RもしくはC上の有限次元ベクトル空間上のノルムが同値であることの証明p24 A.5

1. Analysis III Functional Analysis
III
25 10 3 2 (10:40-12:10)
1 1
1.1 n Rn
or Cn
. . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 ( ) (Linear sp. (Vector sp.)) . . . . . . . . . . . . . . . . . 1
2 (Normed Spaces) 2
2.1 (Norm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3 (Banach Spacses) 2
3.1 Banach (Examples of Banach sps) . . . . . . . . . . . . . . . . . . . . . 3
3.1.1 (Continuous function space) . . . . . . . . . . . . . . . . . 3
3.1.2 Lp
(Lp
-sp.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 (Separable & equivarent norms) . . . . . . . . . . . . . . . 6
3.3 (Completion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 (Hilbert Spaces) 7
4.1 ( ) (Pre-Hilbert sp. (Inner prod. sp.)) . . . . . . . 7
4.2 (Hilbert sp.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.3 (Projection theorem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.4 (ONS=orthonormal system) . . . . . . . . . . . . . . . . . . . . . . . 10
5 (Linear Operators) 12
5.1 (Examples of bounded operators) . . . . . . . . . . . . . . . . . 13
5.2 (Inverse operators) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 ( , , ) 15
6.1 (Uniform bounded principle) . . . . . . . . . . . . . . . . . . . . 16
6.2 (Open mapping theorem) . . . . . . . . . . . . . . . . . . . . . . . . 16
6.3 (Closed graph theorem) . . . . . . . . . . . . . . . . . . . . . . . . 17
7 (Linear Functionals) 18
7.1 (Dual spaces) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
7.2 (Hahn-Banach’s extension thoerem) . . . . . . . . . 19
1

2. A 21
A.1 Lp
. . . . . . . . . . . . . . . . . . . . . . . . . . . 21
A.2 . . . . . . . . . . . . . . . . . . . . . . 21
A.3 Hk,p
(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
A.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A.5 . . . . . . . . . . . . . . . . . 24
A.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
A.7 A2
(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
B , 26
B.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
B.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
C 27
D 28
1 , , , , , ,
.
, .
Rn
, Cn
ℓp
, C([a, b]), Lp
(Ω)
(Ω ⊂ Rn
) ∥ · ∥ , ,
.
, (= ) ,
, ,
. , , .
, , ,
, .
, ,
. , , ,
, , , .
, , .
, , ,
, . , .

3. Functional Analysis 1
1
1.1 n Rn
or Cn
n ∈ N. x = (x1, . . . , xn), y = (y1, . . . , yn) ∈ Rn
(or Cn
), : x + y = (x1 + y1, . . . , xn + yn),
: α ∈ R (or C) , αx = (αx1, . . . , αxn) def. Rn
(or Cn
)
.
(x, y) =
j=1
xjyj in Rn
or (x, y) =
j=1
xjyj in Cn
, , x |x| = (x, x)1/2
.
Rn
, Cn
, n (Euclid spaces)
. ( Hilbert sp. , Banach sp. .)
1.2 ( ) (Linear sp. (Vector sp.))
K = R or C .
1.1 ( ) X K ( )
, i.e., ∀
x, y ∈ X, x + y ∈ X, ∀
α ∈ K, ∀
x ∈ X, αx ∈ X; .
(i) ( ) (x + y) + z = x + (y + z) (x, y, z ∈ X)
(ii) ( ) x + y = y + x (x, y ∈ X)
(iii) ( ) ∃
θ ∈ X; ∀
x ∈ X, x + θ = x (θ = 0 )
(iv) ( ) ∀
x ∈ X, ∃
x′
∈ X; x + x′
= 0 (x′
= −x )
(v) ( ) α(x + y) = αx + αy, (α + β)x = αx + βx (x, y ∈ X, α, β ∈ K)
(vi) ( ) (αβ)x = α(βx) (x ∈ X, α, β ∈ K)
(vii) ( ) 1x = x (∀
x ∈ X)
1.1 . (∃
θ ∈ X; ∀
x ∈ X, x + θ = x θ )
[ ] ∃
θ′
∈ X; ∀
x ∈ X, x + θ′
= x θ = θ + θ′
= θ′
+ θ = θ′
, .
1.2 . (x ∈ X , ∃
x′
∈ X; x + x′
= 0 x′
)
[ ] x ∈ X , ∃
x′′
∈ X; x + x′′
= 0 0 = x + x′
= x + x′′
, ,
x′
= x′
+ (x + x′′
) = (x′
+ x) + x′′
= (x + x′
) + x′′
= x′′
.
(1) X x1, . . . , xn ∈ X
( ) (linear independent)
def
⇐⇒ [α1x1 + · · · + αnxn = 0 (α1, . . . , αn ∈ K) =⇒ α1 = · · · = αn = 0]
( )(linear dependent)
def
⇐⇒ , i.e., ∃
(α1, . . . , αn) ̸= 0; α1x1 + · · · + αnxn = 0.
X n (n-dimensional)
def
⇐⇒ n , n+1
. dim X = n .
X (infinite dimensional)
def
⇐⇒ ∀
n ∈ N, n .
x ∈ X x1, . . . , xn ∈ X (linear combination)
def
⇐⇒ ∃
α1, . . . , αn ∈ K; x = α1x1 + · · · + αnxn

4. Functional Analysis 2
1.3 X n n ,
, i.e.,
dim X = n =⇒ ∃
x1, . . . , xn ∈ X; lin. indep., ∀
x ∈ X, ∃
α1, . . . , αn ∈ K; x = α1x1 + · · · + αnxn.
(2) X K .
Y ⊂ X (subspace)
def
⇐⇒ ∀
x, y ∈ Y, x + y ∈ Y, , ∀
α ∈ K, αx ∈ Y .
2 (Normed Spaces)
, .
.
2.1 (Norm)
2.1 ( ) X ∥ · ∥ : x → ∥x∥ ∥x∥
X (norm) .
(i) ∥x∥ ≥ 0 (x ∈ X) ( )
(ii) ∥x∥ = 0 ⇐⇒ x = 0 ( )
(iii) ∥αx∥ = |α|∥x∥ (α ∈ K, x ∈ X)
(iv) ∥x + y∥ ≤ ∥x∥ + ∥y∥ (x, y ∈ X) ( )
(X, ∥ · ∥) (normed space) .
(X, ∥ · ∥) d(x, y) = ∥x − y∥ . , .
(d.1) d(x, y) ≥ 0 (x, y ∈ X) ( )
(d.2) d(x, y) = 0 ⇐⇒ x = y ( )
(d.3) d(x, y) = d(y, x) (x, y ∈ X) ( )
(d.4) d(x, z) ≤ d(x, y) + d(y, z) (x, y, z ∈ X) ( )
(X, d) .
{xn} ⊂ X , xn → x (n → ∞)
def
⇐⇒ ∥xn − x∥ → 0 (n → ∞)
x {xn} .
2.1 , i.e.,
(i) xn → x, yn → y =⇒ xn + yn → x + y, (ii) αn → α, xn → x =⇒ αnxn → αx.
2.2 , i.e., xn → x =⇒ ∥xn∥ → ∥x∥.
3 (Banach Spacses)
3.1 (Banach sp.) . ,
, . , (X, ∥ · ∥) ,
{xn} ⊂ X : (Cauchy sequence)
def
⇐⇒ ∥xn − xm∥ → 0 (m, n → ∞)
X (complete)
def
⇐⇒ Cauchy {xn} ⊂ X , i.e., ∃
x ∈ X; xn → x.

5. Functional Analysis 3
3.1 Banach (Examples of Banach sps)
3.1 Rn
, Cn
Banach sp.
, , ,
, R1
, Cauchy , ,
Cauchy . 2 .
3.2 Pn: n (n ∈ N)
x(t) = antn
+ an−1tn−1
+ · · · + a0 ∈ Pn (ak ∈ C)
(x + y)(t) = x(t) + y(t), (αx)(t) = αx(t), , y(t) = bntn
+ bn−1
n−1 + · · · + b0
, (x + y)(t) = (an + bn)tn
+ · · · + (a0 + b0), (αx)(t) = αantn
+ · · · + αa0 Pn
. dim Pn = n + 1 ({1, t, t2
, . . . , tn
} )
Pn ∋ x(t) =
n
j=0 ajtj
, ∥x∥ =
n
j=0 |aj|.
3.1 Pn Banach .
3.1.1 (Continuous function space)
3.3 Ω ⊂ Rn
C(Ω)
[(x + y)(t) = x(t) + y(t), (αx)(t) = αx(t)] . dim C[0, 1] = ∞.
∥x∥∞ = supt∈Ω |x(t)| , Banach sp. (x(t) = tn−1
(n ≥ 1) )
{xn} Cauchy in C(Ω) . t ∈ Ω ,
|xn(t) − xm(t)| ≤ ∥xn − xm∥∞ → 0(m, n → ∞)
, {xn(t)} R Cauchy . , ∃
x∗
(t) ∈ R; xn(t) → x∗
(t).
n → ∞ ,
|x∗
(t) − xm(t)| ≤ lim
n→∞
∥xn − xm∥∞
t ∈ Ω , supt∈Ω , m → ∞
lim
m→∞
sup
t∈Ω
|x∗
(t) − xm(t)| ≤ lim
m,n→∞
∥xn − xm∥∞ = 0
x∗
{xn} x∗
. xn → x∗
in C(Ω).
Cb(R) = {x ∈ C(R); ∥x∥∞ < ∞} . , (Cb, ∥· ∥∞) Banach sp.
3.1 . Pn ∋ x(t)
n
j=0 ajtj
, ∥x∥∞ =
max |αj|, x(t) ∈ C([0, 1]) , ∥x∥L1 =
[0,1]
|x(t)|dt .
.
3.2 Pn[0, 1]: Pn [0, 1] C[0, 1]
( ). ( ∥x∥∞ = t∈[0,1] |x(t)|)
.
.

