- 2
2.1 : (A:XY , .. $ . A-1:YX) Û (A:XY , .. ).
-. Þ? $ . A-1:YX Þ A:XY ?
1) Ax1 = Ax2 =( A-1, )Þ A-1(Ax1) = A-1(Ax2) =( -1)Þ A-1Ax1 = A-1Ax2
=(-1=1X)Þ x1=x2 =[ : (Ax1 = Ax2 Þ x1=x2) Û x1¹x2 Þ Ax1 ¹ Ax2)]Þ .!
2) "yÎY =(A-1:YX)Þ x = A-1yÎX =( )Þ
x = (A-1y) =( A-1)Þ x = A-1y =(A-1 = 1Y)Þ x = y Ü( . , imA)Þ imA = A(X) = YÜ( .)Þ .!
, 1) 2) !
Ü? A:XY Þ $ . A-1:YX?
YÈX:
A-1 = {<y,x>ÎY´X | <x,y>ÎA}ÍY´X,
.. x = A-1y Û y = Ax,
. A-1:YX:
1 A-1 Y X? (A-1:D(A-1)ÍYX?)
2 D(A-1) = Y? (A-1:YX?)
3 -1=1X & A-1 = 1Y? (..-1 ?).
1 Ü( )Þ x = A-1y & x¢ = A-1y =(?)Þ x=x¢.
x = A-1y & x¢ = A-1y Ü( -1)Þ y = Ax & y = Ax¢ =( =Y)Þ Ax=Ax¢ =(A )Þ x=x¢!
2 Ü( )Þ D(A-1)ÍY & YÍ D(A-1)? Ü( D(A-1) = {yÎY | $xÎX: x=A-1y}ÍY)Þ YÍ D(A-1)? Ü( )Þ yÎY Þ yÎ D(A-1)?
yÎY =( )Þ $xÎX | y = Ax Ü( -1)Þ $xÎX | x = A-1y =( D(A-1))Þ yÎ D(A-1)!
3 Ü( )Þ
3.1. xÎX =(?)Þ x = A-1Ax
3.2. yÎY =(?)Þ y = AA-1y
xÎX =(A:XY)Þ y =AxÎY Ü( -1)Þ y =Ax & x = A-1y =(
y = Ax )Þ x = A-1Ax!
yÎY =(A-1:YX)Þ x = A-1yÎX Ü( -1)Þ y =Ax & x = A-1y =(
x = A-1y )Þ y = AA-1y!
1,2,3 ¨
1. A, B, C . .
1. : () = ().
2. .
3. .
4. .
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5. , , .
6. ()-1 = -1 1.
7. S[a, b] É B[a, b] É C[a,b] É Ck[a,b] É ∞ [a,b] ( , , ).
8. 1 < p < q Þ l1 Ì lp Ì lq Ì l∞ Ì l ( , , ).
9. Û Ax = y " y ÎY.
10. Û Ax = y " y ÎY.
11. Û Ax = y " y ÎY , x = A-1y, A-1 - .
12. .
(1 ). : () = ().
-. "x (AB)Cx =( )= [()] =( )= A(B[C(x)]) =( )= A(BC(x)) =( B )= A(BC)x =( )Þ () = ()
2. () . .
.
1. X= l 2; Ax = {x3,x4,}:XX; Bx = 
Fx = x1 + x2:X R;
x0 = 
. FABx0 =? (: 
)
2. X=C[0,1]; (Ax)(t) = t2x(t):XX; (Bx)(t) = 
Fx = 
x0(t) = t; FABx0 =? (: 
)
3. X=C[0,1]; (Ax)(t) = x(t2):XX; (Bx)(t) = x2(t):XX; Fx = 
x0(t) = 
FABx0 =? (: 
)
4. X=Ã([0,2]); Ax = xÇ[0,1]:XX; Bx = [0,2]\x:XX; Fx = sup x:X R; x0 = {0,1,2}; FABx0 =? (: 1)
3. "" ( , , - 1). () (mod12) , .
