Let c : Θ ( A ) → Θ ( A ) ⊗ Θ ( A ) c : \Theta(A) \to \Theta(A) \otimes \Theta(A) c : Θ ( A ) → Θ ( A ) ⊗ Θ ( A ) η \eta η A A A ( Θ ( A ) , c ) (\Theta(A), c) ( Θ ( A ) , c )
Θ ( A ) → c Θ ( A ) ⊗ Θ ( A ) ∣ ∣ A ∗ → t m A ( A ⊗ A ) ∗
\begin{array}{ccc}
\Theta(A) & \xrightarrow{c} & \Theta(A) \otimes \Theta(A) \\
| & & | \\
A^* & \xrightarrow{t m_A} & (A \otimes A)^*
\end{array}
Θ ( A ) ∣ A ∗ c t m A Θ ( A ) ⊗ Θ ( A ) ∣ ( A ⊗ A ) ∗ Indeed, η \eta η A A A ( Θ ( A ) , c ) (\Theta(A), c) ( Θ ( A ) , c ) η ( a b ) = m K ∘ ( η ( a ) ⊗ η ( b ) ) ∘ c \eta(ab) = m_K \circ (\eta(a) \otimes \eta(b)) \circ c η ( ab ) = m K ∘ ( η ( a ) ⊗ η ( b )) ∘ c a , b ∈ A a, b \in A a , b ∈ A
η ( a b ) ( f ) = m K ∘ ( η ( a ) ⊗ η ( b ) ) ( c ( f ) )
\eta(ab)(f) = m_K \circ (\eta(a) \otimes \eta(b))(c(f))
η ( ab ) ( f ) = m K ∘ ( η ( a ) ⊗ η ( b )) ( c ( f )) for a , b ∈ A a, b \in A a , b ∈ A f ∈ Θ ( A ) f \in \Theta(A) f ∈ Θ ( A )
η ( a b ) ( f ) = f ( a b ) = ⟨ t m A ( j 1 ( f ) ) , a ⊗ b ⟩
\eta(ab)(f) = f(ab) = \langle t m_A(j_1(f)), a \otimes b \rangle
η ( ab ) ( f ) = f ( ab ) = ⟨ t m A ( j 1 ( f )) , a ⊗ b ⟩ and, by (14),
m K ∘ ( η ( a ) ⊗ η ( b ) ) ( c ( f ) ) = ⟨ j 2 ( c ( f ) ) , a ⊗ b ⟩
m_K \circ (\eta(a) \otimes \eta(b))(c(f)) = \langle j_2(c(f)), a \otimes b \rangle
m K ∘ ( η ( a ) ⊗ η ( b )) ( c ( f )) = ⟨ j 2 ( c ( f )) , a ⊗ b ⟩ for all a , b ∈ A a, b \in A a , b ∈ A f ∈ Θ ( A ) f \in \Theta(A) f ∈ Θ ( A )
Since j 2 j_2 j 2 c c c t m ∘ j 1 t m \circ j_1 t m ∘ j 1 j 2 j_2 j 2 f f f Θ ( A ) \Theta(A) Θ ( A ) n n n f 1 ′ , … , f n ′ , f 1 ′ ′ , … , f n ′ ′ f'_1, \ldots, f'_n, f''_1, \ldots, f''_n f 1 ′ , … , f n ′ , f 1 ′′ , … , f n ′′ Θ ( A ) \Theta(A) Θ ( A )
f ( a b ) = ∑ i = 1 n f i ′ ( a ) f i ′ ′ ( b )
f(ab) = \sum_{i=1}^n f'_i(a) f''_i(b)
f ( ab ) = i = 1 ∑ n f i ′ ( a ) f i ′′ ( b ) for a , b ∈ A a, b \in A a , b ∈ A
c ( f ) = ∑ i = 1 n f i ′ ⊗ f i ′ ′ .
c(f) = \sum_{i=1}^n f'_i \otimes f''_i.
