113

Sp, p
K2.
EKn≥1.
L(E)E, GL (E)
E.
n, m Mm,n (K)m n
K.
m=n, Mn(K)nK,
GLn(K)Mn(K).
Id In
E∗=L(E, K)E
p2.
p E φ :Ep→K
k1p(xi)1≤i≤p
i̸=k
Ep−1
φi:x∈E7→ φ(x1,··· , xk−1, x, xk+1,··· , xp)
E.
φ φ (x1,··· , xp) = 0 (xi)1≤i≤p∈Ep
j̸=k1p xi=xj.
p≥1,Lp(E, K)p E
p= 1,L1(E, K) = E∗
p≥1,Lp(E, K)Knp.
Lp(E, K)K
B= (ei)1≤i≤nE.

φ∈ Lp(E, K) (x1,··· , xp)∈Ep, j 1p
xj=
n
i=1
xi,jei
p φ,
φ(x1,··· , xp) = φp
i1=1
xi1,1ei1, x2,··· , xp=
p
i1=1
xi1,1φ(ei1, x2··· , xp)
=
p
i1=1
p
i2=1
xi1,1xi2,2φ(ei1, ei2, x3··· , xp)
=
n
i1=1 ···
n
i1n=1
xi1,1···xip,pφei1,··· , eip
=
1≤i1,···,ip≤n
xi1,1···xip,pφei1,··· , eip
Φ : φ∈ Lp(E, K)7→ φei1,··· , eip1≤i1,···,ip≤n∈Knp
α=αi1,···,ip1≤i1,···,ip≤n∈Knp,
φ: (x1,··· , xp)7→ 
1≤i1,···,ip≤n
αi1,···,ipxi1,1···xip,p
pΦLp(E, K)Knp.
dim (Lp(E, K)) = np.
φ∈ Lp(E, K)φ(x1,··· , xp) = 0 (xi)1≤i≤p
E.
φ(x1,··· , xp)xk
xjj̸=k
p φ E φ xσ(1),··· , xσ(p)=
ε(σ)φ(x1,··· , xp) (x1,··· , xp)∈Epσ∈ Sp.
φxτ(1),··· , xτ(p)=−φ(x1,··· , xp)
τ, Sp
φ∈ Lp(E, K)τ= (j, k) 1 ≤j <
k≤p.
0 = φ(x1,··· , xj+xk,···, xj+xk,··· , xp)
=φ(x1,··· , xj,···, xj,··· , xp) + φ(x1,··· , xp)
+φxτ(1),··· , xτ(p)+φ(x1,··· , xk,··· , xk,··· , xp)
=φ(x1,··· , xp) + φxτ(1),··· , xτ(p)
φxτ(1),··· , xτ(p)=−φ(x1,··· , xp).
xj=xk1≤j < k ≤p,
τ= (j, k)
φ(x1,··· , xp) = φxτ(1),··· , xτ(p)=−φ(x1,··· , xp)
φ(x1,··· , xp) = 0 K2.

E n ≥1,B= (ei)1≤i≤nE
x∈E, X = (xi)1≤i≤nKx,
x=
n
i=1
xiei
An(E, K)n
1 detB:En→K
detB(x1,··· , xn) = 
σ∈Sn
ε(σ)
n
i=1
xσ(i),i
xj=
n
i=1
xijeij1n.
detBn
j1n,
πj:x=
n
i=1
xiei7→ xj
j
detB(x1,··· , xn) = 
σ∈Sn
ε(σ)
n
i=1
πσ(i)(xi)
πσ(i)(x1,··· , xn)7→
n
i=1
πσ(i)(xi)n
detBn
τ, k =τ(i),
detBxτ(1),··· , xτ(n)=
σ∈Sn
ε(σ)
n
i=1
πσ(i)xτ(i)
=
σ∈Sn
ε(σ)
n
k=1
πσ(τ−1(k)) (xk)
=
σ∈Sn
ε(σ)
n
k=1
πσ◦τ−1(k)(xk)
σ′7→ σ=σ′◦τSn
detBxτ(1),··· , xτ(n)=
σ′∈Sn
ε(σ′◦τ)
n
k=1
πσ′(k)(xk)
=ε(τ)
σ′∈Sn
ε(σ′)
n
k=1
πσ′(k)(xk)
=ε(τ) detB(x1,··· , xn)

detB
φ∈ An(E, K) (x1,··· , xn)∈En,
φ(x1,··· , xn) = 
1≤i1,···,in≤n
xi1,1···xin,nφ(ei1,··· , ein) = 
γ∈Fn
n
i=1
xγ(i),iφeγ(1),··· , eγ(n)
Fn{1,··· , n} {1,··· , n}.
φ φ xγ(1),···, xγ(n)= 0 γ
φ(x1,··· , xn) = 
σ∈Sn
n
i=1
xσ(i),iφeσ(1),··· , eσ(n)
=
σ∈Sn
ε(σ)
n
i=1
xσ(i),iφ(e1,··· , en)
φ=λdetBλ=φ(e1,··· , en)∈KdetB∈ An(E, K)\ {0}.
An(E, K) 1 detB.
detBn E φ (e1,··· , en) = 1.
detB(x1,··· , xn)
n(xi)1≤i≤nB.
n φ E, φ =λdetBλ=φ(e1,··· , en),
φ(x1,··· , xn) = φ(e1,··· , en) detB(x1,··· , xn)
(xi)1≤i≤n∈En.
B′= (e′
i)1≤i≤nE, (xi)1≤i≤n∈
En
detB′(x1,··· , xn) = detB′(e1,···, en) detB(x1,··· , xn)
= detB′(B) detB(x1,···, xn)
detB′(B) detB(B′) = detB′(B′) = 1
(xi)1≤i≤nn E.
(xi)1≤i≤n
BE, detB(x1,··· , xn) = 0
BEdetB(x1,··· , xn) = 0.

(1) ⇒(2) (xi)1≤i≤nφ(x1,··· , xn)=0
ndetB,
BE.
(2) ⇒(3)
(3) ⇒(1) BEdetB(x1,··· , xn) = 0.
B′= (xi)1≤i≤nE1 = detB′(B′) = detB′(B) detB(B′) = 0,
(xi)1≤i≤nn E. E
BEdetB(x1,··· , xn)̸= 0.
E.
u∈ L(E), λu
φ∈ An(E, K)\ {0}(xi)1≤i≤n∈En,
φ(u(x1),··· , u (xn)) = λuφ(x1,··· , xn)
λu= detB(u(e1),··· , u (en))
B= (ei)1≤i≤nE.
u∈ L(E)n
φ∈ An(E, K)\ {0},
φu: (x1,··· , xn)∈En7→ φ(u(x1),··· , u (xn))
n λuφu=λuφ
dim (An(E, K)) = 1
ψ∈ An(E, K)\ {0}n ψ =ρφ
ψu=ρφu=ρλuφ=λuψ
λun
φ= detBφ= detB′,B= (ei)1≤i≤nB′= (e′
i)1≤i≤n
E,
detB(u(e1),··· , u (en)) = λu= detB′(u(e′
1),··· , u (e′ n))
λu
udet (u).
det (u)u E.
(xi)1≤i≤n∈Enu∈ L(E)u(ei) = xii1n,
det (u) = detB(u(e1),··· , u (en)) = detB(x1,··· , xn)
B= (ei)1≤i≤nE.