Functional Analysis Note 1 Baire’s Category Theorem, Uniform Boundedness Theorem, Banach-Steinhaus Theorem, Open Mapping Theorem and Closed Graph Theorem

Functional Analysis

Note 1

Baire’s Category Theorem, Uniform Boundedness Theorem, Banach-Steinhaus Theorem, Open Mapping Theorem and Closed Graph Theorem 

TANAKA Akio

[Baire’s Category Theorem]

Complete distance space     X

Countable closed sets of X     X₁, X₂, …, X_n, …

∪^∞_n=1X_n = X

At least one X_n has open sphere.

[Account]

Distance at X      d (x, y)

Assumption     Any X_n has not open sphere.

X₁ ≠X

Complementary set of X₁     X^C₁ is open set that is not null.

X^C₁ has open sphere.

X₂ has not open sphere S.

X^C₂ ⋂ S (x₁, ε₁/2) ≠Ø

Sequence of open sphere    {S(x_n, ε_n)}

For natural number n>m, {x_n} is Cauchy sequence.

X is complete.

Arbitrary natural number that is convergent at point x∈X     m

 d (x_n, x) → 0  (n→∞)

Existence m’ that is d (x_m’, x)<ε_m/2

m’>m

x ∉ X_m (m = 1, 2,…)

x ∉ ∪^∞_m=1X_m

The result is against ∪^∞_m=1 X_m= X.

[Uniform Boundedness Theorem]

Infinity set     A

Bounded linear operator from Banach space X to norm space Y     Ta, a∈A

x∈X

sup_a∈A ||T_ax|| < ∞ → sup_a∈A ||T_ax|| < ∞

[Account]

Natural number     n

X_n = {x∈X ; sup_a∈A ||T_ax|| ≦n}

{x∈X ; sup_a∈A ||T_ax|| ≦n}     Closed set

X_n     Open set

X     Complete

At least one of X_n ( n=1, 2, …) has open sphere by Baire’s category theorem.

Open sphere to be had     S ( x₀, r ) = { x∈X ; || x – x0 || < r }  ( r > 0 )

x∈S ( x₀, r ) →  ||T_ax|| ≦n₀ ( a∈A )

|| T_ax₀ ||≦4n₀/r || x || ( x∈X, x≠0 )

[Banach-Steinhaus Theorem]

Bounded linear operator’s sequence from Banach space X to Banach space Y     T_n (n = 1, 2, …)

Dense subset of X     X₀

Sup_n || T_nx || < ∞ and for x∈X₀, there exists lim_n→∞T_nx, next are concluded.

(i) For all of x∈X, there exists lim_n→∞T_n

(ii) When Tx = lim_n→∞T_n ( x∈X, ) , T is bounded linear operator from X to Y, || T || ≦lim_n→∞inf ||T_n|| is concluded.

[Account]

(i)

By uniform boundedness theorem, there exists constant M ( >0 ),

||T_n || ≦M ( n = 1,2,…)

x∈X, ε>0

y∈X₀

|x-y| <ε/3M

Adequate natural number     n₀

|Tny – T_my| <ε/3 (n, m≧n₀ )

|T_nx – T_mx| <ε

{ T_nx} is Cauchy sequence ay Y.

Y is complete, there exists lim_n→∞T_n .

(ii)

sup_n||T_ax|| < ∞

By uniform boundedness theorem, || T_nx || is bounded sequence.

||Tx|| = lim_n→∞||T_n x||≦(lim_n→∞inf||T_n|| )||x||  (x∈X)

[Open Mapping Theorem]

Banach space     X, Y

Upper bounded linear operator from X to Y     T

Map of X’s arbitrary open set G by T     TG

TG is open set of Y.

[Account]

<1>

Arbitrary ρ>0

TS_x(0, ρ) ⊃ Sphere S_r(0, ρ’) (ρ’>0)

Y_n =   ( n = 1,2,…)

S_Y(0,r / 2n₀) ⊂TS_x(0, 1)

TS_x(0,ρ) ⊃ S_Y (0, ρ’)

<2>

Open set of X     G  

x∈G  

G⊃Open sphere S_x(x, ρ ) ( ρ >0 )

TG ⊃ S_r(Tx, ρ’)

[Closed Graph Theorem]

Banach space     X, Y

Closed Operator     T

D(T)⊂X,  R(T)⊂Y

When D(T) = X, T is bounded.

[Account]

Graph G(T) is closed linear subspace.

Operator from G(T) to X     J

||J([x, Tx])|| ≦ ||[x, Tx]||

Bounded linear operator from X to G(T)     J ^-1

Adequate constant     c > 0

||x|| + ||Tx|| = ||[x, Tx]|| = ||J ^-1x||≦c||x|| (x∈X)

||Tx||≦c||x|| (x∈X)

Tokyo May 9, 2008

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