c ( f ) = i = 1 ∑ n f i ′ ⊗ f i ′′ . By the corollary of VIII, p. 379, there exists a left A A A E E E K K K f f f ( e 1 , … , e n ) (e_1, \ldots, e_n) ( e 1 , … , e n ) E E E ( e 1 ∗ , … , e n ∗ ) (e_1^*, \ldots, e_n^*) ( e 1 ∗ , … , e n ∗ ) x x x E E E x ∗ x^* x ∗ E ∗ E^* E ∗ f = c E ( x , x ∗ ) f = c_E(x, x^*) f = c E ( x , x ∗ ) f i ′ = c E ( e i , x ∗ ) f'_i = c_E(e_i, x^*) f i ′ = c E ( e i , x ∗ ) f i ′ ′ = c E ( x , e i ∗ ) f''_i = c_E(x, e_i^*) f i ′′ = c E ( x , e i ∗ ) i ∈ [ 1 , n ] i \in [1, n] i ∈ [ 1 , n ]
f ( a b ) = ⟨ x ∗ , a b x ⟩ = ⟨ x ∗ a , b x ⟩ = ⟨ ∑ i ⟨ x ∗ a , e i ⟩ e i ∗ , b x ⟩
f(ab) = \langle x^*, abx \rangle = \langle x^* a, bx \rangle = \left\langle \sum_i \langle x^* a, e_i \rangle e_i^*, bx \right\rangle
f ( ab ) = ⟨ x ∗ , ab x ⟩ = ⟨ x ∗ a , b x ⟩ = ⟨ i ∑ ⟨ x ∗ a , e i ⟩ e i ∗ , b x ⟩ = ∑ i ⟨ x ∗ a , e i ⟩ ⟨ e i ∗ , b x ⟩ = ∑ i ⟨ x ∗ , a e i ⟩ ⟨ e i ∗ , b x ⟩ = ∑ i f i ′ ( a ) f i ′ ′ ( b ) ,
= \sum_i \langle x^* a, e_i \rangle \langle e_i^*, bx \rangle = \sum_i \langle x^*, ae_i \rangle \langle e_i^*, bx \rangle = \sum_i f_i'(a) f_i''(b),
= i ∑ ⟨ x ∗ a , e i ⟩ ⟨ e i ∗ , b x ⟩ = i ∑ ⟨ x ∗ , a e i ⟩ ⟨ e i ∗ , b x ⟩ = i ∑ f i ′ ( a ) f i ′′ ( b ) , and therefore (16).
Let us prove the coassociativity of c. For this, consider the K-linear mappings
c ′ = ( c ⊗ 1 Θ ( A ) ) ∘ c and c ′ ′ = ( 1 Θ ( A ) ⊗ c ) ∘ c
c' = (c \otimes 1_{\Theta(A)}) \circ c \quad \text{and} \quad c'' = (1_{\Theta(A)} \otimes c) \circ c
c ′ = ( c ⊗ 1 Θ ( A ) ) ∘ c and c ′′ = ( 1 Θ ( A ) ⊗ c ) ∘ c from
Θ ( A ) \Theta(A) Θ ( A ) to
Θ ( A ) ⊗ 3 \Theta(A)^{\otimes 3} Θ ( A ) ⊗ 3 . We have the relations
⟨ j 3 ( f ⊗ c ( g ) ) , a ⊗ b ⊗ c ⟩ = ⟨ f , a ⟩ ⟨ j 2 ∘ c ( g ) , b ⊗ c ⟩
\langle j_3(f \otimes c(g)), a \otimes b \otimes c \rangle = \langle f, a \rangle \langle j_2 \circ c(g), b \otimes c \rangle
⟨ j 3 ( f ⊗ c ( g )) , a ⊗ b ⊗ c ⟩ = ⟨ f , a ⟩ ⟨ j 2 ∘ c ( g ) , b ⊗ c ⟩ = ⟨ f , a ⟩ ⟨ g , b c ⟩
= \langle f, a \rangle \langle g, bc \rangle
