Left and right integrals and integral functionals

Let $H$ be a Hopf algebra over a field $k$ as in Hopf algebras and antipodes, with multiplication, unit $1$, counit $\epsilon$, and antipode. The left-integral space and right-integral space of $H$ are ${\int }_H^{\mathrm{\ell }}=\{\mathrm{\Lambda }\in H:h\mathrm{\Lambda }=\epsilon (h)\mathrm{\Lambda }\text{ for every }h\in H\},{\int }_H^r=\{\mathrm{\Lambda }\in H:\mathrm{\Lambda }h=\epsilon (h)\mathrm{\Lambda }\text{ for every }h\in H\}\mathrm{.}$ Both are linear subspaces and include zero. When $H$ is finite-dimensional, its dual $H^{\ast }$ is a Hopf algebra. Its product is convolution $(f\ast g)(h)=\sum f(h_1)g(h_2)$ and its counit is ${\epsilon }_{H^{\ast }}(f)=f(1)$. A left integral functional on $H$ means an element $\lambda \in H^{\ast }$ satisfying $f\ast \lambda =f(1)\lambda$ for every $f\in H^{\ast }$; a right integral functional satisfies $\lambda \ast f=f(1)\lambda$. Zero is allowed in both definitions; existence of a nonzero integral is a theorem, not part of the definition.