1. - 1.
| I(x) = x | |x1 ¹ x2 Þ Ax1¹Ax2 | ||
| . | . | ||
-Fx = 
| Fx = x(0) | ||
| . . Í S[a,b], | .. - , ¥[a,b] | ||
| x Î X, = | |||
| Lp[a,b] | L1[a,b] | , p>1 | |
| C3[a,b] | C2[a,b] | , a<b | |
| a(t) < 0 "t | A-1a = A1/a | . -, aÎS(T,R) | |
| Ax = y | - -, Ax = y | ||
| AB | - -, A,BÎS(X) | ||
| B | BA | - -, A,BÎS(X) | |
| A- | A | - -, AÎS(X) |
2. - 2.
| R Í C´U | : A(X) = Y | ||
| . U, Aú U | Aú U Î S(U,Y): Aú U (x) = A(x) " x Î X | , AÎS(X,Y) U Í X | |
| Ax = dx/dt | (.) | ||
| . . ÍS[a,b], k | . . Í S[a,b], | ||
. . ÍS[a,b], (..  < ¥)
| S(X,Y) | ||
| S(P, PR) | , Ax = y | ||
| B[a,b] | C[a,b] | , a<b | |
| A-1a = A1/a | a(t) > 0 "t | - -, aÎS(T,R) | |
| xÎC[-1, 1] | xÎC1[-1, 1] | -, x(t)=|t| | |
| Ax = y | - -, Ax = y | ||
| A | BA | - -, A,BÎS(X) | |
| A- | A | - -, AÎS(X) |
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3. - 3.
| . | . | Ax1 = Ax2 Þ x1 = x2 | ||
| . | A, | ||
| . | . | ||
| . . Í S[a,b], | .. , B[a,b] | ||
| .. , L1[a,b] | . . ÍS[a,b], p- (.. x,  p<¥)
| ||
| lq | lp | , p<q | |
| FABx, x(t) º 1 | FBAy, y(t) º 1 | , Bx(t)=tx(t), Ax(t)=ò[0,1]tsx(s)ds, Fx = x(1) | |
| A | - =y 1 | - -, Ax = y | |
| Ax = y | - -, Ax = y | ||
| AB | - -, A,BÎS(X) | ||
| AB | - -, A,BÎS(X) | ||
| A | A- | - -, AÎS(X) |
4. - 4.
| . | ., -1 | |||||
| . | . | x1 ¹ x2 Þ Ax1 ¹ Ax2 | |||||
| ., X Y - .. | , X - , - | |||||
| (Ax)(t) = =ò[a,b]k(t,s)x(s)ds, t Î [c,d] | ||||||
| S(X,Y) | . . Í S[a,b], | |||||
. . ÍS[a,b], (..  < ¥)
| S(X,Y) | |||||
| lp | l1 | , p>1 | ||||
| FABx, x(t) = t | FBAy, y(t) = t2 | , Bx(t)=tx(t),Ax(t)=ò[0,1]tsx(s)ds, Fx = x(1) | ||||
| a(t) < 0 "t | A-1a = A1/a | . -, aÎS(T,R) | ||||
| xÎC2[0, 2] | xÎB[0, +¥) | -, x(t)=|t| | ||||
| x = A-1y | - -, Ax = y | |||||
| AB | A B | - -, A,BÎS(X) | ||||
5. - 5.
| , . | . A:DA Í XY ½ DA = X | ||
| Aú U Î S(U,Y): Aú U (x) = A(x) " x Î U | . U, Aú U | ||
| A Î S(X): (x) (t) = a(t) x(t) | ., X Y - .. | ||
| S(X,Y) | |||
| . . Íl, (.. x = {xk }Î l, sup{ôxkô} < ¥) | .. , l¥ | ||
| y Î Y | |||
| C[a,b] | S[a,b] | , a<b | |
| FABx, x(t) = t | FBAy, y(t) = t2 | , Bx(t)=tx(t), Ax(t)=ò[0,1]tsx(s)ds, Fx = x(1) | |
| a(t) < 0 "t | A-1a = A1/a | . -, aÎS(T,R) | |
| xÎC[-1, 1] | xÎC1[-1, 1] | -, x(t)=|t| | |
| x = A-1y | - -, Ax = y | ||
| A B | AB | - -, A,BÎS(X) |
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6. - 6.
| , | () = (()) | , AÎS(X,Y), BÎS(Y,Z) | |
| A, | . | ||
| AÎ S(X,R) | |||
| d - | |||
| . . Íl, x = ={xk}Î l | å1,¥|xk|< ¥ | . . Íl, p- (.. x = {xk}Î l, å1,¥|xk|p < ¥) | ||
| .. , L1[a,b] | . . ÍS[a,b], p- (.. x, p<¥)
| ||
| y Î Y | x Î X, = | ||
| S[a,b] | C¥[a,b] | , a<b | |
| xÎl2 | xÎl¥ | -, x={xk=1} | |
| Ax = y | - -, Ax = y | ||
| x = A-1y | - -, Ax = y | ||
| B | BA | - -, A,BÎS(X) |
ã .. 2005. m_o.doc
