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Im α ∗ = (ker α) ◦ <br />
V ∗ × V → F, (ε, v) ↦→ ε(v) 〈ε|v〉 <br />
U α −→ V V ∗ α∗<br />
−→ U ∗ <br />
〈α ∗ (ε)|u〉 = 〈ε|α(u)〉<br />
u ∈ U ε ∈ V ∗ ˆ : V → V ∗∗ , v ↦→ ˆv ˆv(ε) = ε(v)<br />
V F ˆ : V → V ∗∗ , v ↦→ ˆv <br />
ˆv(ε) = ε(v) <br />
ˆv : V ∗ → F ˆ <br />
<br />
(λ1v1 + λ2v2)(ε) = ε(λ1v1 + λ2v2) = λ1ε(v1) + λ2ε(v2)<br />
= λ1ˆv1(ε) + λ2ˆv2(ε)<br />
= (λ1ˆv1 + λ2ˆv2)(ε)<br />
ε ∈ V ∗ ˆ V <br />
dim V = dim V ∗∗ e1 = 0, e1 ∈ V e1, . . . , en V ε1, . . . , εn <br />
V ∗ ê1(ε1) = ε1(e1) = 1 ê1 = 0<br />
<br />
ε1, . . . , εn V ∗ E1, . . . , En V ∗∗ <br />
Ej = êj ej ∈ V ε1, . . . , εn V ∗ <br />
e1, . . . , en V <br />
V U ≤ V V V ∗∗ <br />
U = U ◦◦ Û = U ◦◦ <br />
U ≤ U ◦◦ u ∈ U ε(u) = 0 ε ∈ U ◦ û(ε) = 0<br />
ε ∈ U ◦ û ∈ U ◦◦ dim U = dim U ◦◦ U = U ◦◦ <br />
U1, U2 ≤ V dim V <br />
(U1 + U2) ◦ = U ◦ 1 ∩ U ◦ 2 <br />
(U1 ∩ U2) ◦ = U ◦ 1 + U ◦ 2 <br />
V V =<br />
P (R) V ∗ = R N <br />
P (R) = 〈p0, p1, . . .〉 ε ∈ V ∗ <br />
(ε(p0), ε(p1), . . . )
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