functional analysis - tokyo metropolitan …1 functional analysis 1.1 normed spaces 1.1.1...

FUNCTIONAL ANALYSIS

Okazaki Junichiro

平成 16 年 4 月 17 日

概要

SPACES

Linear spaces : just a module (algebraic system)SNormed spaces : a module with geometrical property, normSBanach spaces : completenessSHilbert spaces : inner product (orthogonality)S

Euclidean spaces :

i

目次1 Functional analysis 1

1.1 Normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Definition of LINEAR SPACE . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Definition of NORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Some Normed Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Definition of lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Definition of l∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Definition of c0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.4 Definition of Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.5 Definition of L∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.6 Definition of C([0, 1]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.7 Definition of C0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.8 Definition of Cc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.9 Definition of C∞([0, 1]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Banach space & Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Definition of BANACH SPACE . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Theorem; Normed space l2 is complete . . . . . . . . . . . . . . . . . . . . 71.3.3 Definition of INNER PRODUCT . . . . . . . . . . . . . . . . . . . . . . . 81.3.4 Lemma (Schwartz’s inequality) . . . . . . . . . . . . . . . . . . . . . . . . 91.3.5 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.6 Definition of HILBERT SPACE . . . . . . . . . . . . . . . . . . . . . . . . 111.3.7 Theorem (von Neumann) . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.8 Lemma (Holder’s inequality) . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.9 Lemma (Minkowski’s inequality) . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Linear Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.1 Definition of LINEAR OPERATOR . . . . . . . . . . . . . . . . . . . . . 161.4.2 Definition of CONTINUOUS . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.3 Definition of BOUNDED . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.4 Definition of OPERATOR NORM . . . . . . . . . . . . . . . . . . . . . . 161.4.5 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4.6 Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4.7 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4.8 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5 Banach-Bernstein’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.5.1 Banach-Bernstein’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.6 Baire’s Category Theorem 1, 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.6.1 Baire’s Category Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 221.6.2 Baire’s Category Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 231.6.3 Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.6.4 Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.6.5 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

ii

1.6.6 Corollary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.6.7 Pettis-Plesner’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.7 Hahn-Banach’s theorem on R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.7.1 Hahn-Banach’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.7.2 Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.7.5 Riesz’s representation theorem . . . . . . . . . . . . . . . . . . . . . . . . 331.7.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7.21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

iii

1 Functional analysis

1.1 Normed spaces

1.1.1 Definition of LINEAR SPACE

Review; R-module

R: a ring 1 (commutative 1 ∈ R) (← defined with operation , +, × )

M : R-module def⇐⇒(1) M is an abelian2 w.r.t. addition.(2) R×M −→ M : scalar multiplication s.t.for any r, r′ ∈ R, m,m′ ∈M

1. (r + r′)m = rm+ r′m

2. (rr′)m = r(r′m)

3. r(m+m′) = rm+ rm′

4. 1 ·m = m

If R is a field, then M is called a vecter space over R.

1G; a group

G×Ginner operation−−−−−−−−−−→ G

(g1, g2) 7−→ g1g2

1. associativity

2. an identity element

3. the inverse element

R; a ring

R×Rinner operation−−−−−−−−−−→ R¡

(r1, r2) 7−→ r1 + r2 ; say addition(r1, r2) 7−→ r1r2 ; say multiplication

and R is an abelian group w.r.t. additionR is a monoid w.r.t. multiplication (Semi-group ⊂ Monoid ⊂ Group)

moreover, R satisfies the distributivity;r1(r2 + r3) = r1r2 + r1r3

M ; R-module

M ×Minner operation−−−−−−−−−−→ M

(m1, m2) 7−→ m1 + m2 ; say addition

R×Mouter operation−−−−−−−−−−−→ M

(r, m) 7−→ rm ; say scalar multiplication

2(M ,+):an abelian group ⇔ Addition is definded on M , and satisfies the following conditions;

1. associativity; m + (m′ + m′′) = (m + m′) + m′′

2. an identity element; ∃0 ∈ M s.t. 0 + m = m

3. the inverse element; ∀m ∈ M, ∃ −m ∈ M s.t. m + (−m) = 0

4. commutativty; m + m′ = m′ + m

1

1. Functional analysis

Example

When M = R, M is a module over R.R×M −→ M

ExampleA; an abelian grp.R = Z; the ring of intergersn ∈ Z, a ∈ A

na =def

a+ a+ · · ·+ a if n > 00 if n = 0

(−a) + (−a) + · · ·+ (−a) if n < 0

Then A is a module in Z.

abelian group = Z-module (module over Z)

2

1.1. Normed spaces

E is called a vector space, or linear space, if it is a module over fieldK, on which addition(inner operation)& scalar multiplication(outer operation) is defined.