6. Functional Analysis 4
3.1.2 Lp
(Lp
-sp.)
3.4 1 ≤ p < ∞, Ω ⊂ Rn
, Lp
(Ω) Ω u
∥u∥Lp :=
Ω
|u(t)|p
dt
1/p
< ∞
. u = v a.e. , Lp
(Ω) Banach sp. .
p or Lp
.
3.5 Ω ⊂ Rn
, L∞
(Ω) Ω u ∃
α < ∞; |u(t)| ≤ α
a.e. .
∥u∥∞ ≡ ess.supt∈Ω|u(t)| := inf{α; |u(t)| ≤ α a.e.}
. |u| ≤ ∥u∥∞ a.e. . u = v a.e. ,
L∞
(Ω) Banach sp. .
or L∞
.
2 , , normed sp. .
[ (H¨older’s inequality)]
1 ≤ p ≤ ∞ , 1 ≤ q ≤ ∞ 1/p + 1/q = 1 , p = 1 q = ∞ , p = ∞
q = 1 (q p ). ∥uv∥L1 ≤ ∥u∥Lp ∥v∥Lq . ,
Ω
|u(t)v(t)|dt ≤
Ω
|u(t)|p
dt
1/p
Ω
|v(t)|q
dt
1/q
(1 < p < ∞),
Ω
|u(t)v(t)|dt ≤
Ω
|u(t)|dt ∥v∥∞ (p = 1, q = ∞).
p = 1, ∞ . 1 < p < ∞ . ∥u∥Lp = 0 or ∥v∥Lq = 0 uv = 0 a.e.
, ∥u∥Lp ̸= 0 and ∥v∥Lq ̸= 0 . ab ≤ ap
/p + bq
/q (a, b ≥ 0)
. (log , log(ap
/p + bq
/q) ≥ (log ap
)/p + (log bq
)/q = log a + log b = log(ab))
.) a = |u(t)|/∥u∥Lp , b = |v(t)|/∥v∥Lq , .
∥uv∥L1
(∥u∥Lp ∥v∥Lq )
≤
∥u∥p
Lp
p∥u∥p
Lp
+
∥v∥q
Lq
q∥v∥q
Lq
=
1
p
+
1
q
= 1
[ (Minkovsky’s inequality)] ( )
1 ≤ p < ∞. u, v ∈ Lp
(Ω) u + v ∈ Lp
(Ω) , ∥u + v∥Lp ≤ ∥u∥Lp + ∥v∥Lp .
p = 1 . p = ∞ (→ ). 1 < p < ∞ .
. (u + v ∈ Lp
(Ω) .) |u + v|p
≤ (|u| + |v|)|u + v|p−1
H¨older
. , 1/q = 1 − 1/p = (p − 1)/p, i.e., q = p/(p − 1)
∥u + v∥p
Lp ≤
Ω
|u||u + v|p−1
dt +
Ω
|v||u + v|p−1
dt ≤ (∥u∥Lp + ∥v∥Lp )
Ω
|u + v|p
dt
1/q
|u + v|p
dt = 0 , ̸= 0
Ω
|u + v|p
dt
1/q
= ∥u + v∥
p/q
Lp
p/q = p − 1 , .

7. Functional Analysis 5
3.2 ∥u + v∥∞ ≤ ∥u∥∞ + ∥v∥∞ .
Lp
(Ω) . Banach
.
3.3 (X, ∥ · ∥) , {un} ⊂ X: Cauchy . ∃
{unk
} ⊂ {un}; unk
→ u in X
un → u in X .
∥uk − u∥ ≤ ∥uk − unk
∥ + ∥unk
− u∥ → 0 (nk ≥ k → ∞) .
3.4 Lebesgue , Lebesgue .
[Lp
(Ω) ]
{un} Cauchy in Lp
(Ω) . ∃
{unk
}; ∥unk+1
− unk
∥Lp < 1/2k
.
,
∞
j=1
|unj+1 − unj |
Lp
= lim
m→∞
m
j=1
|unj+1 − unj |
Lp
≤
∞
j=1
∥unj+1 − unj ∥Lp ≤ 1 < ∞.
∞
j=1
|unj+1 − unj | ∈ Lp
(Ω).
∞
j=1
|unj+1 (t) − unj (t)| < ∞ for a.e. t ∈ Ω.
k < m ,
|unm (t) − unk
(t)| ≤
m−1
j=k
|unj+1 (t) − unj (t)| → 0 (m > k → ∞) a.e.
a.e. t ∈ Ω , {unk
(t)} R Cauchy , unk
(t) → ∃
u∗
(t).
u∗
{un} Lp
(Ω) . ,
|unk
(t)| ≤ |un1 (t)| +
k−1
j=1
|unj+1 (t) − unj (t)| ≤ |un1 (t)| +
∞
j=1
|unj+1 (t) − unj (t)| ∈ Lp
(Ω)
, g(t) , k → ∞ a.e. t ∈ Ω , |u∗
(t)| ≤ g(t) ∈ Lp
(Ω), i.e.,
u∗
(t) ∈ Lp
(Ω). k < m ,
∥unm − unk
∥Lp ≤
m−1
j=k
∥unj+1 − unj ∥Lp ≤
∞
j=k
1
2j
=
1
2k−1
.
|unm (t) − unk
(t)| ≤ 2g(t) ∈ Lp
(Ω) Lebsgue , m → ∞
∥u∗
− unk
∥Lp ≤
1
2k−1
→ 0 (k → ∞).
unk
→ u∗
in Lp
(Ω) , un → u∗
in Lp
(Ω) .
(X, F, µ) (X a set, F σ-field on X, µ = µ(dx) on X) ,
f ∈ Lp
(X) or Lp
(X, dµ) or Lp
(X, F, µ)
def
⇐⇒ ∥f∥p
Lp := |f|p
dµ =
X
|f(x)|p
µ(dx) .
L∞
(X) . H¨older , Minkovsky ,
Lp
(X) Banach . . ( .)

8. Functional Analysis 6
3.6 1 ≤ p < ∞. x = (x1, x2, . . . ) ∥x∥lp =
∞
n=1
|xn|p
1/p
< ∞
lp
Banach sp. .
3.7 x = (x1, x2, . . . ) ∥x∥∞ = sup{|xn|; n ≥ 1} < ∞ l∞
Banach sp. .
3.2 (Separable & equivarent norms)
X Banach sp. .
L ⊂ X X (dense)
def
⇐⇒ L = X , L L (L ; x ∈ L ⇐⇒
∃
{xn} ⊂ L; xn → x).
X (separable)
def
⇐⇒ , i.e., ∃
L ⊂ X; L = X, ♯L ≤ ℵ0 = ♯N.
3.8 Rn
Banach sp. , Qn
.
3.9 Ω ⊂ Rn
, C(Ω) .
C[0, 1] L [0, 1] , ,
, L = C[0, 1] .
X , 2 ∥ · ∥1, ∥ · ∥2
∥ · ∥1 ∥ · ∥2 ; ∥ · ∥1 ∼ ∥ · ∥2
def
⇐⇒ ∃
0 < c, c′
< ∞; c∥x∥2 ≤ ∥x∥1 ≤ c′
∥x∥2.
normed sp. (X, ∥ · ∥1) (X, ∥ · ∥2) . ,
. (X, ∥ · ∥1) (X, ∥ · ∥2) .
3.5 X X 2 . ( )
3.3 (Completion)
X . ,
X , X , X = X .
, X X Cauchy , {xn}, {yn} ∈ X , {xn} ∼ {yn} ⇐⇒ xn −yn → 0
(n → ∞) . , x = [{xn}] ∈ X := X/ ∼
, x = [{xn}] ∈ X ∥x∥ := lim
n→∞
∥xn∥ ({∥xn∥} R Caushy ,
) , (X, ∥ · ∥) , i.e., Banach sp. . (
.)
x ∈ X [{xn ≡ x}] ∈ X , X ⊂ X X = X
. X X (completion) .
3.10 X0 := {x = (x1, x2, . . . , xn, 0, 0, . . . ); xi ∈ R, n ∈ N} ( 0
) . ∥x∥ = ∥x∥lp , X0 . X0 lp
dense , . , x(n)
= (1, 1/2, 1/22
. . . . , 1/2n
, 0, 0, . . . ) m > n
, 1 ≤ p < ∞ ∥x(n)
− x(m)
∥lp = (
m
k=n+1 2−kp
)1/p
→ 0 (n → ∞) , Cauchy

9. Functional Analysis 7
, x = (1, 1/2, 1/22
. . . . , 1/2n
, . . . ) /∈ X0. X0 . (p = ∞ .)
X0 X0 lp
, i.e., lp
.
X0 lp
.
) (X, ∥ · ∥X), (Y, ∥ · ∥Y ) , ∃
f : X → Y ; , ∥f(x)∥Y = ∥x∥X
X Y . ( X Y .)
3.11 [0, 1] X0 , i.e., X0 = n≥1 Pn[0, 1]. x ∈ X0 ,
∥x∥ = supt∈[0,1] |x(t)| , X0 . X0 C[0, 1] .
4 (Hilbert Spaces)
(inner product) ⟨x, y⟩ or
(inner prod. sp. or pre-Hilbert sp.) , ∥x∥ = ⟨x.x⟩ ,
(Hilbert sp.) . Rn
Cn
, ,
.
4.1 ( ) (Pre-Hilbert sp. (Inner prod. sp.))
4.1 X C , x, y ∈ X , ⟨x, y⟩ ∈ C (inner product)
( ) ⟨x, x⟩ ≥ 0 (x ∈ X). ⟨x, x⟩ = 0 ⇐⇒ x = 0.
( ) ⟨x, y⟩ = ⟨y, x⟩ (x, y ∈ X)
( ) ⟨x1 + x2, y⟩ = ⟨x1, y⟩ + ⟨x2, y⟩, ⟨αx, y⟩ = α⟨x, y⟩ (x1, x2, y ∈ X, α ∈ C).
4.2 X or (X, ⟨·, ·⟩) (pre-Hilbert
sp.) or (inner prod. sp.) .
4.1 X , ∥x∥ := ⟨x, x⟩1/2
(x ∈ X) ,
. ( ) ⊂ ( ) .
.
4.1 (Schwartz ) |⟨x, y⟩| ≤ ∥x∥∥y∥ (x, y ∈ X).
∥y∥ = 0 y = 0 , ⟨x, y⟩ = 0 (→ ), , ∥y∥ ̸= 0 .
∀
α ∈ C , 0 ≤ ⟨x + αy, x + αy⟩ , α := −⟨x, y⟩/∥y∥2
.
, 0 ≤ ⟨x + αy, x + αy⟩ = ∥x∥2
+ α⟨x, y⟩ + α⟨x, y⟩ + |α|2
∥y∥2
= ∥x∥2
− |⟨x, y⟩|2
/∥y∥2
.
y = 0 ⟨x, y⟩ = 0 Schwartz . (⟨x, y⟩ = ⟨x, 0y⟩ = 0⟨x, y⟩ = 0)
[ 4.1 ] . , ∥x + y∥2
=
∥x∥2
+ ⟨x, y⟩ + ⟨y, x⟩ + ∥y∥2
≤ ∥x∥2
+ 2∥x∥∥y∥ + ∥y∥2
= (∥x∥ + ∥y∥)2
.

