Note 3
Space
1 Linear space (Vector space)
Complex number field or real number field Φ
Set X
When X satisfies next condition, X is called linear space over Φ.
It is also called vector space over Φ.
Arbitrary elements of X x, y
Sum of x and y x + y
(i)
Arbitrary elements of X x, y, z
x + y = y + x
(x+y)+z = x+(y+z)
(ii)
There exists one element that is called 0 in X.
Arbitrary x∈X
x+0 = x
(iii)
Arbitrary x∈X
There exists one element that is called –x in X.
x+(-x) = 0
(iiii)
Arbitrary α∈Φ
Arbitrary x∈X
αx that is called product of α and x is uniquely defined.
Arbitrary α, β∈Φ
Arbitrary x, y∈X
1x = x
(v)
(αβ)x = α(βx)
(vi)
α(x+y) = αx+αy
(α+β)x = αx+βx
1-1 Complex linear space
When Φ is complex number field, linear space over Φ is called complex linear space.
1-2 Real linear space
When Φ is real number field, linear space over Φ is called real linear space.
2 Linear subspace
Linear space X
Subset that is not empty in X M
M that satisfies next condition is called linear subspace of X.
(i)
x∈M
y∈M
x+y∈M
(ii)
x∈M
Arbitrary α∈Φ
αx∈M
3 Linear subspace spanned (Linear subspace generated)
Linear space X
Element of X xi∈X ( i = 1, 2, …, n)
α1x1+α2x2+…+αnxn (αi∈Φ, i = 1, 2, …, n) is called linear combination of x1, x2, …, xn.
Subset that is not empty in X S
All the sets of linear combinations of arbitrary finite elements M
M is called linear subspace spanned by S.
It is also called linear subspace generated by S.
4 Normed space
Linear space X
Element of X x
Real number ||x||
||x|| that satisfies next condition is called norm of x.
X is called normed space.
(i)
||x|| ≧0
||x|| = 0 ⇌ x = 0
(ii)
Arbitrary α∈Φ
Arbitrary x, y∈X
||αx|| =|α| ||x||
(iii)
||x+y|| ≦||x||+||y||
4-1 Complex normed space
When Φ is complex number field, normed space over Φ is called complex normed space.
4-2 Real normed space
When Φ is real number field, normed space over Φ is called real normed space.
5 Distance space
Normed space X
X’s element is also called point.
Arbitrary points of X x, y
When d(x, y) = ||x-y|| is defined, d(x, y) satisfies next 3 conditions for distance.
X is distance space.
(i)
d(x, y) ≧0,
d(x, y) = 0 ⇌ x = y
(ii)
d(x, y) = d(y, x)
(iii)
d(x, y) ≦d(x, z) = d(z, y)
6 Linear subspace
Normed space X
Linear subspace of X M
M is called linear subspace of normed space X.
6-1 Closed linear subspace
When M is linear subspace of normed space X and closed set, M is called closed linear subspace.
7 Closed linear subspace spanned (Closed linear subspace generated)
Normed space X
Subset that is not empty in X S
Linear subspace spanned by S M
Closure of M is called closed linear subspace spanned by S.
It is also called closed linear subspace generated by S.
8 Banach space
Sequence {xn} of normed space X satisfied limm, n →∞ = 0, {xn} is called Cauchy sequence.
When in normed space X, arbitrary Cauchy sequence is convergent to X’s point, X is called complete.
Complete normed space is called Banach space.
8-1 Complex Banach space
When X is complex normed space, Banach space is complex Banach space
8-2 Real Banach space
When X is real normed space, Banach space is real Banach space
9 Sequence space
Set that is consisted from all of n real numbers’ group (ξ1, ξ2, …, ξn) Vn
Two elements of Vn x = (ξ1, ξ2, …, ξn) y = (η1, η2, …, ηn)
x+y := (ξ1+η1, ξ2+η2, …ξn+ηn)
Real number α
Element of Vn x = (ξ1, ξ2, …, ξn)
αx := (αξ1, αξ2, …, αξn)
Vn becomes real linear space.
10 Euclid space
Set that is consisted from all of n real numbers’ group (ξ1, ξ2, …, ξn) Vn
x = (ξ1, ξ2, …, ξn)∈Vn
||x|| := 
Vn becomes complete eral normed space, i.e. real Banach space.
Vn is called n dimensional Euclid space. Notation is Rn.
11 Unitary space
Set that is consisted from all of n complex numbers’ group (ξ1, ξ2, …, ξn) Kn
In Kn, 2 points x and y, sum x+y, product αx and norm ||x|| is similarly defined as Rn.
Kn becomes complex Banach space.
Kn is called n dimensional unitary space.
12 Functional space
Bounded closed interval [a, b]
Real valued continuous function defined at [a, b] x(t)
All of x(t) C[a, b]
By norm and completeness, C[a, b] is real Banach space.
Complex valued continuous function defined at [a, b] x(t)
All of x(t) C[a, b]
By operation, norm and completeness, C[a, b] is complex Banach space.
13 p power Lebesgue integrable real valued function
Interval (a, b)
Real valued measurable function at (a, b) x(t)
x(t) satisfies next condition, x(t) is called p power Lebesgue integrable.
∫ba| x(t)|pdt < ∞
All of x(t) LP(a, b)
By operation, norm completeness, LP(a, b) is real Banach space.
All of complex valued measurable functions at (a, b) is complex Banach space.
14 Essentially bounded real valued function
Interval (a, b)
Real valued measurable function at (a, b) x(t)
Set of measure 0 N⊂(a, b)
Complement of N at (a, b) (a, b)∖N
When x(t) is supt∈(a, b)∖N | x(t)| < ∞, x(t) is essentially bounded.
All of measurable at (a, b) and essentially bounded real valued functions L∞(a, b)
By operation, norm completeness, L∞ (a, b) is real Banach space.
All of measurable at (a, b) and essentially bounded complex valued functions is complex Banach space.
15 Inner product space
Complex linear space X
Arbitrary 2 elements pair of X {x, y}
Complex corresponded with {x, y} (x, y)
When (x, y) satisfied next condition, (x, y) is called inner product between x and y.
(i) (x, x)≧0 ; x=0 and only itself (x, x) = 0
(ii) (x, y) = 
(iii) (x+z, y) = (x, y) + (z, y)
(iiii) (ax, y) = a (x, y) (a; complex number)
Space defined by inner product is called inner product space.
When a at (iiii) is real number and corresponded number pair is real number (x, y), X is called real product space.
16 Hilbert space
When inner product space X is complete on norm ||x|| =
, X is called Hilbert space.
, X is called Hilbert space.
When X is real inner product space, X is called real Hilbert space.
Tokyo May 23, 2008
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