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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? Is there something inelegant about the traditional treatment of finite dimensional manifolds and differential forms on them?

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It depends on what you mean by "traditional treatment". There are in fact many different "traditional treatments" of tensors on manifolds, and some of them are indeed quite messy. – Deane Yang May 18 '13 at 14:58
Maybe this is just me, but from my experience as a grad student, I would suggest using a different book, unless you are already familiar with finite dimensional manifolds, or interested mainly in a few very general notions. I.e. I did not find that book a useful place to begin. If you too are a beginner, people like the books by Michael Spivak and John Lee. I see now that Lang expanded the book more than double (by adding finite dimensional topics), over the 1st edition which consisted mostly of theorems on differential equations now found in chapters IV and VI. So it is now a hybrid. – roy smith May 18 '13 at 15:15
I second Roy's comment. Lang's book is suitable (but not necessarily the best) only if you are already familiar with finite-dimensional manifolds and have a specific need for learning about infinite-dimensional manifolds. Even then, it might be easier to focus first only on the specific infinite-dimensional spaces that you want to work with, rather than learning the general theory first. – Deane Yang May 18 '13 at 15:27

I'm not sure what Lang had in mind with "unnecessary double dualization", but here's an example that occurred to me in ancient times when I was trying to understand differential geometry better. Some (many? most?) people define tangent vectors to a manifold to be certain derivations on the smooth functions, and they define the cotangent space to be the dual of the tangent space. So a cotangent vector is a function taking as arguments tangent vectors, which are themselves functions taking as arguments smooth functions. In this sense, a cotangent vector is a doubly-dualized function. But one can avoid these dualizations by defining a cotangent vector at a point $p$ to be an element of $m/(m^2)$ where $m$ is the maximal ideal in the ring of germs of smooth functions at $p$. In other words, $m$ consists of the smooth germs that vanish at $p$ and $m^2$ consists of those that vanish to second order, so the quotient is "first-order data about a germ at $p$, omitting the value (zero-order data) at $p$." That picture captures pretty well my intuition of what a cotangent vector should be. (I think that the $m/(m^2)$ definition is used more in algebraic geometry than in differential geometry.)

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