Note 1
Baire’s Category Theorem, Uniform Boundedness Theorem, Banach-Steinhaus Theorem, Open Mapping Theorem and Closed Graph Theorem
[Baire’s Category Theorem]
Complete distance space X
Countable closed sets of X X1, X2, …, Xn, …
∪∞n=1Xn = X
At least one Xn has open sphere.
[Account]
Distance at X d (x, y)
Assumption Any Xn has not open sphere.
X1 ≠X
Complementary set of X1 XC1 is open set that is not null.
XC1 has open sphere.
X2 has not open sphere S.
XC2 ⋂ S (x1, ε1/2) ≠Ø
Sequence of open sphere {S(xn, εn)}
For natural number n>m, {xn} is Cauchy sequence.
X is complete.
Arbitrary natural number that is convergent at point x∈X m
d (xn, x) → 0 (n→∞)
Existence m’ that is d (xm’, x)<εm/2
m’>m
x ∉ Xm (m = 1, 2,…)
x ∉ ∪∞m=1Xm
The result is against ∪∞m=1 Xm= X.
[Uniform Boundedness Theorem]
Infinity set A
Bounded linear operator from Banach space X to norm space Y Ta, a∈A
x∈X
supa∈A ||Tax|| < ∞ → supa∈A ||Tax|| < ∞
[Account]
Natural number n
Xn = {x∈X ; supa∈A ||Tax|| ≦n}
{x∈X ; supa∈A ||Tax|| ≦n} Closed set
Xn Open set
X Complete
At least one of Xn ( 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 ) → ||Tax|| ≦n0 ( a∈A )
|| Tax0 ||≦4n0/r || x || ( x∈X, x≠0 )
[Banach-Steinhaus Theorem]
Bounded linear operator’s sequence from Banach space X to Banach space Y Tn (n = 1, 2, …)
Dense subset of X X0
Supn || Tnx || < ∞ and for x∈X0, there exists limn→∞Tnx, next are concluded.
(i) For all of x∈X, there exists limn→∞Tn
(ii) When Tx = limn→∞Tn ( x∈X, ) , T is bounded linear operator from X to Y, || T || ≦limn→∞inf ||Tn|| is concluded.
[Account]
(i)
By uniform boundedness theorem, there exists constant M ( >0 ),
||Tn || ≦M ( n = 1,2,…)
x∈X, ε>0
y∈X0
|x-y| <ε/3M
Adequate natural number n0
|Tny – Tmy| <ε/3 (n, m≧n0 )
|Tnx – Tmx| <ε
{ Tnx} is Cauchy sequence ay Y.
Y is complete, there exists limn→∞Tn .
(ii)
supn||Tax|| < ∞
By uniform boundedness theorem, || Tnx || is bounded sequence.
||Tx|| = limn→∞||Tn x||≦(limn→∞inf||Tn|| )||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
TSx(0, ρ) ⊃ Sphere Sr(0, ρ’) (ρ’>0)
Yn = ( n = 1,2,…)
SY(0,r / 2n0) ⊂TSx(0, 1)
TSx(0,ρ) ⊃ SY (0, ρ’)
<2>
Open set of X G
x∈G
G⊃Open sphere Sx(x, ρ ) ( ρ >0 )
TG ⊃ Sr(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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