Note 2
Orthogonal Decomposition
1 <Linear operator>
Normed space V
Subset of V D
Element of D x
Element of normed space V1 Tx
Mapping T : x → Tx∈V1
The mapping T is called operator.
Domain of T D(T)
Range of T R(T)
T satisfies next condition, T is called continuous at point x0.
Arbitrary ε> 0
Adequate δ> 0
x∈D, ||x-x0|| < δ ⇒ ||Tx – Tx0|| <ε
Operators that are continuous at every point of D are called continuous.
T satisfies next condition, T is called linear operator.
T(αx+βy) =αTx+βTy
T satisfies next condition, T is called bounded linear operator.
(i) D(T) =V1
(ii) ||Tx||≦γ||x|| γ>0
Infimum of γ is called norm of T, that is notated by ||T||.
||T|| = inf { γ : ||Tx||≦γ||x|| (x∈V)}
2 <Inverse operator>
Linear operator from normed space V to Normed space V1 T
N ( T ) := { x : x∈D(T), Tx = 0 }
When T is bounded linear operator, N ( T ) is closed subset of V.
When T has one versus one correspondence between D(T) and R(T), inverse mapping T-1 is linear operator from V1 to V.
D(T-1) = R(T) , R(T-1) = D(T)
(T-1)-1 = T
When T satisfies next condition, there exists inverse operator T-1.
x∈D(T), Tx = 0 ⇒ x = 0 or N ( T ) = { 0 }
When T satisfied next condition, there exists continuous inverse operator T-1.
x1, x2, …∈D(T), limn→∞Tx= 0 ⇒ limn→∞xn = 0
3 <Orthogonal decomposition>
(1)
Two elements of linear space V x, y
0≦α≦1
Segment between x and y [x, y] : = αx +(1-α)y
When 0<α<1, [x, y] is called inner point
Set of V A
x, y ∈A satisfies segment [x, y]∈A, a is called vertex set.
Minimum vertex set containing A is called vertex hull that is notated by C (A ).
Minimum strongly closed vertex set is called closed vertex hull.
As strongly closed and weakly closed are equal, closed vertex hull is called closed vertex set.
Hilbert space H
Closed vertex set of H A
x∉A
Element that is the nearest from x xA
xA∈A
y∈A, y≠xA
||x-xA|| <||x-y||
(2)
Hilbert space H
Arbitrary subset of H A
When A satisfies next condition, A is called orthogonal complement, that is notated by A⊥.
A⊥ : = { x ; (x, y) = 0, (y∈A)}
A⊥ is closed subspace of H.
(3)
Closed subspace of H K
Arbitrary x∈H
x is uniquely decomposed to the next.
x = xK + x’, xK∈K, x’∈K⊥
4 <Functional>
Hilbert space H
Subspace of H D
Linear functional is defined by the next.
α,β∈C, x, y∈D
f (αx+βy) =αfx +βfy
When f satisfies next condition, f is called bounded linear functional.
|f(x)| ≦γ||x|| (x∈H)
γ >0
Functional’s norm || f || is defined by the next.
|| f || = inf { γ : | f (x) |≦γ||x|| (x∈H)}
5 <Riesz Theorem>
Bounded linear functional over H f
There uniquely exists x0∈H that is satisfies the next.
f (x) = ( x, x0 ) (x∈H)
|| f || = || x0 ||
5 Adjoint operator
Bounded linear operator over H T
T : x →(Tx, y)∈C
T is bounded linear functional.
y* is uniquely defined by Riesz theorem.
(Tx, y) = (x, y*)
T ’s adjoint operator is defined by the next.
(Tx, y) = (x, T*y) (x, y∈H)
T* satisfies the next.
(i) || T* || = || T ||
(ii) ( T + S )* = T* + S*
(iii) (αT )* = 
T*
T*
(iiii) (TS)* = S*T*
(v) (T*)* = T
Bounded linear operator that is A = A* is called self-adjoint operator.
Bounded linear operator that is UU* = U*U = I is called unitary operator.
Bounded linear operator that is next condition is called isometric operator.
(i) || U || = 1
(ii) || Ux || = || x || ( x∈H )
(iii) (Ux, Uy ) = ( x, y ) ( x, y∈H )
(iiii) U*U = I
6 Projection operator
Hilbert space H
Closed subset K
Decomposition x = xK + x’ ( xK∈K, x’∈K⊥)
Operator that is corresponded with x∈H to xK is called projection operator over K.
Projection operator PK has next quality.
(i) PKH = R ( PK ) = K
(ii) x⊥K ⇌ PKx = 0
7 Weak convergent
Hilbert space H
Sequence of H x1, x2, …
x0∈H
lim n→∞(xn, x) = (x0, x) Notation is also w-lim n→∞xn = x0
x1, x2, … is called weakly convergent to x0.
x0 is called weak limit.
Normal orthogonal system is weakly convergent to 0.
Strongly convergent and weakly convergent are equal in complex Hilbert space.
Sequence of bounded linear operators T1, T2, …
(i) Uniformly convergent lim n→∞||Tn –T|| = 0
(ii) Strongly convergent lim n→∞Tnx = Tx (x∈H)
(iii) Weakly convergent w- lim n→∞Tnx = Tx (x∈H)
[References]
<Normal orthogonal system>
Tokyo June 3, 2008
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