Currently I am reading the book "Hopf Algebras. An Introduction" by S. Dascalescu, C. Nastasescu, S. Raianu. There is a Definition 1.5.20 on page 44 (boldface is mine):
An algebra $A$ is called proper (or residually finite dimensional) if $i_A:A \to A^{\circ*}$ is injective, it is called weakly reflexive if $i_A$ is injective, and it is called reflexive if $i_A$ is bijective.
This definition is stated as if being proper and weakly reflexive are two different things, but according to their definition these two concepts are equivalent. I'm wondering if there is a typo in the text. Or the two concepts are actually identical?
Here $A$ is assumed to be a unital, associative algebra over some field $k$. $$A^\circ = \Big\{ f \in A^* : \exists I : \text{Ideal of} \; A \; \text{such that } \; I \subset \ker f \; \text{and}\; \dim_k \frac{A}{I} < \infty \Big\}$$ is a finite-dual coalgebra and $i_A$ is a canonical map defined by $i_A(a)(f) = f(a)$. The ideal $I$ in the above definition is assumed to be two-sided.
I don't want to learn wrong definitions, so I will be grateful if somebody more experienced than me can tell me if the definition is correct.