10. Functional Analysis 8
4.2 X ∥x∥ = ⟨x, x⟩1/2
, .
∥x + y∥2
+ ∥x − y∥2
= 2(∥x∥2
+ ∥y∥2
).
.
⟨x, y⟩ =
1
4
∥x + y∥2
− ∥x − y∥2
+ i∥x + iy∥2
− i∥x − iy∥2
.
X , , ⟨x, y⟩ ,
. , X ⇐⇒ .
.
, .
4.1 ⟨x, y⟩ x, y , i.e., xn → x, yn → y ⟨xn, yn⟩ → ⟨x, y⟩.
Schwartz .
4.2 (Hilbert sp.)
4.3 X (Hilbert
sp.) . K = R , K = C .
[Hilber sps ]
4.1 Rn
, Cn
: , Hilbert .
4.2 l2
: x = (xn), y = (yn) ∈ l2
, ⟨x, y⟩ =
n≥1
xnyn , ,
, ∥x∥2
2 = |xn|2
= ⟨x, x⟩. Hilbert .
4.3 L2
(Ω) (Ω ⊂ Rn
open): u, v ∈ L2
(Ω), ⟨u, v⟩ =
Ω
u(t)v(t)dt Hilbert.
4.4 A2
(Ω) (Ω ⊂ Cn
open, f ∈ A2
(Ω) ,
Ω
|f(z)|2
dxdy < ∞ (z = x + iy))
f, g ∈ A2
(Ω) , ⟨f, g⟩ =
Ω
f(z)g(z)dxdy Hilbert. ( )
4.5 C(Ω) (Ω ⊂ Rn
) L2
(Ω) ,
Hilbert .
4.3 (Projection theorem)
H Hilbert sp. x, y ∈ H, A, B ⊂ H ,
x ⊥ y
def
⇐⇒ ⟨x, y⟩ = 0.
A ⊥ B
def
⇐⇒ ∀
a ∈ A, ∀
b ∈ B, a ⊥ b. ( x ⊥ B
def
⇐⇒ {x} ⊥ B.)
L ⊂ H ,
L⊥
:= {x ∈ H; x ⊥ L} L (orthogonal complement) .

11. Functional Analysis 9
4.1 L ⊂ H , L⊥
H .
x ⊥ y =⇒ ∥x + y∥2
= ∥x∥2
+ ∥y∥2
( )
L1, L2 ⊂ H , L1, L2 (direct sum) L1 ⊕L2 := L1 +L2; L1 ∩L2 = {0}.
[L1 ∩ L2 = {0} ⇐⇒ x = x1 + x2 ∈ L1 + L2 ] .
.
[ ] (⇒) x1 + x2 = x′
1 + x′
2 (xi, x′
i ∈ Li) x1 − x′
1 = x′
2 − x2 ∈ L1 ∩ L2 = {0} ,
xi = x′
i. (⇐) x ∈ L1 ∩ L2 x = x + 0 = 0 + x ∈ L1 + L2 , x = 0.
4.3 ( ) L ⊂ H H L L⊥
; H =
L ⊕ L⊥
, i.e., ∀
x ∈ H, ∃
y ∈ L, ∃
z ∈ L⊥
; x = y + z , .
y ∈ L x ∈ H L or or . PLx = y ,
PL L ( ) .
x ∈ L ∩ L⊥
⟨x, x⟩ = 0 , x = 0, i.e., L ∩ L⊥
= {0}. .
. ∀
x ∈ H , δ := infy∈L ∥x − y∥ . inf
∃
{yn} ⊂ L; ∥x − yn∥ → δ.
2(∥x − yn∥2
+ ∥x − ym∥2
) = ∥(x − yn) + (x − ym)∥2
+ ∥(x − yn) − (x − ym)∥2
= ∥2x − (yn + ym)∥2
+ ∥yn − ym∥2
.
(yn + ym)/2 ∈ L , δ ≤ ∥x − (yn + ym)/2∥.
∥yn − ym∥2
= 2(∥x − yn∥2
+ ∥x − ym∥2
) − 4 x −
yn + ym
2
2
≤ 2(∥x − yn∥2
+ ∥x − ym∥2
) − 4δ2
( )→ 0 (m, n → ∞). H ∃
y ∈ H; yn → y. L closed y ∈ L.
δ = ∥x − y∥ . z = x − y (∥z∥ = δ). z ⊥ L . ξ ∈ L , γ = ⟨z, ξ⟩
, ϕ(t) = ∥z −γtξ∥2
= ∥x−(y +γtξ)∥2
(t ∈ R) . y +γtξ ∈ L , ϕ(t) ≥ δ2
= ϕ(0)
(δ ).
ϕ(t) = ∥z∥2
− γt⟨z, ξ⟩ − γt⟨ξ, z⟩ + |γ|2
t2
∥ξ∥2
= δ2
− 2|γ|2
t + |γ|2
t2
∥ξ∥2
= δ2
− |γ|2
t(2 − t∥ξ∥2
).
γ ̸= 0 , t > 0 0 , 2 − t∥ξ∥2
> 0 , ϕ(t) < ϕ(0) = δ2
.
γ = 0.
4.6 H = L2
(Ω) (Ω ⊂ Rn
: bdd open) , u ∈ L
def
⇐⇒ u ∈ H;
Ω
u(t)dt = 0
, L , PLu(t) = u(t) −
1
|Ω| Ω
u(t)dt. L⊥
= { }.
4.7 H = L2
(−1, 1) u ∈ L
def
⇐⇒ u ∈ H; u(−t) = u(t) L
, L⊥
= {v ∈ H; v(−t) = −v(t)}.
4.2 2 . ( , M = { } , L ⊂ M⊥
, L ⊃ M⊥
. , [0, 1) , u(t) = v(t) + v(−t) .)

12. Functional Analysis 10
4.4 (ONS=orthonormal system)
{xk} ⊂ H: (ONS)
def
⇐⇒ ⟨xj, xk⟩ = δjk.
4.8 L2
(0, 1) {
√
2 sin(πkt)}∞
k=1, {e2πkti
}∞
k=0 .
4.2 {xk} ⊂ H: ONS ∀
x ∈ H,
k
|⟨x, xk⟩|2
≤ ∥x∥2
(Bessel ).
x ∈ H , αk = ⟨x, xk⟩ . ∀
n ∈ N ,
0 ≤ ∥x −
n
k=1
αkxk∥2
= ∥x∥2
−
n
k=1
αk⟨x, xk⟩ −
n
k=1
αk⟨xk, x⟩ +
n
j,k=1
αjαk⟨xj, xk⟩
= ∥x∥2
−
n
k=1
|αk|2
n
k=1
|αk|2
≤ ∥x∥2
. n → ∞ .
{xk} ⊂ H , ⟨{xk}⟩ :=
n
k=1
αkxk; αk ∈ K, n ∈ N , L := ⟨{xk}⟩ {xk}
. ( upper bar .)
4.4 {xk} ⊂ H: ONS, L = ⟨{xk}⟩: {xk} . .
(i) L = H
(ii) ∀
x ∈ H, x = k⟨x, xk⟩xk ( Fourier )
(iii) ∀
x, y ∈ H, ⟨x, y⟩ = k⟨x, xk⟩⟨y, xk⟩ ( ).
(iv) ∀
x ∈ H, ∥x∥2
= k |⟨x, xk⟩|2
(Perseval ).
(v) ∀
k, ⟨x, xk⟩ = 0 x = 0.
(i) L = H ⇐⇒ L⊥
= {0} (by Proj. Th.) .
[(i) ⇒ (ii)] x ∈ H , αk = ⟨x, xk⟩ Bessel , k |αk|2
≤ ∥x∥2
, ,
. m > n ,
m
k=1
αkxk −
n
k=1
αkxk
2
=
m
k=n+1
αkxk
2
=
m
k=n+1
|αk|2
→ 0 (m > n → ∞).
{
n
k=1
αkxk} Cauchy in H. ∃
y =
∞
k=1
αkxk ∈ H.
⟨x − y, xk⟩ = ⟨x, xk⟩ − ⟨
n
αnxn, xk⟩ = αk −
n
αn⟨xn, xk⟩ = αk − αk = 0.
(x − y) ⊥ ⟨{xk}⟩, , (x − y) ⊥ ⟨{xk}⟩ = L = H.
x − y = 0, i.e., x = y.
[(ii) ⇒ (iii)] x, y ∈ H , αk = ⟨x, xk⟩, βk = ⟨y, xk⟩ . Schwartz Bessel
n
k=1
|αkβk| ≤
n
k=1
|αk|2
1/2 n
k=1
|βk|2
1/2
≤ ∥x∥∥y∥.

13. Functional Analysis 11
n → ∞
∞
k=1
αkβk .
⟨x, y⟩ = lim
n→∞
⟨
n
k=1
αkxk,
n
k=1
βkxk⟩ =
∞
k=1
αkβk.
[(iii) ⇒ (iv)], [(iv) ⇒ (v)] .
[(v) ⇒ (i)] x ∈ L⊥ ∀
k ≥ 1, ⟨x, xk⟩ = 0. x = 0 , L⊥
= {0} .
H = L.
4.4 H {xk} , (com-
plete ONS=CONS) .
4.9 l2
ej = (δj,n)n≥1 (j 1, 0) {ej} CONS.
4.10 H = L2
(−π, π) ,
1
√
2π
,
1
√
π
sin nt,
1
√
π
cos nt
∞
n=1
CONS.
∀
x ∈ H ,
x(t) =
1
2
a0 +
∞
n=1
(an cos nt + bn sin nt) an =
1
π
π
−π
x(t) cos ntdt, bn =
1
π
π
−π
x(t) sin ntdt
, Fourier , an, bn Fourier .
4.3 2 .
[Schmidt ] {yk} ⊂ H .
e1 ≡ x1 := y1/∥y1∥, en = xn/∥xn∥ with xn = yn −
n−1
k=1
⟨yn, ek⟩ek (n ≥ 2)
. {ek} ONS . Schmidt .
{ek} ONS .
4.5 Hilbert H CONS .
H {zk} {yn} . Schmidt
, {en} . , ∀
n, ⟨x, en⟩ = 0 x = 0
, CONS .
4.6 .
H . H
. x, y ∈ H , ∃
xn, yn ∈ H; xn → x, yn → y. {⟨xn, yn⟩} Cauchy
(Schwartz {∥xn∥}, {∥yn∥} ). ⟨x, y⟩ := lim⟨xn, yn⟩
, . ( , {xn}, {yn} ⊂ H
, H .)

