{fn} E f E /
x E {fn(x)} f (x)#
" @ 3 % 5A #
, E
#
) D &
0 a b -
ta + (1 − t)b& t ! 0 J ?# $
ta + (1 − t)b, 0 t 1, & 0 ab#
H 3 D /
/ ! D#
H 3 D / 5
# F 3 / 3 3
3 3 # 3 3 /
3 # ) 3 /
@ ! J / 3
3 #
! Z4[6/ 3 D/ / a, b D 0 t 1
F (at + (1 − t)b) tF (a) + (1 − t)F (b).
R / 7
& # ' & 0
7 & D&
#
( ) ) ! / !
F (x)
F (α1x1 + α2x2 + ... + αnxn) α1F (x1) + α2F (x2) + ... + αnF (xn),
α1, α2, ..., αn > 0, α1 + α2 + ... + αn = 1/ x1, x2, ...xn D#
? F J ! 3 R = {x D, F (x) r}
# H #
) / a, b R# H F (at + (1 − t)b) tF (a) + (1 − −t)F (b) tr + (1 − t)r = r# H 3 3 3 3 a b
3 R / @ G #
?# $ #&
#
@# B(x0, r) #
A# 7
- #
/
/ x0 ! F (x) = x − x0 E
#
) / O6 3 4 6 3 3
F (at + (1 − t)b) = at + (1 − t)b − x0 E = (a − x0)t + (1 − t)(b − x0) E
(a − x0)t E + (1 − t)(b − x0) E = t a − x0 E + (1 − t) b − x0 E =
=tF (a) + (1 − t)F (b)# N #
H / ! Ax F / A J
/ ! #
M 3 3/ |
B(x0, r) |
S(x0, r) = {x, |
x − x0 E = r}# 23 G 3 3 4B#O6 3 3 |
, ! 3 #
0
D # 9
0 x D
a1, a2, ..., am t1, t2, ..., tm, t1 + t2 + ... + tm = = 1& x = t1a1 + t2a2 + ... + tmam#
! 3 / 5
3 · 1 4B#,6/ / 3 5
33 ,# R /
3 3 ! 3 3 (±1, 0, ..., 0), |
(0, ±1, 0, |
..., 0), ..., (0, 0, ...0, ±1)# 9 |
|
· 1 0 & #
? G 3 A n53 3 E/ 5
3 3 · 1/ 3 e/ 3
3 3 7
A = max{ ± Aej 1, j = 1, 2, ..., n} = sup{Aej 1, j = 1, 2, ..., n}.
? (aij ) = [A]e→e J 3 ! / ! 3 / /
7 [Aej ]e = (aij )n=1# R
i
|
|
|
n |
|
|
|
|
|
|
|
Aej 1 |
= i=1 |aij | / / 3 / 3 |
|||||||||
|
1/ |
G 3 3 ! 3 |
||||||||
· |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
n |
|
|
|
|
|
|
|
|
A |
1 |
max |
| |
a |
ij | |
. |
|
|
|
|
= 1 j n 1 |
|
|
) 3 · ∞ /
! # R /
0 zk & ±1# ( ) F 0 - A ∞#
F 3 ·p/ 3 4,6/ @ 3 5
p = 1, 2, ∞# ? 3 3 5
A/ @ 3 3 3
E/ 3 G 3#
? [A] = [A]e→e = (aij )ni,j=1 J 3 ! A [x]e = = (x1, x2, ..., xn)T J ! E# H
/ [Ax]e = [A][x]e#
E 3 3 3 · 1# ?
Ax 1 = [A][x]e 1 = |
n |
|
n |
|
i=1 |
k=1 |
aikxk .
3 / 3 33
33 @ 3 / 3
|
n |
n |
|
|
n |
n |
|
|
|
n |
|
n |
Ax 1 |
k |
|
|
|
|
|
|
|
||||
|
|aik||xk| = |
|
|
|aik||xk| = |xk| |aik| |
||||||||
|
i=1 |
=1 |
|
|
k=1 i=1 |
|
k=1 |
|
i=1 |
|||
|
|
n |
|
|
n |
|
|
|
n |
|
|
|
|
|
k |
k| |
k |
|
ik| |
|
k |
|
ik| |
1 |
|
|
| |
| |
|
| |
. |
|||||||
|
|
x |
|
max |
a |
|
= max |
a |
x |
|
||
|
|
=1 |
|
|
i=1 |
|
|
|
i=1 |
|
|
|
R / A 1 |
|
n |
|
|
|
|
|
|
|
|||
max |aik|# " / |
ki=1
ek/ / k5
" / 3 3 |
|
|
|
n |
|
|
|
|
|
|
|
|
|
|
|
n |
|
|
||||||
|
|
|
|
a |
|
|
|
|
|
A |
|
max |
|
|
|
|||||||||
! / 3 3 |
e |
k 1 |
= 1, |
|
Ae |
|
|
= i=1 | |
ik| |
/ |
|
1 |
a |
ik| |
# |
|||||||||
|
|
|
|
k 1 |
|
|
|
|
|
k |
i=1 | |
|
||||||||||||
|
|
|
|
|
|
|
3 3 3 |
|
|
|
|
|
|
|||||||||||
|
|
|
|
|
|
|
|
|
|
n |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
A |
1 |
= |
max |
i |
| |
a |
ik| |
. |
|
|
|
|
|
(5) |
||||||
|
|
|
|
|
|
|
|
|
||||||||||||||||
|
|
|
|
|
k |
=1 |
|
|
|
|
|
|
|
|
||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
M 3 3 3 · ∞# ) x ∞ = max |xk|/
k
n |
n |
n |
|
|
|
∞ |
|
i |
k |
|
ik |
|
|
|
|
| |
ik|| |
|
k| |
i |
|
ik| |
k | |
|
k| |
|
|
|
Ax |
|
= max |
|
n |
a |
x |
|
|
max |
a |
|
x |
|
max |
a |
|
max |
x |
|
. |
|||
|
|
|
|
|
|
|
=1 |
|
|
|
# |
|
k=1 |
|
|
|
|
|
k=1 |
|
|
|
|
|
H |
|
A |
|
max |
|
a |
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
|
|
∞ |
n |
|
k |
| |
ik| |
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
|
|
i |
=1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
? max |aik| i = j# 3 3 x
ik=1
3 xk = |ajk|/ajk/ ajk = 0 , 3 # R /
x ∞ = 1/
∞ |
i |
|
n |
ik k |
|
|
jk k |
= |
| |
|
jk| |
|
||||
|
|
k=1 |
|
|
|
k=1 |
|
|
|
k=1 |
|
|
|
|||
Ax |
= max |
|
a x |
|
|
|
|
|
|
a |
|
, |
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
n |
|
|
|
|
|
|
|
|
|
|
|
|
∞ = |
|
k |
|
ik| |
|
|
|
|
|
|
||
|
|
|
a |
|
|
max |
|
a |
|
. |
|
|
|
|
(6) |
|
|
|
|
|
|
|
|
|
=1 |
|
|
|
|
|
|
|
|