Bài tập toán cao cấp chương II

1
Ch
’
u
’
ong 2. Khˆong gian ¯di
.
nh chu
’
ˆan
10. Cho A, B l`a hai to´an t
’
’
u t´ıch phˆan trong C
[a,b]
v
´
’
oi ha
.
ch l
`
ˆan l
’
u
’
o
.
t l`a K(t, s), H(t, s)
Ax(t) =
b


a
K(t, s)x(s)ds, Bx(t) =
b

a
H(t, u)x(u)du.
Ch
´
’
ung minh B ◦ A c˜ung l`a to´an t
’
’
u t´ıch phˆan v`a ha
.
ch l`a
b

a
H(t, u)K(t, s)ds.
Gi
’
ai
(BA)x(t) = B(Ax(t)) =
b

a
H(t, u)Ax(u)du
=
b


a
H(t, u)

b

a
K(u, s)x(s)ds

du =
b

a

b

a
H(t, u)K(u, s)x(s)ds

du.
V`ı c´ac h`am H(t, u), K(u, s) liˆen tu
.
c trˆen {a ≤ t, s ≤ b} v`a h`am x(s) liˆen tu
.
c trˆen
[a, b] nˆen h`am H(t, u)K(u, s)x(s) liˆen tu
.
c trˆen {a ≤ u, s ≤ b}. Thay ¯d
’
ˆoi th

´
’
u t
’
u
.
l
´
ˆay t´ıch
phˆan ta ¯d
’
u
’
o
.
c
BAx(t) =
b

a

b

a
H(t, u)K(u, s)x(s)du

ds =
b

a

(H(t, u)K(u, s)du) x(s)ds.
Vˆa
.
y B ◦ A l`a to´an t
’
’
u t´ıch phˆan v`a ha
.
ch
b

a
H(t, u)K(t, s)ds.
Ch
’
u
’
ong 4. C´ac nguyˆen l´ı c
’
o b
’
an c
’
ua gi
’
ai t´ıch h`am
5. Cho X, Y l`a c´ac khˆong gian ¯di
.
nh chu
’

ˆan, M ⊂ X, v`a f : M → Y sao cho f(M)
compact. Ch
´
’
ung minh n
´
ˆeu G
A
= {(x, f(x)) : x ∈ M} ¯d´ong trong M × Y th`ı f liˆen tu
.
c
trˆen M.
Gi
’
ai
Gi
’
a s
’
’
u f khˆong liˆen tu
.
c trˆen M. T
`
’
u ¯d´o t
`
ˆon ta
.
i x ∈ M sao cho f khˆong liˆen tu

.
c ta
.
i
M, t
´
’
uc l`a t
`
ˆon ta
.
i d˜ay {x
n
} ⊂ M sao cho x
n
→ x nh
’
ung f(x
n
)  f(x).
Suy ra t
`
ˆon ta
.
i s
´
ˆo ε
0
> 0 sao cho ∀n, ∃n
k

> n (n
k
> n
k−1
) sao cho f(x
n
k
) − f(x) ≥
ε
0
.
Ta c´o {f(x
n
k
)}
k
⊂ f(M) v`a f (M) nˆen t
`
ˆon ta
.
i d˜ay con {f(x
n
k
j
)}
j
, v
´
’
oi

f(x
n
k
j
) → y ∈ f(M).
Khi ¯d´o
2
(x
n
k
j
, f(x
n
k
j
)) → (x, y).
Do G
f
¯d´ong nˆen (x, y) ∈ G
f
. Suy ra y = f(x). Do ¯d´o f(x
n
k
j
) → f(x) (vˆo l´y).
Ch
’
u
’
ong 5. Khˆong gian Hilbert

4. Cho X l`a khˆong gian Hilbert th
’
u
.
c v`a A : X → X l`a to´an t
’
’
u tuy
´
ˆen t´ınh liˆen tu
.
c.
To´an t
’
’
u A go
.
i l`a x´ac ¯di
.
nh d
’
u
’
ong n
´
ˆeu ∀x ∈ X ta c´o Ax, x ≥ αx, x, trong ¯d´o α > 0.
Ch
´
’
ung minh n

´
ˆeu A x´ac ¯di
.
nh d
’
u
’
ong th`ı A l`a song ´anh v`a A
−1
 ≤
1
α
.
Gi
’
ai
* A l`a ¯d
’
on ´anh.
* A l`a to`an ´anh ⇔ ImA = X.
A : X → ImA l`a song ´anh.
∀x ta c´o αx
2
= αx, x ≤ Ax, x ≤ Ax.x ⇒ αx ≤ Ax. Do ¯d´o A
−1
:
ImA → X liˆen tu
.
c v`a A
−1