14. Functional Analysis 12
5 (Linear Operators)
5.1 X, Y , D ⊂ X , T : D → Y (linear)
(i) T(x1 + x2) = Tx1 + Tx2 (x1, x2 ∈ D), (ii) T(αx) = αTx (α ∈ K, x ∈ D)
(linear operator) . D(T) := D T
(domain), R(T) := T(D) T (range) . Y = X T X
.
5.2 X, Y normed sps, lin. op. T : D(T) ⊂ X → Y
(i) (bounded operator)
def
⇐⇒ ∃
M; ∥Tx∥ ≤ M∥x∥ (x ∈ D(T)).
(ii) (conti. operator)
def
⇐⇒ xn → x in D(T) Txn → Tx.
5.1 X, Y normed sps, T : D(T) ⊂ X → Y ⇐⇒ T
(⇒) . , ∀
n ≥ 1, ∃
xn ∈ D(T); ∥Txn∥ > n∥xn∥ yn :=
xn/(
√
n∥xn∥) , ∥yn∥ = 1/
√
n , yn → 0. , ∥Tyn∥ = ∥Txn∥/(
√
n∥xn∥) >
√
n → ∞
, T . T .
(⇐) xn → x in D(T) . , ∥Txn − Tx∥ = ∥T(x − xn)∥ ≤ M∥xn − x∥ → 0
, Txn → Tx , T .
X, Y normed sps ,
T ∈ L(X, Y )
def
⇐⇒ D(T) = X T : X → Y ; bdd lin. op. L(X) := L(X, X).
, , T : X → Y D(T) = X .
X∗
:= L(X, K) X (conjucate sp.), (lin. conti.
functional) .
T ∈ L(X, Y ) , (operator norm) .
∥T∥ := sup
x∈X{0}
∥Tx∥
∥x∥
= sup
∥x∥=1
∥Tx∥.
∥Tx∥ ≤ ∥T∥∥x∥ . (→ : .)
5.2 X normed sp., Y Banach sp. . L(X, Y ) ∥T∥
Banach sp. .
{Tn} ⊂ L(X, Y ) Cauchy , i.e., ∥Tn − Tm∥ → 0 (m, n → ∞). ∀
x ∈
X, ∥Tnx−Tmx∥ ≤ ∥Tn−Tn∥∥x∥ → 0 , {Tnx} Cauchy in Y . Y : , ∃
y ∈ Y ; Tnx → y.
y x Tx = y , . ,
∀
ε > 0 , n, m , ∥Tn − Tm∥ < ε , ∥Tnx − Tmx∥ ≤ ∥Tn − Tn∥∥x∥ ≤ ε∥x∥ ,
m → ∞ ∥Tnx−Tx∥ ≤ ε∥x∥ . , ∥Tx∥ ≤ ∥Tx−Tnx∥+∥Tnx∥ ≤ (ε+∥Tn∥)∥x∥
, T ∈ L(X, Y ). ∥Tn − T∥ ≤ ε , Tn → T in L(X, Y ). L(X, Y )
.

15. Functional Analysis 13
5.1 (Examples of bounded operators)
5.1 Ω ⊂ Rn
, k(t) ∈ L∞
(Ω), 1 ≤ p ≤ ∞ . x ∈ Lp
(Ω) ,
(Tx)(t) = k(t)x(t) (t ∈ Ω) T Lp
(Ω) , ∥T∥ = ∥k∥∞.
5.1 . ( , ∥k∥∞ = 0 . > 0 . ∥T∥ ≤ ∥k∥∞ .
∀
ε > 0, Ωε = {|k| > ∥k∥∞ −ε} , Ωε , |Ωε| > 0 , x(t) := |Ωε|−1/p
1Ωε (t)
. , |Ωε| = ∞ , {|t| ≤ n}; n: , .)
5.2 Ω ⊂ Rn
, k(t, s) ∈ L2
(Ω2
), i.e.,
Ω2
|k(t, s)|2
dtds < ∞ . x ∈ L2
(Ω)
, (Tx)(t) =
Ω
k(t, s)x(s)ds T , ∥T∥ ≤
Ω2
|k(t, s)|2
dtds
1/2
.
k , T .
Schwartz ,
|(Tx)(t)| ≤
Ω
|k(t, s)||x(s)|ds ≤
Ω
|k(t, s)|2
ds
1/2
Ω
|x(s)|2
ds
1/2
.
|(Tx)(t)|2
≤
Ω
|k(t, s)|2
ds · ∥x∥2
L2 . t ∥Tx∥2
L2 ,
.
5.3 ( ) ρ ∈ L1
(Rn
) . 1 ≤ p ≤ ∞ , x ∈ Lp
(Rn
) ,
(Tx)(t) = (ρ ∗ x)(t) :=
Rn
ρ(t − s)x(s)ds =
Rn
ρ(s)x(t − s)ds
T Lp
(Rn
) Lp
(Rn
) , ∥Tx∥Lp ≤ ∥ρ∥L1 ∥x∥Lp .
T = ρ ∗ x ρ (convolution op.) .
p = 1, ∞ . 1 < p < ∞ q p (1/p + 1/q = 1)
. |ρ(t − s)x(s)| = (|ρ(t − s)|1/p
|x(s)|)|ρ(t − s)|1/q
H¨older ,
|(Tx)(t)| ≤ |ρ(t − s)||x(s)|p
ds
1/p
|ρ(s)|ds
1/q
= |ρ(t − s)||x(s)|p
ds
1/p
∥ρ∥
1/q
L1 .
p , t ,
∥Tx∥p
Lp ≤ ∥ρ∥
p/q
L1 dt |ρ(t − s)||x(s)|p
ds = ∥ρ∥
p/q+1
L1 ∥x∥p
Lp .
p/q + 1 = p(1/q + 1/p) = p , .
5.2 (Inverse operators)
T ∈ L(X, Y ) , ∃
S ∈ L(Y, X); TS = IY , ST = IX S T
(inv. op.) , T−1
.
T ∈ L(X) y ∈ X , (I − T)x = y x ∈ X
, (I − T)−1
.

16. Functional Analysis 14
5.3 X: Banach, T ∈ L(X) . ∥T∥ < 1 R(I−T) = X , ∃
(I−T)−1
∈
L(X); (I − T)−1
=
∞
n=0
Tn
= I + T + T2
+ · · · (T0
= I). (Neumann
series) , L(X) . ∥(I − T)−1
∥ ≤
∞
n=0
∥Tn
∥ ≤ 1/(1 − ∥T∥)
. ∥T∥ < 1
∞
n=0
∥Tn
∥ < ∞ , ,
. ∥(I − T)−1
∥ ≤
∞
n=0
∥Tn
∥ < ∞ .
∥T∥ < 1
∞
n=0
∥Tn
∥ ≤
∞
n=0
∥T∥n
= 1/(1 − ∥T∥) < ∞ ,
. X , L(X) .
n
k=0
Tk
−
m
k=0
Tk
≤
n
k=m+1
∥Tk
∥ → 0 (n > m → ∞)
,
n
k=0
Tk
L(X) Cauchy , ∃
S =
∞
k=0
Tk
∈ L(X).
TS = ST =
∞
n=0
Tn+1
=
∞
n=0
Tn
− I = S − I
, (I−T)S = S(I−T) = I, i.e., ∃
(I−T)−1
= S =
∞
n=0 Tn
. ∥(I − T)−1
∥ ≤
∞
n=0
∥Tn
∥
.
5.4 ( ) −∞ < a < b < ∞, y ∈ C[a, b] .
y(t) = x(t) −
b
a
k(t, s)x(s)ds
x ∈ C[a, b] . k(t, s) ∈ C([a, b]2
) , M := maxt,s∈[a,b] |k(t, s)| ,
M(b − a) < 1 . (X, ∥ · ∥) := (C[a, b], ∥ · ∥∞) x ∈ X ,
(Kx)(t) =
b
a
k(t, s)x(s)ds
, K ∈ L(X) , ∥Kx∥ ≤ M(b − a)∥x∥, i.e., ∥K∥ ≤ M(b − a) < 1 .
y = (I − K)x , ∃
(I − K)−1
; x = (I − K)−1
y = y + Ky + K2
y + · · · .
k1(t, s) = k(t, s), kn(t, s) =
b
a
k1(t, r)kn−1(r, s)dr (n ≥ 2)
|kn(t, s)| ≤ Mn
(b − a)n−1
, Kn
y(t) =
b
a
kn(t, s)y(s)ds
. h(t, s) := n≥1 kn(t, s) , ,
h(t, s) ∈ C([a, b]2
) ,
x(t) = y(t) +
∞
n=1
b
a
kn(t, s)y(s)ds = y(t) +
b
a
h(t, s)y(s)ds

17. Functional Analysis 15
5.2 x ∈ C[a, b] Kx ∈ C[a, b] .
5.5 ( ) , t .
y(t) = x(t) −
t
a
k(t, s)x(s)ds
x ∈ C[a, b] . , M(b − a) < 1 . x
. , (Kx)(t) =
t
a
k(t, s)x(s)ds
k1(t, s) = k(t, s), kn(t, s) =
t
s
k1(t, r)kn−1(r, s)dr (n ≥ 2)
|kn(t, s)| ≤ Mn (t − s)n−1
(n − 1)!
≤ Mn (b − a)n−1
(n − 1)!
Kn
y(t) =
t
a
kn(t, s)y(s)ds
.
∞
n=1
∥Kn
∥ ≤
∞
n=1
Mn (b − a)n−1
(n − 1)!
< ∞ ,
∃
(I − K)−1
; x = (I − K)−1
y = y + Ky + K2
y + · · · . h(t, s) := n≥1 kn(t, s)
, h(t, s) ∈ C([a, b]2
) , x(t) = y(t) +
t
a
h(t, s)y(s)ds.
5.3 |kn(t, s)| ≤ Mn
(t − s)n−1
/(n − 1)! .
6 ( , , )
3 , , ,
. .
6.1 ( (Baire’s category theorem)) (X, d)
, Xn ⊂ X (n ≥ 1). X =
∞
n=1 Xn , 1 Xn X
, i.e., X ∃
B ⊂ ∃
Xn.
. Xn . X1
X1 ̸= X (X ). ∃
x1 ∈ X X1. X1 closed , d1 := d(x1, X1) =
infx∈X1 d(x1, x) > 0. ρ1 := 1 ∧ (d1/2) ≤ 1, B1 := B(x1, ρ1) B1 ∩ X1 = ∅.
X2 , ∃
x2 ∈ B1 X2, d2 := d(x2, X2) > 0. x2 /∈ X B1
( ) , d′
2 := d(x2, X B1) > 0. ρ2 := min{1/2, d2/2, d′
2/2} ≤ 1/2, B2 := B(x2, ρ2)
B2 ⊂ B1, B2 ∩ X2 = ∅. B1 ⊃ B2 ⊃ · · · , Bk ∩ Xk = ∅, ρk ≤ 1/k
{Bk} . Bk xk k < m d(xk, xm) ≤ ρk ≤ 1/k
, Cauchy . X ∃
x ∈ X; xk → x. ∀
k, x ∈ Bk Bk ∩ Xk = ∅ ,
x /∈ Xk, i.e., x /∈
∞
k=1 Xk ,
∞
k=1 Xk = X .

