Differentiable manifolds, Serge Lang

Question

I have started reading "Introduction to differentiable manifolds" by Serge Lang. In this book, Lang takes a different approach, by immediately introducing manifolds on arbitrary Banach spaces. His approach uses little to no multilinear algebra and he states the following in the foreword: "The orgy of multilinear algebra in standard treatises arises from unnecessary double dualization and an abusive use of the tensor product." What exactly does he mean by this? How is the use of the tensor product "abusive"?.Is there something inelegant about the traditional treatment of finite dimensional manifolds and differential forms on them?