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 question on Grothendieck space
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.