18. Functional Analysis 16
6.1 (Uniform bounded principle)
6.2 ( ) X Banach sp., Y normed sp. . {Tλ}λ∈Λ ⊂
L(X, Y ) ,
∀
x ∈ X, sup
λ∈Λ
∥Tλx∥ < ∞ =⇒ sup
λ∈Λ
∥Tλ∥ < ∞.
Xn := λ∈Λ{x ∈ X; ∥Tλx∥ ≤ n} Tλ Xn ,
∞
n=1 Xn = X . X Baire , Xn0
,
i.e., ∃
x0 ∈ X, ρ0 > 0; B(x0, ρ0) ⊂ Xn0 . ∀
y ∈ B(0, ρ0), y + x0 ∈ B(x0, ρ0) , y = (y + x0) − x0
∥Tλy∥ ≤ ∥Tλ(y + x0)∥ + ∥Tλx0∥ ≤ 2n0. ∀
x ∈ X , µ = 2∥x∥/ρ0
∥µ−1
x∥ = ρ0/2 < ρ0 µ−1
x ∈ B(0, ρ0) , ∥Tλx∥ = µ∥Tλ(µ−1
x)∥ ≤ 2n0µ = 4n0∥x∥/ρ0.
, supλ ∥Tλ∥ ≤ 4n0/ρ0 .
6.3 (Banach-Steinhaus theorem) X Banach sp., Y normed sp. {Tn} ⊂
L(X, Y ) . ∀
x ∈ X, {Tnx} , Tx := lim
n→∞
Tnx T ∈ L(X, Y )
∥T∥ ≤ limn→∞ ∥Tn∥.
γ := limn→∞ ∥Tn∥ . {Tnx} , supn ∥Tnx∥ < ∞. X Banach ,
supn ∥Tn∥ < ∞. γ < ∞. ∀
ε > 0 , ∃
{nk}; ∥Tnk
∥ ≤ γ + ε.
∀
x ∈ X , ∥Tx∥ = limk→∞ ∥Tnk
x∥ ≤ (γ + ε)∥x∥. T ∈ L(X, Y ) ,
∥T∥ ≤ γ + ε. ε ∥T∥ ≤ γ.
6.2 (Open mapping theorem)
6.4 ( ) X, Y Banach sps . T ∈ L(X, Y ) , R(T) = Y
T , i.e., ∀
U ⊂ X; open, T(U) ⊂ Y ; open.
(1st Step) ∃
ρ > 0; BY (0, ρ) ⊂ TBX(0, 1) . R(T) = Y ,
Y = T(X) =
∞
n=1
TBX(0, n) =
∞
n=1
TBX(0, n) , Y Baire ,
∃
n ≥ 1, a ∈ Y, δ > 0; BY (a, δ) ⊂ TBX(0, n). ∀
y ∈ BY (0, δ) . y + a, a ∈ BY (a, δ) ⊂
TBX(0, n) , ∃
yk, y′
k ∈ TBX(0, n); yk → y + a, y′
k → a. yk − y′
k ∈ TBX (0, 2n) ,
, y = (y + a) − a = lim(yk − y′
k) ∈ TBX(0, 2n). BY (0, δ) ⊂ TBX (0, 2n).
ρ = δ/(2n) T BY (0, ρ) ⊂ TBX (0, 1).
(2nd Step) ρ , η = ρ/2 > 0 BY (0, η) ⊂ TBX(0, 1) .
BY (0, ρ) ⊂ TBX (0, 2), i.e, ∀
y ∈ BY (0, ρ) , ∃
x ∈ BX(0, 2); y = Tx .
εk = 2−k
(k ≥ 0) , BY (0, εkρ) ⊂ TBX(0, εk). y ∈ BY (0, ρ) ⊂
TBX(0, 1) ∃
x0,n ∈ BX (0, 1); Tx0,n → y , ∃
x0 ∈ BX(0, 1); ∥y − Tx0∥ < ε1ρ.
y − Tx0 ∈ BY (0, ε1ρ) , ∃
x1 ∈ BX(0, ε1); ∥y − Tx0 − Tx1∥ < ε2ρ. ,
∃
xk ∈ BX (0, εk); ∥y −
k
j=0
Txj∥ < εk+1ρ. ∥
m
j=k
xj∥ ≤
m
j=k
∥xj∥ ≤
m
j=k
εj → 0 (m >
k → ∞) , {
k
j=0
xj} Cauchy in X , ∃
x =
∞
k=0
xk ∈ X. T

19. Functional Analysis 17
Tx =
∞
k=0
Txk. ∥x∥ ≤ ∥x0∥ +
∞
k=1
∥xk∥ < ∥x0∥ +
∞
k=1
εk < 1 + 1 = 2. x ∈ BX (0, 2).
∥y −
k
j=0
Txj∥ < εk+1ρ k → ∞ y =
∞
k=0
Txk = Tx.
(3rd Step) T , i.e., ∀
U ⊂ X; open, T(U) ⊂ Y ; open . T
, ∀
α > 0, BY (0, αη) ⊂ TBX (0, α). ∀
y0 ∈ T(U) . ∃
x0 ∈ U; y0 = Tx0. U open ,
∃
δ > 0; x0 + BX (0, δ) = BX(x0, δ) ⊂ U. ∀
y ∈ BY (y0, δη) , y′
:= y − y0 ∈ BY (0, δη)
. BY (0, δη) ⊂ TBX(0, δ) ∃
x′
∈ BX (0, δ); y′
= Tx′
. x0 + x′
∈ U ,
y = y0 + y′
= Tx0 + Tx′
= T(x0 + x′
) ∈ T(U). BY (y0, δη) ⊂ T(U) , T(U) open.
6.5 ( (range theorem)) X, Y Banach sps. . T ∈ L(X, Y )
, R(T) = Y T 1 to 1 T−1
∈ L(Y, X).
T−1
, Y X .
. ∀
U ⊂ X; open , S := T−1
S−1
(U) = {y ∈ Y ; Sy = T−1
y ∈
U} = {y ∈ Y ; y ∈ T(U)} T(U) . , T(U) open. , S−1
(U)
open , S = T−1
. T−1
∈ L(Y, X).
6.3 (Closed graph theorem)
, T D(T) ̸= X .
6.1 (X, ∥ · ∥X), (Y, ∥ · ∥Y ) normed sps. . T X Y
(closed op.) T : D(T) ⊂ X → Y , T G(T) = {(x, Tx) ∈
X × Y ; x ∈ D(T)} ∥(x, Tx)∥G = ∥x∥X + ∥Tx∥Y ,
xn ∈ D(T) → x in X, Txn → y in Y (x, y) ∈ G(T), i.e., x ∈ D(T) y = Tx.
D = D(T) , T
def
⇐⇒ xn → x in D Txn → Tx in Y ,
T D .
6.6 (i) D(T) closed , T : D(T) ⊂ X → Y (= ) T .
D(T) = X .
(ii) Y Banach , T : D(T) ⊂ X → Y , T D(T) T
, , T .
(iii) X, Y Banach , T ⇐⇒ D(T) ∥x∥G := ∥x∥X + ∥Tx∥Y
.
(i) xn ∈ D(T) → x ∈ X, Txn → y ∈ Y D(T) closed , x ∈ D(T) ,
y = Tx, T .
(ii) xn ∈ D(T) → x ∈ X . {Txn} Y Cauchy , Y
Txn → ∃
y ∈ Y . Tx := y , T D(T) , T = T on D(T),
∥T∥ = ∥T∥ . (i) , T .
(iii) (⇒) {xn} ⊂ D(T); ∥xn − xm∥G → 0 (m, n → ∞) . X, Y xn → ∃
x in
X, Txn → ∃
y in Y . x ∈ D(T), y = Tx. ∥xn − x∥G = ∥xn − x∥X +
∥Txn − Tx∥Y → 0 , D(T) .

