Le dual en dimension infinie 1 Un peu de théorie des ensembles

k E k
E∗E
{ei}i∈I
E k(I)I
k E I
dim(E)<dim(E∗)
EA={X∈E, X 6∈ X}E
A∈A
A6∈ A A ∈A
A
A B A →B A
Bcard(A) = card(B)A→B
B→Acard(A)6card(B) card(B)>
card(A)A B card(A)<card(B)
card(A)6card(B) card(B)6card(C) card(A)6card(C)
card(A)6card(B) card(B)6card(A)
card(A) = card(B)
P(A)Acard(P(A)) >card(A)
A→ P(A)a{a}card(P(A)) >card(A)
f:A→ P(A)B={a∈A, a 6∈ f(a)}a0∈A
f B a0∈B a06∈ f(a0) = B
a06∈ B a0∈f(a0) = B f
n>1Ann A A
card(An) = card(A)
P∗(A)A A card(P∗(A)) =
card(A)

A, B z ∈B B(A)f:A→B
f−1(B\ {z})Acard(A)>card(B)>2 card(B(A)) = card(A)
z B(A)
card(B)>2b∈B\ {z}A→B(A)a
{a}f(a) = b f(x) = z x 6=a
B(A)→A f ∈B(A)
Sf=f−1(B\ {z})f Sf
card(A)>card(B)
B(A)' q
S∈P∗(A)BS↓
' q
a∈AB=A×B↓
→A×A↓ 'A.
k E k E∗
dim(E)<dim(E∗)E∗E
{ei}i∈IE E
k(I)I k x =Pxiei
f f(i) = xiE
I E∗kI
I k ϕ :E→k ei
 k
Q Fp F =(I)F∗
I
card(F∗) = card(I)>card({0,1}I) = card(P(I)) >card(I)
{0,1}I' P(I)
NIcard(I)>card()>2
card(F) = card(I)<card(F∗) dim`(F)<dim`(F∗)
dim`F∗6dimkE∗
ϕs:I→ s ∈S ϕ0
s=ϕs:I→⊂k k
Psαsϕ0
s= 0 αs∈k{fj}j∈Jk
 αs=Pjαsj fjαsj ∈ i
0 = X
s,j
αsj fjϕs(i) = X
jX
s
αsj ϕs(i)fj
Psαsj ϕs(i)=0 j{fj}Psαsj ϕs= 0
I  ϕs αsj = 0 s, j
αs= 0 ϕ0
s
dimk(E) = dim`(F)<dim`(F∗)6dimk(E∗)