Saturday 28 April 2018

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


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     X1X2, …, Xn, …
n=1Xn X
At least one Xn has open sphere.
[Account]
Distance at X      (xy)
Assumption     Any Xn has not open sphere.
XX
Complementary set of X    XCis open set that is not null.
XChas open sphere.
X2 has not open sphere S.
XC (x1ε1/2) Ø
Sequence of open sphere    {S(xnεn)}
For natural number n>m, {xn} is Cauchy sequence.
X is complete.
Arbitrary natural number that is convergent at point xX     m
 (xn, x)  0  (n∞)
Existence m’ that is d (xmx)<εm/2
m’>m
∉ Xm (m = 1, 2,…)
x  m=1Xm
The result is against m=1 XmX.

[Uniform Boundedness Theorem]
Infinity set     A
Bounded linear operator from Banach space X to norm space Y     TaaA
xX
supaA ||Tax|| < ∞  supaA ||Tax|| < ∞
[Account]
Natural number     n
Xn = {xX ; supaA ||Tax|| n}
{xX ; supaA ||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 ( x0r ) = { xX ; || x – x0 || < r }  ( r > 0 )
xS ( x0r ) →  ||Tax|| n0 ( a)
|| Tax0 ||4n0/r || x || ( xX, x0 )

[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 xX0, there exists limn→∞Tnx, next are concluded.
(i) For all of xX, there exists limn→∞Tn
(ii) When Tx = limn→∞TxX, ) , T is bounded linear operator from to Y, || T || limn→∞inf ||Tn|| is concluded.
[Account]
(i)
By uniform boundedness theorem, there exists constant M ( >0 ),
||Tn || M ( = 1,2,…)
xX, ε>0
yX0
|x-y| <ε/3M
Adequate natural number     n0
|Tny – Tmy| <ε/3 (n, mn)
|Tn– 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||  (xX)

[Open Mapping Theorem]
Banach space     XY
Upper bounded linear operator from 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 =   ( = 1,2,…)
SY(0,/ 2n0TSx(0, 1)
TSx(0,ρ S(0, ρ’)
<2>
Open set of X      
xG  
GOpen sphere Sx(xρ ) ( ρ >0 )
TG ⊃ Sr(Txρ’)

[Closed Graph Theorem]
Banach space     XY
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([xTx])||  ||[x, Tx]||
Bounded linear operator from to G(T)     -1
Adequate constant     > 0
||x|| + ||Tx|| = ||[xTx]|| = ||-1x||c||x|| (xX)
||Tx||c||x|| (xX)

Tokyo May 9, 2008

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