20. Functional Analysis 18
(⇐) xn ∈ D(T) → x ∈ X, Txn → y ∈ Y {xn} D(T) ∥·∥G Cauchy
. ∃
x∗
∈ D(T); ∥xn − x∗
∥G → 0. x∗
= x, Tx = Tx∗
= y
, T closed .
D(T) closed , Y Banach , T
.
6.1 X = C[0, 1], D(T) = C1
[0, 1] (Tx)(t) = x′
(t) , ,
.
6.7 ( ) X, Y Banach sps, T X Y .
D(T) = X T ∈ L(X, Y ).
Z = G(T) = {(x, Tx) ∈ X × Y ; x ∈ D(T) = X} . T
X, Y Banach , ∥(x, Tx)∥Z := ∥x∥ + ∥Tx∥ Banach
. S : Z → X; S(x, Tx) = x S ∈ L(Z, X) ,
R(S) = X S 1 to 1 . X Banach , S−1
. ∥Tx∥ ≤ ∥(x, Tx)∥Z = ∥S−1
x∥Z ≤ ∃
M∥x∥. T , i.e., T ∈ L(X, Y )
.
7 (Linear Functionals)
normed sp X K = R or C
(bdd lin. functional) . , (conti. lin. functional)
. X∗
:= L(X, K) , X (dual sp.) .
K = R or C , X∗
. f ∈ X∗
,
i.e., f : X → K; f(αx + βy) = αf(x) + βf(y) (α, β ∈ K, x, y ∈ X), ∥f∥ < ∞.
7.1 (Dual spaces)
X Hilbert sp. H , H∗
= H . Rn
, Cn
, L2
(Ω), l2
dual sps
. .
7.1 ( (Riesz’s representation theorem)) X = H Hilbert
sp. . ∀
f ∈ H∗
, ∃1
y ∈ H; f(x) = ⟨x, y⟩ (∀
x ∈ H). ∥f∥ = ∥y∥ .
H∗
= H .
f ≡ 0 y = 0 . f ̸≡ 0 . N = {x ∈ H; f(x) = 0}
. ( , x ∈ N, α ∈ K f(αx) = αf(x) = 0 , αx ∈ N. x, x′
∈ N
f(x + x′
) = f(x) + f(x′
) = 0 , x + x′
∈ N. . xn ∈ N → x ∈ H
f(x) = lim f(xn) = 0 x ∈ N. N .) H = N ⊕ N⊥
. f ̸≡ 0
, N⊥
̸= ∅. y0 ∈ N⊥
; y0 ̸= 0 1 . f(y0) ̸= 0. y := (f(y0)/∥y0∥2
)y0 , y ∈ N⊥
, . , ∀
x ∈ H , f(y0)x − f(x)y0 ∈ N ,
0 = ⟨f(y0)x − f(x)y0, y⟩ = f(y0)⟨x, y⟩ − f(x)⟨y0, y⟩ = f(y0)(⟨x, y⟩ − f(x))

21. Functional Analysis 19
, f(x) = ⟨x, y⟩. ∃
y′
∈ H; f(x) = ⟨x, y⟩ = ⟨x, y′
⟩ ∀
x ∈ H, ⟨x, y−y′
⟩ = 0
, x = y − y′
∥y − y′
∥ = 0, i.e., y = y′
. y . ∥x∥ = 1
, Schwartz , |f(x)| = |⟨x, y⟩| ≤ ∥x∥∥y∥ = ∥y∥. ∥f∥ ≤ ∥y∥. x0 = y/∥y∥
f(x0) = ⟨x0, y⟩ = ∥y∥ ∥y∥ = f(x0) ≤ sup
∥x∥=1
|f(x)| = ∥f∥. ∥f∥ = ∥y∥.
Banach sp. X , X∗
, .
7.1 1 ≤ p < ∞ . q p , i.e., 1/p + 1/q = 1 ( , p = 1
q = ∞). (i) Ω ⊂ Rn
(Lp
(Ω))∗
= Lq
(Ω). (ii) (lp
)∗
= lq
.
7.2 l∞
0 := {(xn) ∈ l∞
; limn→∞ xn = 0} l∞
0 l∞
∥ · ∥∞
Banach , (l∞
0 )∗
= l1
.
7.2 (Hahn-Banach’s extension thoerem)
7.2 ( ) X , L ⊂ X . f
L . ∃
p : X → R; , p(λx) = λp(x) (λ > 0, x ∈ X),
p(x + y) ≤ p(x) + p(y) (x, y ∈ X) , f ≤ p on L , ∃
F ∈ X∗
= L(X, R); F = f
on L, F ≤ p on X . , f X , f ≤ p
.
L = X , L ̸= X . x0 ∈ X L ,
L1 = L + Rx0 . .
x = y + tx0 ∈ L1 .
f L1 , c ∈ R , F(x) = F(y + tx0) := f(y) + tc .
F L1 .
c F ≤ p on L1.
Zorn f X F ≤ p .
L1 , x = y+tx0 = y′
+t′
x0 (y, y′
∈ L, t, t′
∈ R)
0 = (y−y′
)+(t−t′
)x0, i.e, (t−t′
)x0 = y′
−y ∈ L. t ̸= t′
x0 = (y′
−y)/(t−t′
) ∈ L
, . t = t′
, y = y′
.
xi = yi + tix0 ∈ L1 (yi ∈ L, ti ∈ R) αi ∈ R , F(α1x1 + α2x2) = F((α1y1 +
α2y2)+(α1t1 +α2t2)x0) = f(α1y1 +α2y2)+(α1t1 +α2t2)c = α1(f(y1)+t1c)+α2(f(y2)+t2c) =
α1F(x1) + α2F(x2).
y, y′
∈ L , f(y) + f(y′
) = f(y + y′
) ≤ p(y + y′
) = p(y + x0 + y′
− x0) ≤
p(y + x0) + p(y′
− x0) , f(y′
) − p(y′
− x0) ≤ p(y + x0) − f(y). β1 := supy′∈L(f(y′
) −
p(y′
− x0)), β2 := infy∈L(p(y + x0) − f(y)) , β1 ≤ β2 β1 ≤ c ≤ β2 c
, f(y) + c ≤ p(y + x0), f(y′
) − c ≤ p(y′
− x0) (y, y′
∈ L). t > 0
F(x) = F(y + tx0) = f(y) + tc = t(f(y/t) + c) ≤ tp(y/t + x0) = p(y + tx0) = p(x). t < 0
F(x) = F(y + tx0) = f(y) + tc = (−t)(f(−y/t) − c) ≤ −tp(−y/t − x0) = p(y + tx0) = p(x).
t = 0 F(x) = F(y) = f(y) ≤ p(y) = p(x). F ≤ p on L1.
g ∈ Φ
def
⇐⇒ g : Lg → R; , L ⊂ Lg, g = f on L, g ≤ p on Lg .
Φ ̸= ∅ . Φ ( ) . g, g′
∈ Phi , g ≼ g′ def
⇐⇒
Lg ⊂ Lg′ , g = g′
on Lg . Φ {gλ} . Lλ := Lgλ

22. Functional Analysis 20
. L0 := Lλ , g0 on L0 g0 = gλ on Lλ , g0 ∈ Φ {gλ}
. Φ . Zorn , ∃
F ∈ Φ
, i.e, g ∈ Φ; F ≼ g g = F. F X .
, , F Φ . F
.
7.3 ( ) X , L ⊂ X . f L
. ∃
p : X → C; , p(λx) = |λ|p(x) (λ ∈ C, x ∈ X),
p(x + y) ≤ p(x) + p(y) (x, y ∈ X) , |f| ≤ p on L , ∃
F ∈ X∗
= L(X, C);
F = f on L, |F| ≤ p on X . , f X , f ≤ p
.
. f(x) = g(x) + ih(x)
. g, h L .
g, h ≤ |f| ≤ p on L. g X G; G ≤ p on X
. −G(x) = G(−x) ≤ p(−x) = p(x) |G| ≤ p. , g(ix) + ih(ix) =
f(ix) = if(x) = ig(x) − h(x) , h(x) = −g(ix). F(x) := G(x) − iG(ix)
, . , F = f on L F(x1 + x2) = F(x1) + F(x2) ,
F(ix) = G(ix)−iG(−x) = i(−iG(ix)+G(x)) = iF(x) a ∈ R F(ax) = aF(x) , α ∈ C
F(αx) = αF(x) . F(x) = reiθ
F(e−iθ
x) = e−iθ
F(x) ∈ R ,
G(e−iθ
x) , |F(x)| = |eiθ
F(x)| = |G(e−iθ
x)| ≤ p(e−iθ
x) = |e−iθ
|p(x) = p(x).
7.1 X ( ) , L ⊂ X . f L ( )
. ∃
F ∈ X∗
; F = f on L, ∥F∥X = ∥f∥L.
p(x) = ∥f∥L∥x∥ , .
7.2 X . ∀
x0 ∈ X, ̸= 0, ∃
g ∈ X∗
; g(x0) = ∥x0∥, ∥g∥ = 1.
L := ⟨x0⟩ = {tx0; t ∈ K} f(x) = f(tx0) := t∥x0∥ (x = tx0 ∈ L) ,
. |f(x)| = |t|∥x0∥ = ∥tx0∥ = ∥x∥ ∥f∥ = 1.
7.3 X , L X . x0 ∈ X L , d := infy∈L ∥x0 −
y∥ > 0 . ∃
f ∈ X∗
; f = 0 on L, f(x0) = 1, ∥f∥ ≤ 1/d.
L1 := L + Rx0 g(x) = t (x = y + tx0 ∈ L1) g = 0 on L, g(x0) = 1,
∥g∥L1 ≤ 1/d. .
, . 2 , Banach sp., ,
, , , (= compact ) ,
, . ,
, , , .

23. Functional Analysis 21
A
, .
A.1 Lp
(X, F, µ) (X a set, F σ-field on X, µ = µ(dx) on X) , H¨older
, Minkovsky .
f ∈ Lp
(X) (or Lp
(X, dµ) or Lp
(X, F, µ) ) ∥f∥Lp < ∞ .
,
∥f∥Lp =
X
|f(x)|p
µ(dx)
1/p
(1 ≤ p < ∞),
∥f∥∞ = ess.supx∈X|f(x)| := inf{α; |f(x)| ≤ α µ-a.e}.
, 1 ≤ p ≤ ∞ , 1 ≤ q ≤ ∞ 1/p + 1/q = 1 . ,
p = 1 q = ∞, p = ∞ q = 1 . H¨older .
∥fg∥L1 ≤ ∥f∥Lp ∥g∥Lq , ∥fg∥L1 ≤ ∥f∥L1 ∥g∥∞ (∥fg∥L1 ≤ ∥f∥∞∥g∥L1 ).
Minkovsky ∥f + g∥Lp ≤ ∥f∥Lp + ∥g∥Lp , Lp
(X)
.
(X, F) = (N, 2N
) µ = n≥1 δn counting measure ( ) , N
f(n) ,
N
f(n)µ(dn) =
n≥1
f(n) . x = (x1, x2, . . . ) ,
f(n) = |xn|p
n≥1
|xn|p
. lp
Banach sp.
.
A.2
A.1 X , ∃
X Banach sp., ∃
J : X → X ; ∥Jx∥ = ∥x∥
(x ∈ X), J(X) dense in X
J(X) X , X ⊂ X X = X . X X
(completion) .
X X Cauchy . {xn}, {yn} ∈ X , {xn} ∼ {yn} ⇐⇒
xn − yn → 0 (n → ∞) . , x = [{xn}] ∈
X := X/ ∼ .
x = [{xn}], y = [{yn}] ∈ X, α ∈ K , αx := [{αxn}], x + y := [{xn + yn}]
, X . x = [{xn}] ∈ X , |∥xn∥ − ∥xm∥| ≤ ∥xn − xm∥ → 0
, {∥x∥} R Cauchy , ∃
lim
n→∞
∥xn∥ =: ∥x∥ . X
( ).
x ∈ X , xn = x , Jx := [{xn = x}] x
, X ⊂ X . , J : X → X , ∥Jx∥ = ∥x∥ . ∀
x ∈ X ,