1.1.2 Definition of NORM

E; a vector space∀ x ∈ E

‖ · ‖ : E −→ R≥0

1. ‖x‖ ≥ 0, ‖x‖ = 0 ⇐⇒ x = 0

2. ‖αx‖ = |α|‖x‖

3. ‖x+ y‖ ≤ ‖x‖+ ‖y‖

Then we call ‖ · ‖ norm , and if it does not satisfy ‖x‖ = 0⇔ x = 0, we call it semi-norm.

The pair (E, ‖ · ‖E) is called a normed space.

3


1.2 Some Normed Space

1.2.1 Definition of lp

lp = {x = (x1, x2, · · · , xn, · · · ) | ‖x‖p ; finite} is a vect. sp. and is normed with

‖x‖p =def

( ∞∑n=1

|xn|p) 1

p

1.2.2 Definition of l∞

l∞ = {x = (x1, x2, · · · , xn, · · · ) | ‖x‖∞ ; finite} is a vect. sp. and is normed with

‖x‖∞ =def

sup1≤n<∞

|xn|

1.2.3 Definition of c0

c0 = {x = (x1, x2, · · · , xn, · · · ) | xj −−−−→j−→∞

0} is a vect. sp. and is normed with

‖x‖∞ =def

sup1≤n<∞

|xn|

4

1.2. Some Normed Space

1.2.4 Definition of Lp

Lp(X,µ) ={f : X −→ C | ∫

X|f(x)|pdµ(x); finite

}/ {f(x) = 0 µ− a.e.} is a vect. sp. and

is normed with

‖[f ]‖p =def

(∫

X

|f(x)|pdµ(x)) 1

p

1.2.5 Definition of L∞

L∞(X,µ) = {f : essentially bounded} / {f(x) = 0 µ−a.e.} is a vect. sp. and is normed with

‖[f ]‖∞ def= ess.sup|f(x)|def= inf

[α | µ

({x ∈ X | |f(x)| > α

})= 0

]

1.2.6 Definition of C([0, 1])

C([0, 1]) = { f : [0, 1] −→ C | continuous} is a vect. sp. and is normed with

‖f‖∞ = max0≤x≤1

|f(x)|

1.2.7 Definition of C0

C0(R) = {f : R −→ R | continuous, |f(x)| −−−−−→|x|−→∞

0} is a vect. sp. and is normed with

‖f‖∞ = supR|f |

1.2.8 Definition of Cc

Cc = {f ∈ C0 | suppf ; compact}

1.2.9 Definition of C∞([0, 1])

C∞([0, 1]) = {f : [0, 1] −→ R | C∞ − class} is a vect. sp. and is normed with

‖f‖∞ = sup0≤x≤1

|f(x)|

5


1.3 Banach space & Hilbert space

1.3.1 Definition of BANACH SPACE

(E, ‖ · ‖E); Banach space

⇐⇒ E; complete3 w.r.t. T‖·‖E

Example;Every finite dimensional normed space is Banach space.

Remark

lp, l∞, c0, Lp, L∞, C0 are Banach spaces.

Cc(R), C∞([0, 1]) are not Banach spaces.

3i.e. Any Cauchy sequence converges in E

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1.3. Banach space & Hilbert space

1.3.2 Theorem; Normed space l2 is complete

7


1.3.3 Definition of INNER PRODUCT

〈·|·〉; inner product on Edef⇐⇒ 〈·|·〉 : E × E −→ C satisfying the following conditions;

1. 〈x|x〉 ≥ 0, 〈x|x〉 = 0 ⇔ x = 0

2. 〈αx|y〉 = α〈x|y〉 for α ∈ C

3. 〈x+ y|z〉 = 〈x|z〉+ 〈y|z〉

4. 〈x|y〉 = 〈y|x〉

Example

x, y ∈ l2〈x|y〉 def=

∑∞n=1 xnyn

8


1.3.4 Lemma (Schwartz’s inequality)

(E, 〈·|·〉E); an inner product space

|〈x|y〉E | ≤ ‖x‖E‖y‖E for any x, y ∈ E

9


1.3.5 Theorem

(E, 〈·|·〉E); an inner product space

Then

1. ‖x‖E def= (〈x|x〉E)12 satisfies the properties of norm.

