泛函分析 孙炯版课后答案 第二章

z = x1 1 + x2 2,
Ù¥ z ∈ Z, z = (x1, x2). y² Z ´ Banach ˜m. 30. (X1, x 1), (X2, x 2) ´D‰˜m, 3¦È‚5˜m X1 × X2 ¥½Â
z 1 = x1 1 + x2 2, z 2 = max{ x1 1, x2 2}.
0
y² x Ø´ Lp[0, 1] þ ‰ê, 0 α 1, K α αp 1.)
d(x, y) = x − y ´ Lp[0, 1] þ ål. (J«: e
12. 3 l∞ ¥, U‹I½Â‚5$Ž…é x ∈ l∞, x = {ξk} ½Â x = supn |ξn|. y² l∞ ´˜‡D‰˜m.
n=1
(1) · ´Ä´ X þ ‰ê? (2) x − y ´ÄŒ½Â• X þ ¿Â.
ål?
beŒ±, `²
x(n) − x
→ 0 (n → ∞)
4. 3 C1[a, b] ¥-
b
x 1 = ( (|x(t)|2 + |x (t)|2)dt)1/2, ∀x ∈ C1[a, b].
a
(1) y² · 1 ´ C1[a, b] þ ‰ê; (2) ¯ (C1[a, b], · 1) ´Ä ?
1
f 1 = |f (t)| dt,
0
f 2=(
1
|f
(t)|2
dt)
1 2
,
1
f 3=(
(1
+
t)|f
(t)|2
dt)
1 2
.
0
0
= ´ d‰ê? Á`²nd.
22. {xn} ´ Banach ˜m X ¥ : . e•3 (0, ∞) þšK4~
xn ≤ g(n) (n = 1, 2, 3, · · · ). y²
·‚k
αx = d(αx, 0) = |α|d(x, 0) = |α| x ( àg);
(iv)dål n Ø ªk
x + y = d(x + y, 0) ≤ d(x + y, y) + d(y, 0),
2Šâ²£ØC5 Œ d(x + y, y) = d(x, 0),
d(x + z, y + z) = d(x, y),
∞1
∞1
αx0 = 2k min(1, |αxk|) = 2k = 1,
k=1
k=1
·6·
1 Ù D‰˜m
∞1
∞1
|α| x0 = 2(
2k min(1, |xk|)) = 2
= 2. 2k
k=1
k=1
(2) d · ½Â´•, x − y ÷všK5! ½5!é¡5Ún Ø ª, K
d(x, y) = x − y
10. M ´ [a, b] þk.¼ê N. ‚5$Ž ½Â† C[a, b] ¥ƒÓ. 3 M ¥½Â‰ê x = supa≤t≤b |x(t)|. y² M ´ Banach ˜m.
11. 0 < p < 1, •Ä˜m Lp[0, 1], Ù¥
1
x = |x(t)|pdt < ∞, x ∈ Lp[0, 1].
y² (1) · ´ C(0, 1] ˜mþ ‰ê; (2) l∞ † C(0, 1] ˜‡f˜m åÓ .
20. X • n ‘D‰‚5˜m, E0 ´ X ý4f˜m, y²•3 x0 ∈ X, ¦
x0 = 1,
d(x0, E0) = inf x0 − x = 1. x∈E0
21. 3 L2[0, 1] þ5½ØÓ‰ê:
X •´ Banach ˜m. 28. (X1, x 1), (X2, x 2) ´D‰˜m, 3¦È‚5˜m Z = X1 × X2 ¥½Â
z = x1 1 + x2 2,
Ù¥ z ∈ Z, z = (x1, x2). {(xn, yn)} ´ Z ¥ ˜S . y² (1) {(xn, yn)} 3 Z ¥Âñu (x, y), …= {xn} 3 X1 ¥Âñu x, {yn} 3 X2 ¥Â ñu y. (2) {(xn, yn)} ´ Z ¥ Cauchy , …= {xn} ´ X1 ¥ Cauchy , {yn} ´ X2 ¥ Cauchy . 29. (X1, x 1), (X2, x 2) ´ Banach ˜m, 3¦È‚5˜m Z = X1 × X2 ¥½Â
x + y ≤ d(x, 0) + d(y, 0) = x + y (n Ø ª).