 ≤
1
α
.
* ImA l`a khˆong gian con ¯d´ong c
’
ua X.
* X = ImA ⊕ (ImA)
⊥
. Ta ch
´
’
ung minh (ImA)
⊥
= {0}.
∀z ∈ (ImA)
⊥
th`ı Az ∈ ImA nˆen 0 = Az, z ≥ αz, z. Do ¯d´o z, z = 0. T
`
’
u ¯d´o
z = 0.
Vˆa
.
y A l`a song ´anh.
12. Gi
’
a s
’
’

u X l`a khˆong gian Hilbert, A : X → X l`a mˆo
.
t to´an t
’
’
u tuy
´
ˆen t´ınh. Ch
´
’
ung
minh r
`
˘
ang n
´
ˆeu v
´
’
oi m
˜
ˆoi u ∈ X, phi
´
ˆem h`am
x → Ax, u, x ∈ X
¯d
`
ˆeu liˆen tu
.
c th`ı A liˆen tu

.
c.
Gi
’
ai
* Ta ch
´
’
ung minh A l`a to´an t
’
’
u ¯d´ong ⇔ G(A) = {(x, Ax) : x ∈ X} l`a khˆong gian con
¯d´ong c
’
ua X × X.
Gi
’
a s
’
’
u {(x
n
, Ax
n
)} ⊂ G(A) v`a lim
n→∞
(x
n
, Ax
n

) = (x, y). Khi ¯d´o
lim
n→∞
Ax
n
, u = y, u (i)
∀u ∈ X. M
˘
a
.
t kh´ac, do phi
´
ˆem h`am c
’
ua ¯d
`
ˆe b`ai liˆen tu
.
c nˆen
lim
n→∞
Ax
n
, u = Ax, u (ii)
3
∀u ∈ X. T
`
’
u (i) v`a (ii), ta suy ra
y, u = Ax, u ⇔ y − Ax, u = 0, ∀u ∈ X

⇔ y − Ax = 0 ⇔ y = Ax.
Do ¯d´o (x, y) = (x, Ax) ∈ G(A).
* V`ı X l`a khˆong gian Banach nˆen A liˆen tu
.
c.
13. Gi
’
a s
’
’
u {e
n
}
n
l`a mˆo
.
t c
’
o s
’
’
o c
’
ua khˆong gian Hilbert X v`a
P
n
x =
n

k=1

x, e
k
e
k
, x ∈X, n = 1, 2, . . .
l`a d˜ay ph´ep chii
´
ˆeu tr
’
u
.
c giao. Ch
´
’
ung minh r
`
˘
ang d˜ay {P
n
}
n
hˆo
.
i tu
.
¯di
’
ˆem ¯d
´
ˆen to´an t

’
’
u ¯d
`
ˆong
nh
´
ˆat I nh
’
ung khˆong hˆo
.
i tu
.
¯d
`
ˆeu ¯d
´
ˆen I.
Gi
’
ai
P
n
l`a ph´ep chi
´
ˆeu tr
’
u
.
c giao lˆen khˆong gian con tuy

´
ˆen t´ınh L{e
1
, . . . , e
n
}. V`ı {e
n
} l`a
mˆo
.
t c
’
o s
’
’
o c
’
ua X nˆen v
´
’
oi mo
.
i x ∈ X ta c´o
x =
∞

n=1
x, e
n
e

n
.
Khi ¯d´o
lim
n→∞
P
n
x = lim
n→∞
n

k=1
x, e
k
e
k
=
∞

n=1
x, e
n
e
n
= x = Ix, ∀x ∈ X.
Vˆa
.
y d˜ay {P
n
} hˆo

.
i tu
.
¯d
´
ˆen I.
Gi
’
a s
’
’
u d˜ay {P
n
} hˆo
.
i tu
.
¯d
`
ˆeu ¯d
´
ˆen I. Khi ¯d´o, lim
n→∞
P
n
− I = 0. Do ¯d´o, P
n
0
− I < 1
v

´
’
oi n
0
¯d
’
u l
´
’
on. L
´
ˆay x = e
n
0
+1
th`ı
(P
n
0
− I)e
n
0
+1
 ≤ P
n
0
− I.e
n
1
+1

 < 1.
M
˘
a
.
t kh´ac, ta c´o
(P
n
0
− I)e
n
0
+1
 = P
n
0
e
n
0
+1
− e
n
0
+1
 = e
n
0
+1
 = 1 (vˆo l´y) .