2. ‖x+ y‖2E + ‖x− y‖2E = 2(‖x‖2E + ‖y‖2E)

10


1.3.6 Definition of HILBERT SPACE

(E, 〈·|·〉E); Hilbert space

def⇐⇒ (E, ‖ · ‖E); Banach space, here ‖ · ‖E = (〈·|·〉E)12

11


1.3.7 Theorem (von Neumann)

∃ inner product 〈·|·〉 on E s.t.√〈x|x〉E = ‖x‖E

⇐⇒ ‖x+ y‖2E + ‖x− y‖2E = 2(‖x‖2E + ‖y‖2E) for any x, y ∈ E

12


proof

13


◎ Inequalties

1.3.8 Lemma (Holder’s inequality)

1 < p <∞

q =p

p− 14

f ∈ Lp(X)

g ∈ Lq(X)

Then, ∣∣∣∣∫

X

f(x)g(x)dx∣∣∣∣ ≤ ‖f‖p‖g‖q

4

1

p+

1

q= 1

14


1.4 Linear Operator

(E, ‖ · ‖E) T−−−→linear

(F, ‖ · ‖F )

finite ⇒ T ;matrix

infinite ⇒ ???

1.4.1 Definition of LINEAR OPERATOR

(E, ‖ · ‖E), (F, ‖ · ‖F ); normed space

T : E −→ F ; linear operator def⇐⇒ T (αx+ βy) = αTx+ βTy

NOTATION

L(E,F ) def= {T | T : E −→ F ; linear}and, if E = F , L(E,F ) is denoted by L(E).

1.4.2 Definition of CONTINUOUS

T ∈ L(E,F )

T ; continuous on E def⇐⇒ ‖xn − x‖ −→n→∞

0 ⇒ ‖Txn − Tx‖ −→n→∞

0

1.4.3 Definition of BOUNDED

T ∈ L(E,F )

T ; bounded on E def⇐⇒ ∃K > 0 s.t. ‖Tx‖F ≤ K‖x‖E for any x ∈ E

NOTATION

B(E,F ) def= {T ∈ L(E,F ) | T ; bounded}

1.4.4 Definition of OPERATOR NORM

∀ T ∈ B(E,F )

‖T‖ def= inf{K > 0 | ||Tx||F ≤ K‖x‖E for any x ∈ E}

; called operator norm, uniform norm

16

1.4. Linear Operator

1.4.5 Theorem

E, F ; normed space

=⇒ (B(E,F ), ‖ · ‖) ; normed space

17


1.4.6 Lemma

γ1 =def

sup0 6=x∈E

‖Tx‖F‖x‖E

γ2 =def

sup‖x‖E=1

‖Tx‖F

γ3 =def

sup‖x‖E<1

‖Tx‖F

Then γ1 = γ2 = γ3 = ‖T‖

18

1.4. Linear Operator

1.4.7 Theorem

T ∈ L(E,F )

Then the following conditions are equivalent;

1. T ; conti. on E

2. T ; conti. at 0

3. T ; bounded

19


1.4.8 Theorem

F ; Banach space =⇒ B(E,F ); Banach space

20

1.5. Banach-Bernstein’s Theorem

1.5 Banach-Bernstein’s Theorem

1.5.1 Banach-Bernstein’s Theorem

(Uniform Bounded Theorem)E,F ; normed spaceE; Banach spaceB(E,F ) 3 T : E −→ F ; bounded linear operatorA ⊂ B(E,F ); a non-empty subset of B(E,F )

∀x ∈ E (fixed.), supT∈A ‖Tx‖F ; finite (strong operator topology) on B(E,F )

=⇒ supT∈A ‖T‖; finite (uniform operator topology) on B(E,F ) 5

Proof∀n ≥ 1Put Cn = { x ∈ E | ‖Tx‖F ≤ n for any T ∈ A }∀x ∈ E, ∃n ∈ N, s.t. supT∈A ‖Tx‖F ≤ n by assumption=⇒ ‖Tx‖F ≤ n for every T ∈ A=⇒ x ∈ Cn

=⇒ E ⊂ ⋃∞n=1 Cn

=⇒ E =⋃∞

n=1 Cn

claim; Cn; closed in E.Furthermore, since E is a Banach space

5A; strongly operator bounded =⇒ A; uniformly operator bounded

Remark;

E, F ; finite dimensional =⇒ (s)-top. = (u)-top.