24. Functional Analysis 22
{xn} ∀
ε > 0, ∃
N; ∀
n, m ≥ N, ∥xm − xn∥ < ε , n ≥ N ,
Jxn ∈ X , x − Jxn = [{xm − xn}m≥1]
∥x − Jxn∥ = lim
m→∞
∥xm − xn∥ ≤ ε
∥x − Jxn∥ → 0 (n → ∞). , J(X) dense in X.
X . {xn} X Cauchy . xn {x
(n)
k }k≥1
Cauchy ∃
kn; ∀
m > kn, ∥x
(n)
m − x
(n)
kn
∥ ≤ 1/n. x := [{x
(n)
kn
}]
, x ∈ X , {xn} . , x ∈ X {x
(n)
kn
} ∈ X (i.e., X
Cauchy ) .
(A.1) ∥xn − Jx
(n)
kn
∥ = lim
m→∞
∥x
(n)
km
− x
(n)
kn
∥ ≤
1
n
.
∥x
(n)
kn
− x
(m)
km
∥ = ∥Jx
(n)
kn
− Jx
(m)
km
∥ ≤ ∥Jx
(n)
kn
− xn∥ + ∥xn − xm∥ + ∥xm − Jx
(m)
km
∥
≤ ∥xn − xm∥ +
1
n
+
1
m
→ 0 (n, m → ∞).(A.2)
{x
(n)
kn
} ∈ X. (A.1) ,
∥x − xn∥ ≤ ∥x − Jx
(n)
kn
∥ + ∥Jx
(n)
kn
− xn∥ ≤ ∥x − Jx
(n)
kn
∥ +
1
n
.
(A.2) ,
∥x − Jx
(n)
kn
∥ = lim
p→∞
∥x
(p)
kp
− x
(n)
kn
∥ ≤ lim
p→∞
∥xp − xn∥ +
1
n
. , 2 ,
lim
n→∞
∥x − xn∥ ≤ lim
n→∞
∥x − Jx
(n)
kn
∥ ≤ lim
n,p→∞
∥xp − xn∥ = 0,
xn → x in X , X .
A.3 Hk,p
(Ω)
Ω ⊂ Rn
, Ck
(Ω) k . α =
(α1, . . . , αn) (multi-index ) , |α| := α1 + · · · + αn, ∂α
x := ∂α1
1 · · · ∂αn
n . ,
∂j = ∂/∂xj .
Ck,p
(Ω) :=
⎧
⎪⎨
⎪⎩
u ∈ Ck
(Ω); ∥u∥k,p :=
⎛
⎝
α;|α|≤k Ω
|∂α
x u(x)|p
dx
⎞
⎠
1/p
< ∞
⎫
⎪⎬
⎪⎭
. (Ck,p
(Ω), ∥ · ∥k,p) Hp,k
(Ω) , Sobolev sp. .
{un} Cauchy in Ck,p
(Ω) ⇐⇒ ∀
α; 1 ≤ |α| ≤ k,
Ω
|∂α
x un(x) − ∂α
x um(x)|p
dx → 0 (m, n →
∞). Lp
(Ω) , ∃
uα
∈ Lp
(Ω);
Ω
|∂α
x un(x) − ∂α
x uα
(x)|p
dx → 0 (n → ∞).
(uα
)|α|≤k Hp,k
(Ω) , uα
= ∂α
x u .

25. Functional Analysis 23
A.4
A.2 [a, b] f(t) Pn(t) ,
i.e., ∃
{Pn(t)}; Pn
→
→ f on [a, b], i.e., lim
n→∞
sup
t∈[a,b]
|Pn(t) − f(t)| = 0.
t′
= (t − a)/(b − a) , [a, b] [0, 1] , (
) , [a, b] = [0, 1] . t ∈ [0, 1] ,
n
k=0
(k − nt)2 n
k
tk
(1 − t)n−k
= nt(1 − t) ≤
1
4
n
. ( t(1 − t) ≤ (t + (1 − t))/2 = 1/2 .) ,
,
n
k=0
n
k
xk
yn−k
= (x + y)n
,
n
k=0
n
k
tk
(1 − t)n−k
= 1
, x , x ,
n
k=0
k
n
k
xk
yn−k
= nx(x + y)n−1
,
n
k=0
k2 n
k
xk
yn−k
= nx(nx + y)(x + y)n−2
x = t, y = 1 − t (k − nt)2
= k2
− 2ntk + n2
t2
.
Pn(t) =
n
k=0
f
k
n
n
k
tk
(1 − t)n−k
Pn [0, 1] , f .
|f(t) − Pn(t)| ≤
n
k=0
f(t) − f
k
n
n
k
tk
(1 − t)n−k
, f [0, 1] , ∀
ε > 0, ∃
δ > 0; ∀
t, t′
∈ [0, 1]; |t−t′
| < δ, |f(t)−f(t′
)| < ε.
t , k |t−k/n| < δ |t−k/n| ≥ δ , S1, S2
,
|f(t) − Pn(t)| ≤ S1 + S2, , S1 ≤ ε
n
k=0
n
k
tk
(1 − t)n−k
= ε.
M = maxt∈[0,1] |f(t)| |f(t) − f(k/n)| ≤ 2M, |t − k/n| ≥ δ 1 ≤
|nt − k|/(nδ) ,
S2 ≤ 2M
n
k=0
|nt − k|
nδ
2
n
k
tk
(1 − t)n−k
≤
M
2nδ2
.
|f(t) − Pn(t)| ≤ ε +
M
2nδ2
.
t ∈ [0, 1] , supt∈[0,1] , n → ∞
lim
n→∞
sup
t∈[0,1]
|f(t) − Pn(t)| ≤ ε
, ε > 0 , ( )= 0 .

26. Functional Analysis 24
A.5
3.5
dim X = n {x1, . . . , xn} basis ∀
x ∈ X, ∃
(α1, . . . , αn); x = αixi.
∥x∥∞ = max |αi| 1 norm . ∥x∥ .
∥x∥ ≤ |αi|∥xi∥ ≤ max |αi| · ∥xi∥ = c1∥x∥∞ (c1 = ∥xi∥).
. . ∀
k ≥ 1, ∃
yk; ∥yk∥ > k∥yk∥ , ∥(yk/∥yk∥∞)∥ < 1/k → 0.
, zk = yk/∥yk∥∞ , ∥zk∥∞ = 1 , ∥zk∥ < 1/k → 0, i.e., zk → 0 under ∥ · ∥.
, zk =
n
i=1 β
(k)
i xi , 1 = ∥zk∥∞ = max |β
(k)
i | , ∃
β
(kj )
i → ∃
βi in K ,
∃
zkj → ∃
z under ∥ · ∥∞, under ∥ · ∥ by ( ). z = 0 ,
1 = ∥zkj ∥ → ∥z∥ . ∃
c0 > 0; ∥x∥ ≥ c0∥x∥∞ (x ∈ X).
.
[ ] ∃
c0 > 0; ∥x∥ ≥ c0∥x∥∞ (x ∈ X), i.e., 0 < c0 ≤ ∥x∥/∥x∥∞ = ∥(x/∥x∥∞)∥
. y = x/∥x∥∞ , f(y) := ∥y∥ , ∃
miny;∥y∥∞=1 f(y) > 0
. yk → y under ∥ · ∥∞ under ∥ · ∥ ,
f(y) conti. under ∥ · ∥. S := {y; ∥y∥∞ = 1} , compact
. , y =
n
i=1 βixi , 1 = ∥y∥∞ = max |βi| , y (β1, . . . , βn) ∈ Kn
, S = {(β1, . . . , βn); max |βi| = 1} Kn
. S (
) compact . , ∀
y ∈ S, f(y) > 0 , compact set
, ∃
y0 ∈ S; minS f = f(y0) = ∥y0∥ > 0 , c0 = ∥y0∥ .
A.6
A.3 X , ∥x + y∥2
+ ∥x − y∥2
= 2(∥x∥2
+ ∥y∥2
)
,
⟨x, y⟩ =
1
4
∥x + y∥2
− ∥x − y∥2
+ i∥x + iy∥2
− i∥x − iy∥2
, .
.
⟨x, x⟩ = ∥x∥2
, ⟨y, x⟩ = ⟨x, y⟩, ⟨ix, y⟩ = i⟨x, y⟩, ⟨x, −y⟩ = −⟨x, y⟩, ⟨x, 0⟩ = ⟨0, x⟩ = 0.
, , .
(1) Re⟨x1, y⟩ + Re⟨x2, y⟩ =
1
2
Re⟨x1 + x2, 2y⟩ = Re⟨x1 + x2, y⟩.
(2) Re .
(3) α ∈ R ⟨αx, y⟩ = α⟨x, y⟩. γ ∈ C , ⟨γx, y⟩ = γ⟨x, y⟩.
(1) , i.e., ⟨x, y⟩ = Re⟨x, y⟩.
⟨x1, y⟩ + ⟨x2, y⟩ =
1
4
(∥x1 + y∥2
+ ∥x2 + y∥2
) − (∥x1 − y∥2
+ ∥x2 − y∥2
)
=
1
8
(∥x1 + x2 + 2y∥2
+ ∥x1 − x2∥2
) − (∥x1 + x2 − 2y∥2
+ ∥x1 − x2∥2
)
=
1
8
(∥x1 + x2 + 2y∥2
− ∥x1 + x2 − 2y∥2
)
=
1
2
⟨x1 + x2, 2y⟩.

