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C(X)→A
C[0,1]
L∞

M∈ Mn(C)M=UDU−1
U D
M
β={e1, e2, ...}
HU:`2→ H
Ux =x1e1+x2e2+...
H`2→ H
T
{αk}kT ek=αkekU
M=U−1T U (Mx)k=αkxk
H`2
f
T
f(M)M2
f:x7→ x2
C(X)→ B(H)

H
AC C
A×A→A
(A, k.k)ACk.k
kxyk6kxk.kyk ∀x, y ∈A
A
A
EB(E)E
Mn(C)n
k(aij )k=
n
X
i,j=1 |aij |
C(X, C)X
Ck.k∞
D={z∈C;|z|61}C(D)
DH(D)C(D)
D
H(D)C(D)k.k∞
H(D)
T
T
(C(T),k.k∞)
W(T).
A1k1k= 1
x∈A y ∈A xy =yx =1
A A
A A
x7→ x−1A

x∈Akxk<11−x
(1−x)−1=X
n>0
xn
k(1−x)−1k61
1− kxk
k1−(1−x)−1k6kxk
1− kxk
kxnk6kxknn z := X
n>0
xn
z(1−x)=(1−x)z= lim
N→∞(1−x)
N
X
k=0
xk= lim
N→∞(1−xN+1) = 1
1−x z
kzk6X
n>0kxnk6X n>0kxkn=1
1− kxk
1−z=−X
n>1
xn=−xz ||1−z|| 6kxk.kzk
x∈(A) (A)
r > 0h∈Akhk< r x +h∈(A)
(x+h) = x(1−(−x−1h)) x+hkx−1hk<1
||h|| <1
kx−1kr=1 kx−1kx∈(A)
h∈A x +h∈(A)khk
h
(x+h)−1−x−1= [x(1+x−1h)]−1−x−1= [(1+x−1h)−1−1]x−1
||h|| <1
||x−1||
||(x+h)−1−x−1|| 6||(1+x−1h)−1−1||.||x−1|| 6||x−1||.||x−1h||
1− ||x−1h||
0||h|| 0
(A)
a∈A a
(a) = {λ∈C;x−λ1/∈(A)}
x−λ x −λ1
a∈A
(a): (C\(A)−→ A
λ7−→ Rλ(a) = (a−λ1)−1