21


1.6 Baire’s Category Theorem 1, 2

1.6.1 Baire’s Category Theorem 1

(X, d); complete metric spaceOn ⊂ X; open dense6 in X

=⇒ ⋂∞n=1On; dense in X

6i.e. On = X

22

1.6. Baire’s Category Theorem 1, 2

1.6.2 Baire’s Category Theorem 2

(X, d); complete metric spaceCn; closed in XX =

⋃∞n=1 Cn

=⇒ ∃m ≥ s.t. Int(Cm) 6= ∅

23


1.6.3 Proposition

(E, ‖ · ‖); Banach space

24


1.6.4 Proposition

p, q ∈ R≥1

p < q =⇒ lp ⊂ lq

25


1.6.5 Theorem

1p

+1q

= 1, (p, q > 1) =⇒ (l∞, ‖ · ‖p)∗ = lp

26


1.6.6 Corollary

p = 0, 1(l1)∗ ∼= l∞

(l∞)∗ ∼= l1

then(C0)∗∗ ∼= l∞

(C0)∗∗∗ ∼= l∞ ⊃ (C0)∗ = l1

1.6.7 Pettis-Plesner’s theorem

(E, ‖ · ‖E); Banach spaceE∗ ⊂ E∗∗∗

=⇒ E∗∗∗ = E∗ ⊕l1 E⊥

where E⊥ = { f ∈ E∗∗∗ | f |E = 0 }

27


1.7 Hahn-Banach’s theorem on R

F ⊂ (E, ‖ · ‖E)ϕ ∈ F ∗

=⇒ ∃ϕ ∈ E∗ s.t.

1. ϕ|F = ϕ

2. ‖ϕ‖ = ‖ϕ‖

1.7.1 Hahn-Banach’s theorem

E; linear space over RF ⊂ E; linear subspace over Rp : E −→ R s.t.

1. p(x+ y) ≤ p(x) + p(y)

2. p(αx) = αp(x) (α ≥ 0, x ∈ E)

p is called Minkowski function on E.

ϕ : F −→ R : R-linear function s.t.ϕ(x) ≤ p(x) for any x ∈ F

=⇒ ∃ϕ : E −→ R ;linear s.t.

1. ϕ|F = ϕ

2. ϕ(x) ≤ p(x) for any x ∈ E

28

1.7. Hahn-Banach’s theorem on R

(proof)

29


1.7.2 Lemma

E, p; as in the last Thm.J ⊂ E; a subset of Eψ : J −→ R; linear

1. ψ : J −→ R ;linaer

2. ψ(x) ≤ p(x) ∀x ∈ J

If J 6= E, 0 6= ∃x0 ∈ E\JJ $ J + Rx0 = { x+ αx0 | x ∈ J, α ∈ R } ;linear subspace of E

=⇒ ∃R-linear Ψ;J + Rx0 −→ R s.t.

1. Ψ|J = ψ

2. Ψ(x) ≤ p(x) (∀x ∈ J + Rx0)

30


(proof)

31


1.7.3

1.7.4

32


1.7.5 Riesz’s representation theorem

H; a Hilbert space

f ; bounded linear functional =⇒ ∃xf ∈ H , uniquely s.t.

f(x) =< x|xf > for any x ∈ H

‖f‖ = ‖xf‖

33


1.7.6

1.7.7

1.7.8

1.7.9

1.7.10

1.7.11

1.7.12

1.7.13

1.7.14

1.7.15

1.7.16

1.7.17

1.7.18

1.7.19

1.7.20

1.7.21

34

functional analysis - tokyo metropolitan …1 functional analysis 1.1 normed spaces 1.1.1...

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