Exercise 5, § 1

Let E be a bigebra and let E' = \mathrm{Hom}(E, K) be its dual with the algebra structure derived by duality from the cogebra structure on E (Algebra, Chapter III, § 11, no. 1). Let m be the kernel of the homomorphism u \mapsto u(1) of

E' onto K; it is an ideal of E'. Show that, if $u \in m^2$ and $x \in P(E)$, then $u(x) = 0$. When K is a field show that $P(E)$ is the orthogonal of $m^2$ in $E^+$; deduce that $\dim P(E) \leq \dim m/m^2$ and that we have equality if E is of finite rank over K.