A question on Grothendieck space A Banach space $X$ is said to be Grothendieck if the weak and the weak* convergence of sequences in $X^{*}$ coincide. I have the following two questions.
Question 1. A Banach space $X$ is Grothendieck if and only if every weak*-Cauchy sequence in $X^{*}$ is weakly Cauchy?
Question 2. If $(x^{*}_{n})_{n}$ is a weak Cauchy sequence and a weak*-null sequence in $X^{*}$, is $(x^{*}_{n})_{n}$ a weak-null sequence?
Thank you!
 A: I find the following criterion useful: A sequence $(x_n)$ is Cauchy iff for all subsequences $(x_{n_{k+1}}-x_{n_k})$ tends to $0$. This works for the norm topology, the weak topology and the weak$^*$ topology. This answers Q1 in the positive.
As for Q2, if $(x_n^*)$ is weakly Cauchy and weak$^*$ null, it has a limit $x^{***}$ for the weak$^*$ topology of $X^{***}$; decompose $x^{***}=x^* + x_s^{***}$, where $x_s^{***}$ is the ``singular part'' in the annihilator of $X$ in $X^{***}$. By, the assumption of Q2, $x^*=0$; i.e., $x^{***}$ is singular. This seems to be as good as it gets in a general Banach space.
A: Q1 is already answered by Prof. Dirk Werner above. I simply list a number of equivalent conditions that seems to be related, but not the same as the Grothendieck property.
The following are indeed equivalent:

*

*every weak*-null sequence in $X^{\ast}$ has a weakly Cauchy subsequence.

*every bounded weak*-sequentially compact subset of $X^{\ast}$ is weakly precompact (every sequence has a weakly Cauchy subsequence).

*for every bounded $T:X\to c_0$, the adjoint $T^{\ast}:\ell^1\to X^{\ast}$ is weakly precompact (i.e., $T^{\ast}$ maps bounded sets onto weakly precompact sets).

*for every bounded $T:X\to Y$, where $Y$ is another Banach space with weak* sequentially compact dual ball, the adjoint $T^{\ast}$ is weakly precompact.

*no weak*-null sequence in the unit ball of $X^{\ast}$ contains an $\ell^1$-subsequence.

*there is no surjective bounded $T:X\to c_0$.

It is well known that $X$ is a Grothendieck space iff $X^{\ast}$ is weakly sequentially complete and (6)