¤± (X, · ) ´D‰‚5˜m.
2. y² d(x, y) ÷všK5!î‚ !é¡5´²w . eyn Ø ª¤á. e x = y, K d(x, y) = 0 ≤ d(x, z) + d(z, y); Šâ‰ên Ø ª, ·‚k:
25. (X, · ) D‰˜m, X0 ´ X ¥ È—f8. y²éuz˜ x ∈ X, •3 {xn} ⊂ X0,
¦
x=
∞ n=1
xn,
¿…
∞ n=1
xn
< ∞.
·4·
1 Ù D‰˜m
26. X ´D‰‚5˜m, M ´ X 4f˜m. y²e X Œ©, K X/M •Œ©. 27. X ´D‰‚5˜m, M ´ X 4f˜m. XJ M ±9 X/M ´ Banach ˜m. y²
∞ n=1
xn
Âñ.
23. (X, · ) ´D‰˜m, Y ´ X f˜m, éu x ∈ X, ·
ŒÈ¼ê g(t), ¦
δ = d(x, Y ) = inf x − y . y∈Y
XJ•3 y0 ∈ Y , ¦ x − y0 = δ, ¡ y0 ´ x •Z%C. (1) y²XJ Y ´ X k¡‘f˜m. Kéz˜ x ∈ X, •3•Z%C. (2) ÁÞ~`², Y Ø´k¡‘˜mž, (1) (Øؤá. (3) ÁÞ~`², ˜„/, •Z%CØ•˜. (4) y²éuz˜: x ∈ X, x 'uf˜m Y •Z%C:8´à8.
ål˜m;
(2) e½Â f = d(f, 0), K (Ck[a, b], · ) ´ Banach ˜m.
9. H ´3†‚ R þ²•ŒÈ, ꕲ•ŒÈ ëY¼ê8Ü, éuz‡ f ∈ H, ½Â
∞
f = ( |f (t)|2 dt +
∞
|f
(t)|2
dt)
1 2
.
−∞
−∞
y² H ´ Banach ˜m.
Ù¥ z ∈ X1 × X2, z = (x1, x2). y² · 1, · 2 ´ X1 × X2 þ d‰ê.
·5·
源自文库
1 ÙSK{‰
1. y²(i) x = d(x, 0) ≥ 0 (šK5); (ii) x = 0 ⇔ d(x, 0) = 0 ⇔ x = 0 ( ½5); (iii)Šâƒq5 d(αx, 0) = d(αx, α0) = |α|d(x, 0),
à5)§2|^ Minkowski Ø
xn(t) =
∞
(t − c)2 + 1 , (a ≤ t ≤ b) n2
, Ù¥ c ∈ (a, b).
n=1
N´y {xn(t)}∞ n=1 U‰ê · 1 ´Ä , …::Âñ x(t) = |t − c|, x(t) ∈
C1[a, b]. =3 (C1[a, b], · 1) ¥•3ØÂñ Ä
5. 3 Cn ¥½Â‰ê x = max |xi|, y²§´ Banach ˜m. i
6. X ´ [0, 1] þ¤këY¼ê x = x(t) 8Ü. y²
1
x = ( |x(t)|2 dt)1/2
0
´ X þ ‰ê, X 3ù«‰êeØ .
7. (X, · ) ´D‰˜m, X = {0}. y² X • Banach ˜m ¿‡^‡´ X ¥ ü
¥¡ S = {x ∈ X x = 1}
.
1
·2·
1 Ù D‰˜m
8. Ck[a, b] ´ [a, b] þäk k ëY ê ¼ê N. ½Â
k
d(f, g) = max |f (i)(x) − g(i)(x)|, f, g ∈ Ck[a, b].
x∈[a,b] i=0
y² (1) (Ck[a, b], d) ´
15. Hp (0 < p 1) L« [a, b] þ N÷v Ho¨lder ^‡
|x(t1) − x(t2)| M |(t1 − t2) | p
¼ê, ‚5$Ž ½Â† C[a, b] ¥ ƒÓ. 3 Hp ¥½Â‰ê
x = |x(a)| + sup |x(t1) − x(t2)| , a t1<t2 b |t1 − t2|p
Œ½Â• X þ ål, = (X, d) •ål˜m. …é?¿ {x(n)} ⊂ X,
x(n) − x → 0 (n → ∞)
du {x(n)} U‹IÂñ x.