= ⟨ f , a ⟩ ⟨ g , b c ⟩ = ⟨ j 2 ( f ⊗ g ) , a ⊗ b c ⟩
= \langle j_2(f \otimes g), a \otimes bc \rangle
= ⟨ j 2 ( f ⊗ g ) , a ⊗ b c ⟩ = ⟨ t ( I d A ⊗ m A ) ∘ j 2 ( f ⊗ g ) , a ⊗ b ⊗ c ⟩
= \langle t(\mathrm{Id}_A \otimes m_A) \circ j_2(f \otimes g), a \otimes b \otimes c \rangle
= ⟨ t ( Id A ⊗ m A ) ∘ j 2 ( f ⊗ g ) , a ⊗ b ⊗ c ⟩ for
f , g ∈ Θ ( A ) f, g \in \Theta(A) f , g ∈ Θ ( A ) and
a , b , c ∈ A a, b, c \in A a , b , c ∈ A . From this, we deduce that the following diagram commutes:
Θ ( A ) ⊗ Θ ( A ) → I d Θ ( A ) ⊗ c Θ ( A ) ⊗ Θ ( A ) ⊗ Θ ( A ) ↓ j 2 ↓ j 3 ( A ⊗ A ) ∗ → t ( I d A ⊗ m A ) ( A ⊗ A ⊗ A ) ∗ .
\begin{array}{ccc}
\Theta(A) \otimes \Theta(A) & \xrightarrow{\mathrm{Id}_{\Theta(A)} \otimes c} & \Theta(A) \otimes \Theta(A) \otimes \Theta(A) \\
\downarrow j_2 & & \downarrow j_3 \\
(A \otimes A)^* & \xrightarrow{t(\mathrm{Id}_A \otimes m_A)} & (A \otimes A \otimes A)^*.
\end{array}
Θ ( A ) ⊗ Θ ( A ) ↓ j 2 ( A ⊗ A ) ∗ Id Θ ( A ) ⊗ c t ( Id A ⊗ m A ) Θ ( A ) ⊗ Θ ( A ) ⊗ Θ ( A ) ↓ j 3 ( A ⊗ A ⊗ A ) ∗ . Because of the commutativity of this diagram and that of (15), for f ∈ Θ ( A ) f \in \Theta(A) f ∈ Θ ( A ) a , a ′ , a ′ ′ ∈ A a, a', a'' \in A a , a ′ , a ′′ ∈ A
⟨ j 3 ∘ c ′ ( f ) , a ⊗ a ′ ⊗ a ′ ′ ⟩ = ⟨ f , ( a a ′ ) a ′ ′ ⟩ ;
\langle j_3 \circ c'(f), a \otimes a' \otimes a'' \rangle = \langle f, (aa')a'' \rangle;
⟨ j 3 ∘ c ′ ( f ) , a ⊗ a ′ ⊗ a ′′ ⟩ = ⟨ f , ( a a ′ ) a ′′ ⟩ ; we can show the relation
⟨ j 3 ∘ c ′ ′ ( f ) , a ⊗ a ′ ⊗ a ′ ′ ⟩ = ⟨ f , a ( a ′ a ′ ′ ) ⟩
\langle j_3 \circ c''(f), a \otimes a' \otimes a'' \rangle = \langle f, a(a'a'') \rangle
⟨ j 3 ∘ c ′′ ( f ) , a ⊗ a ′ ⊗ a ′′ ⟩ = ⟨ f , a ( a ′ a ′′ )⟩ likewise. Since multiplication in A is associative, we have
j 3 ∘ c ′ = j 3 ∘ c ′ ′ j_3 \circ c' = j_3 \circ c'' j 3 ∘ c ′ = j 3 ∘ c ′′ , and therefore
c ′ = c ′ ′ c' = c'' c ′ = c ′′ because
j 3 j_3 j 3 is injective.
Finally, formulas (16) and (17) imply that Θ ( A ) \Theta(A) Θ ( A ) ε \varepsilon ε
Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.