120

E, L(E)E GL (E)
E.
E, E∗E,
n= dim (E).
n≥1,Mn(R)n
GLn(R)Mn(R).
Id In
(E, h· | ·i)n≥1kxk=phx|xi
Rn
∀(x, y)∈Rn×Rn,hx|yi=tx·y=
n
X
k=1
xkyk
X E
X⊥={y∈E| ∀x∈X, hx|yi= 0}
F E {0},
F
F E
F E, E =F⊕F⊥.
Sn(R) = {A∈ Mn(R)|tA=A}n
∀A∈ Mn(R),∀(x, y)∈Rn×Rn,hAx |yi=x|tAy®
hAx |yi=t(Ax)y=txtAy =x|tAy®

A∈ Sn(R)⇔ ∀(x, y)∈Rn×Rn,hAx |yi=hx|Ayi
hAx |yi=hx|Ayix, y, hx|Ayi=hx|tAyi
h· | ·i Ay =tAy y,
A=tA.
` E, a ∈E
∀x∈E, ` (x) = hx|ai.
ϕ a ∈E ϕ (a)
∀x∈E, ϕ (a) (x) = hx|ai
h· | ·i ϕ(a)∈E∗,
h· | ·i ϕ a ϕ
hx|ai= 0 x∈E, a = 0
h· | ·i E E∗ϕ
`∈E∗a∈E ` =ϕ(a),
u∈ L(E),
u∗∈ L(E)
∀(x, y)∈E2,hu(x)|yi=hx|u∗(y)i
y∈E, x 7→ hu(x)|yi
E, u∗(y)∈Ehu(x)|yi=hx|u∗(y)ix∈E.
u∗E E
u∗. y, z E λ, µ
R, x ∈E
hu(x)|λy +µzi=λhu(x)|yi+µhu(x)|zi
=λhx|u∗(y)i+µhx|u∗(z)i
=hx|λu∗(y) + µu∗(z)i
λu∗(y) + µu∗(z) = u∗(λy +µz)u∗(λy +µz).
u∗u.
B= (ei)1≤i≤nE u
E A u∗BtA.

A= ((aij))1≤i,j≤nuB
u(ej) =
n
X
i=1
aijei(1 ≤j≤n)
i, j 1n
aij =hu(ej)|eii
A∗=¡¡a∗
ij¢¢1≤i,j≤nu∗B, i, j
1n
a∗
ij =hu∗(ej)|eii=hej|u(ei)i=aji
A∗=tA.
A= (aij)1≤i,j≤nu
i, j 1n
aij =hu(ej)|eii
aii =hu(ei)|eii
Tr (u) =
n
X
i=1 hu(ei)|eii
u
aij =aji =hu(ei)|eji
u, v L(E),
(u∗)∗=u.
(u◦v)∗=v∗◦u∗.
u∈GL (E), u∗∈GL (E) (u∗)−1= (u−1)∗.
ker (u∗) = (Im (u))⊥Im (u∗) = (ker (u))⊥.
rg (u∗) = rg (u).
u
E
x∈ker (u∗)z=u(y)∈Im (u),
hx|zi=hx|u(y)i=hu∗(x)|yi=h0|yi= 0
x∈(Im (u))⊥.ker (u∗)⊂(Im (u))⊥.
x∈(Im (u))⊥y∈E,
hu∗(x)|yi=hx|u(y)i= 0
u∗(x) = 0, x ∈ker (u∗).
ker (u∗) = (Im (u))⊥.

Im (u∗)⊂(ker (u))⊥
dim (Im (u∗)) = n−dim (ker (u∗)) = n−dim ³(Im (u))⊥´
= dim (Im (u)) = n−dim (ker (u)) = dim ³(ker (u))⊥´
Im (u∗) = (ker (u))⊥.
rg (u∗) = rg (u)
rg ( tA) = rg (A)
A.
u∈ L(E)u∗=u.
S(E)E.
u∈ S (E)⇔u∗=u⇔ ∀(x, y)∈E2,hu(x)|yi=hx|u(y)i
S(E).
u∈ L(E)
E
BE u ∈ L(E), A
B.
u u∗=utA=A A ∈ Sn(R).
A∈ Sn(R), X x ∈EB,
x, y E
hu(x)|yi=t(AX)Y=tXtAY =tX(AY ) = hx|u(y)i
u∈ S (E).
A∈ Sn(R), u
S(Rn).
S(E)L(E)n(n+ 1)
2.
S(E)L(E)
Sn(R)u7→ A, A u
E
Sn(R)Mn(R)
n(n+ 1)
2, ϕ :Sn(R)→Rn(n+1)
2
∀A= ((aij))1≤i,j≤n∈ Sn(R), ϕ (A) = (aij)1≤i≤j≤n
(Eij)1≤i,j≤nMn(R)
Eij = ((δipδqj))1≤p,q≤n

δrs
A=X
1≤i,j≤n
aijEij =
n
X
i=1
aiiEii +X
1≤i<j≤n
(aijEij +ajiEji)
=
n
X
i=1
1
2aii (2Eii) + X
1≤i<j≤n
aij (Eij +Eji)
(Eij +Eji)1≤i≤j≤nSn(R).
dim (Sn(R)) = card (Eij +Eji)1≤i≤j≤n=n(n+ 1)
2.Sn(R)
n(n+ 1)
2.
u∈ S (E), u +u∗, u∗u uu∗
S(E).hu∗ux |xi=ku(x)k2huu∗(x)|xi=ku∗(x)k2,
u∗u uu∗S+(E)
u∈ S (E), up∈ S (E)p v∗◦u◦v∈ S (E)
v∈ L(E).
A=µ1 1
1 2 ¶B=µ1 2 2 1 ¶S2(R)AB =µ3 3 5 4 ¶/∈ S2(R).
A=µa b
b c ¶A0=µa0b0 b0c0¶S2(R),
(1,2) (2,1) AA0b12 =ab0+bc0b21 =a0b+b0c
b12 6=b21.
u, v E,
u◦v u v
u, v
u◦v
u, v u ◦v x, y
E, hu◦v(x)|yi=hx|u◦v(y)i,hv(x)|u(y)i=hu(x)|v(y)i,
hu◦v(x)|yi=hv◦u(x)|yiu◦v(x) = v◦u(x)x∈Rn,
u◦v=v◦u.
u∈ S (E).ker (u) Im (u)
E= ker (u)⊥ ⊕Im (u)
x∈ker (u)z=u(y)∈Im (u),
hx|zi=hx|u(y)i=hu(x)|yi=h0|yi= 0