The paper proves a general theorem that relates some modes of convergence, such as pointwise a.e. convergence, to weak convergence in any space with the weak Banach-Saks property. Some results that follow immediately from the main theorem are given for specific functional spaces.
"On Some Modes of Convergence in Spaces with the Weak Banach-Saks Property." Real Anal. Exchange 38 (2) 487 - 492, 2012/2013.