4. y² (1) · w,÷v‰êcn‡^‡(šK5, ½5, ªN´ yÙ÷vn Ø ª.
(2) (C1[a, b], · 1) Ø . •ÄC1[a, b]¥ ¼ê :
1 Ù D‰˜m
SK 2
1. 3‚5˜m X ¥½Â ål d ÷v²£ØC5Úƒq5, = d(x + z, y + z) = d(x, y), d(αx, αy) = |α|d(x, y). - x = d(x, 0). y² (X, · ) ´D‰‚5˜m.
2. (X, · ) ´D‰‚5˜m, éu x, y ∈ X, -
24.
(Xk, · k) ´˜ D‰˜m, x = {xk}, xk ∈ Xk(k = 1, 2, · · · ) …÷v^‡
∞ k=1
xk
p k
< ∞, ^ X L«¤k x N. U‹I½Â‚5$Ž ¤ ‚5˜m, 3 X ¥½Â
∞
x =(
xk
p k
)1/p(p
≥
1).
k=1
y² (X, · ) ´˜‡D‰˜m.
{xn(t)}∞ n=1, l (C1[a, b], · 1)
Ø.
5. y² ´y · ´‰ê, e¡y² (Cn, · ) ´
. {xk} ´ Cn ¥ Cauchy , Ù
¥ xk = {xk1 , xk2 , · · · , xkn}. é ∀ε •3 K, k, l > K ž
0,
x = y;
d(x, y) =
x − y + 1, x = y.
y² d ´ål, Ø´d‰êp ål, =Ø•3 X þ ‰ê · 1, ¦
d(x, y) = x − y 1, x, y ∈ X.
3. X L«ES x = (x1, x2, · · · ) N, ½Â
∞
x = 2−n min(1, |xn|).
y² Hp • Banach ˜m. 16. x(t) ´ [a, b] þ ëY¼ê, -
x p=(
b
|x(t)|p
dt)
1 p
,
a
y² limp→∞ x p = x ∞.
x ∞ = max |x(t)|, a≤t≤b
·3·
17. y²‚5˜m X ¥?Û˜xà8 E´à8; é?Û x0 ∈ X, à8 A /£Ä0 x0 ¤ 8Ü A + x0 = {y + x0|y ∈ A} E´à8.
d(x, y) = x − y + 1 ≤ x − z + z − y + 1.
du z ØUÓž u x Ú y, Ø”b½ x = z, þª¤•:
d(x, y) = x − y + 1 ≤ x − z + z − y + 1 = d(x, z) + z − y ≤ d(x, z) + d(z, y).
13. y² lp (1 ≤ p < ∞) ´Œ© Banach ˜m.
14. Hp (1 p < ∞) L«äk5Ÿ
f p = sup {
2π
|f
(reiθ
p
)|
dθ}1/p
<
∞
0<r<1 0
)Û¼ê f (z) (|z| < 1)
N. y² f p ´‰ê¿…(Hp, · p) ´ Banach ˜m.
=n Ø ª¤á. ¤± d(x, y) ´ål. Ï•
αx + 1, x = 0;
d(αx, 0) =
0,
x = 0.
w,e α ∈/ {1, −1, 0}, x = 0, K d(αx, 0) = |α|d(x, 0), =Ø÷v àg5. ¤± d(x, y) Ø´‰êp ål.
3. y² (1) · Ø´ X þ ‰ê, Ï• · Ø÷vàg5. ¯¢þ, é α = 2 ∈ C, x0 = {i, 2i, · · · , ni, · · · } k
18. M ‚5˜m X ¥ f8, y²
n
C0(A) = {α1x1 + · · · + αnxn ∈ X | n ´?¿g,ê, xk ∈ A, αk ≥ 0 … αk = 1}.
k=1
19. C(0, 1] L« (0, 1] þëY…k. ¼ê x(t) N. -
x = sup{|x(t)| |0 < t ≤ 1}.