27. Functional Analysis 25
⟨x1, y⟩ + ⟨x2, y⟩ = ⟨x1 + x2, 2y⟩/2. x2 = 0 , ⟨0, y⟩ = 0 ,
⟨x, y⟩ = ⟨x, 2y⟩/2. , ⟨x1, y⟩ + ⟨x2, y⟩ = ⟨x1 + x2, 2y⟩ = ⟨x1 + x2, y⟩/2.
Re⟨x1, y⟩ + Re⟨x2, y⟩ = Re⟨x1 + x2, y⟩.
(2) ⟨x1 + x2, y⟩ + ⟨y, x1 + x2⟩ = (⟨x1, y⟩ + ⟨y, x1⟩) + (⟨x2, y⟩ + ⟨y, x2⟩) y iy
, ⟨x1 + x2, y⟩ − ⟨y, x1 + x2⟩ = (⟨x1, y⟩ − ⟨y, x1⟩) + (⟨x2, y⟩ − ⟨y, x2⟩) ,
⟨x1 + x2, y⟩ = ⟨x1, y⟩ + ⟨x2, y⟩.
(3) y ∈ X α ∈ R , f(α) := ⟨αx, y⟩ . . (2)
, n ∈ N , f(1) = f(n/n) = nf(1/n), i.e., f(1/n) = (1/n)f(1).
f(m/n) = (m/n)f(1) (m, n ∈ N). f(−α) = −f(α) , , ∀
r ∈ Q, f(r) = rf(1).
, ∀
α ∈ R, f(α) = αf(1), i.e., ⟨αx, y⟩ = α⟨x, y⟩. β ∈ R , ⟨iβx, y⟩ =
i⟨βx, y⟩ = iβ⟨x, y⟩. ∀
γ ∈ C, ⟨γx, y⟩ = γ⟨x, y⟩ .
A.7 A2
(Ω)
Ω ⊂ Cn
open ,
f ∈ A2
(Ω)
def
⇐⇒ f on Ω,
Ω
|f(z)|2
dxdy < ∞ (z = x + iy).
f, g ∈ A2
(Ω) , (f, g) =
Ω
f(z)g(z)dxdy Hilbert.
. {fn} A2
(Ω) Cauchy . z = x+iy ∈ C (x, y) ∈ R2
, fn ∈ L2
(Ω) Cauchy ∃
f ∈ L2
(Ω); ∥fn −f∥L2 → 0.
f . z ∈ Ω , Cauchy ,
fn(z) =
1
2πi |ζ−z|=r
fn(ζ)
ζ − z
dζ (0 < r ≤ ε)
, ε > 0 ; {|ζ − z| < ε} ⊂ Ω. r , r 0 ε
(A.3)
1
2
ε2
fn(z) =
1
2πi
ε
0
drr
|ζ−z|=r
fn(ζ)
ζ − z
dζ =
1
2π |ζ−z|≤ε
fn(ζ)dξdη (ζ = ξ + iη).
( |ζ − z| = r ζ − z = r(cos θ + i sin θ) , dζ = r(− sin θ + i cos θ)dθ =
ir(cos θ + i sin θ)dθ = i(ζ − z)dθ ,
ε
0
drr
|ζ−z|=r
dζ
ζ − z
=
ε
0
drr
2π
0
idθ = i
|ζ−z|≤ε
dξdη.)
fn L2
(Ω) (A.3) z Ω . (∀
K ⊂ Ω
compact 1 , ε > 0 Kε := {ζ; |ζ − z| ≤ ε, z ∈ K} ⊂ Ω .
sup
z∈K |ζ−z|≤ε
|fn(ζ) − f(ζ)|dξdη ≤
Kε
|fn(ζ) − f(ζ)|dξdη ≤ ∥fn − f∥L2 |Kε|1/2
→ 0).
(A.3) 2/ε2
f∗
. fn → f in L2
(Ω) , f = f∗
a.e. f ∈ A2
(Ω).

28. Functional Analysis 26
B ,
X , X∗∗
:= (X∗
)∗
X 2 . X ⊂ X∗∗
. X∗∗
= X X Banach sp. . 1 < p < ∞ , LP
(Ω)
lp
Banach sps .
B.1
X normed ps. . ,
.
xn → x (strong) in X
def
⇐⇒ ∥xn − x∥ → 0. s-limn→∞ xn = x .
xn → x (weak) in X
def
⇐⇒ ∀
f ∈ X∗
, f(xn) → f(x). w-limn→∞ xn = x
.
.
B.1 xn → x (weak) , .
[ ] xn → x′
(weak) . x ̸= x′
, ∃
f ∈ X∗
; f(x − x′
) =
∥x − x′
∥ ̸= 0, ∥f∥ = 1. f(x − x′
) = f(x) − f(x′
) = w- lim(f(xn) − f(xn)) = 0 ,
. qed
B.1 X = l2
f ∈ l2
, ∃1
y = (yk) ∈ l2
; f(x) = xkyk (x = (xk) ∈ l2
).
x(n)
= (x
(n)
k = δn,k) ∈ l2
f(x(n)
) = yn → 0 , x(n)
→ 0 (weak).
∥x(n)
− x(m)
∥l2 =
√
2 (m ̸= n). {x(n)
} .
B.1 X , xn → x (weak) in X . {∥xn∥}
∥x∥ ≤ lim inf ∥xn∥ .
∀
f ∈ X∗
, Tn(f) := f(xn) Tn ∈ X∗∗
. , {Tn(f)} . X∗
Banach , , sup ∥Tn∥ < ∞. T(f) := f(x) = lim f(xn) = lim Tn(f)
T ∈ X∗∗
, ∥T∥ = ∥x∥. ∥Tn∥ = ∥xn∥ , Banach-Steinhaus
, ∥x∥ = ∥T∥ ≤ lim inf ∥Tn∥ = lim inf ∥xn∥.
B.2 H Hilbert sp. . xn → x (weak) in H, , ∥xn∥ → ∥x∥
xn → x (strong) in H.
xn → x (weak) in H Riesz , ∀
y ∈ H, ⟨xn, y⟩ → ⟨x, y⟩
. ∥xn − x∥2
= ∥xn∥2
+ ∥x∥2
− 2ℜ⟨xn, x⟩ →
∥x∥2
+ ∥x∥2
− 2ℜ⟨x, x⟩ = 0.
X .
fn → f (weak*) in X∗
: ( * )
def
⇐⇒ ∀
x ∈ X, fn(x) → f(x).
B.3 X , fn → f (weak*) in X∗
. ∥f∥ ≤ lim inf ∥fn∥.
B.2 .

29. Functional Analysis 27
B.2
B.1 X, Y . T : D(T) ⊂ X → Y , D(T)
dense in X . g ∈ D(T∗
) ⊂ Y ∗ ∃
f ∈ X∗
; g ◦ T = f on D(T) ,
T∗
: D(Y ∗
) ⊂ Y ∗
→ X∗
T∗
(g) = f . , T∗
(g) = g ◦ T. T∗
T (adjoint op.) .
g ◦ T = f f g , . , g ◦ T = f′
, f = f′
on D(T) , D(T) dense , f, f′
f = f′
on X .
T , T X ,
D(T) = X . ∀
g ∈ Y ∗
, f := g ◦ T f ∈ X∗
,
D(T∗
) = Y ∗
.
X = H, Y = H′
Hilbert ⟨x, T∗
y⟩ = ⟨Tx, y⟩ (x ∈ D(T), y ∈ D(T∗
)) .
D(T∗
) = D(T) T = T∗
on D(T) T (self-adjoint
op.) . D(T) ⊂ D(T∗
) T = T∗
on D(T) , , ⟨x, Tx⟩ = ⟨Tx, y⟩
(x, y ∈ D(T)) T (symmetric op.) .
B.2 X = Y = Rn
, T Rn
Rn
tj,k := ⟨Tej, ek⟩
, T = (tj,k) . T∗
= t
T; T . X = Y = Cn
T∗
= t
T; .
B.3 H = L2
(0, 1) . k(t, s) [0, 1]2
, x ∈ H ,
Tx(t) =
[0,1]
k(t, s)x(s)ds T H , D(T) = D(T∗
) = H.
T∗
y(t) =
[0,1]
k(t, s)y(s)ds .
C
T n , ∃
λ ∈ C, ∃
x ∈ Rn
; x ̸= 0, Tx = λx λ
T , x T . λ ⇐⇒ det(λI − T) = 0.
λ , , (λI − T)x = 0 x 0 (Ker(λI − T) = {0}, i.e.,
λI − T 1 to 1) , det(λI − T) ̸= 0 , ∃
(λI − T)−1
.
, T X ,
z ∈ ρ(T)
def
⇐⇒ z ∈ C; Ker(zI − T) = {0} (zI − T 1 to 1) (zI − T)−1
∈ L(X).
, ρ(T) T (resolvent set) . R(z) := (zI − T)−1
T
. σ(T) := C ρ(T) T (spectrum) .
∃
x ∈ D(T); x ̸= 0, Tx = zx, i.e., (zI − T)x = 0 z ∈ C T ,
σp(T) , (point spectrum) . N(zI − T) := Ker(zI − T)
z , z . σp(T) ⊂ σ(T).
z1, z2 ∈ ρ(T) R(z1) − R(z2) = (z1 − z2)R(z1)R(z2) (resolvent eqution) .
ρ(T) . σ(T) .
R(z): holmorphic on ρ(T).
T ∈ L(X) , r(T) := lim sup n
∥Tn∥ (spectral radius) .

30. Functional Analysis 28
(i) |z| > r(T) =⇒ z ∈ ρ(T), R(z) =
k≥0
1
zk+1
Tk
=
1
z
+
1
z2
T +
1
z3
T2
+ · · ·.
(ii) ∃
z ∈ σ(T); |z| = r(T).
D
X, Y Banach sps .
T : X → Y compact (or )
def
⇐⇒ ∀
{xn} ⊂ X: bdd, ∃
{xnk
}; Txnk
→ ∃
y ∈ Y .
compact op. . ( , ∃
xn ∈ X; ∥xn∥ = 1, ∥Txn∥ ≥
n → ∞ , compact )
T: compact =⇒ xn → x (weak) in X Txn → Tx (strong) in Y .
D.1 H: Hilbert, T : H → H: compact self adj. op. ,
{λn} ⊂ R ONS {xn} ,
∀
x ∈ H, ∃
ck ∈ K, ∃
x′
∈ H; Tx′
= 0, x = ckxk + x′
, Tx = λkckxk.
λn → 0 dim N(λnI − T) < ∞ (∀
n).