Relationship between functional analysis and differential geometry I am taking courses on functional analysis (through Coursera.com) and differential geometry (textbook author : O'neil) on my university.
I made the following table on my own.

Are the similar concepts linear form and 1-form?
Also I want to know more relationships deeply. Or any recommendation (book or whatever) 
(Modified and added)
I feel that the weak topology is generated by pulling back in dual space so that we get small open sets.
I wonder whether it is the same story as pulling forms back.
 A: certainly they are similar concepts. In functional analysis you have a complete normed space $X$ with an inner product $<, >$ i.e. a Hilbert Space. Your tangent space $T_{p}M$ at a point $p\in M$ in a smooth manifold of dimension $n$ is a finite dimensional Hilbert Space $(T_{p}M, <,>_{g})$ with the inner product $<,>_{g}$ induced by the Riemannian metric $g$ hence all the concepts you learned in functional analysis apply to the tangent space at each point.
As you mentioned the dual space is simple the space the of continuous linear forms on $X$ and the concepts translate immediately to the tangent space $T_{p}M$ creating the dual space i.e. the cotangent space $T^{\ast}_{p}M$. 
Now in differential geometry we learned the importance of globally defined, invariant objects. Let's defined global Hilbert spaces. Consider the space of (measurable w.r.t. $dV_{g}$) vector fields $\mathcal{F}(M, TM)$. For simplicity let's assume that $M$ is compact. The standard way to define an inner product on  $\mathcal{F}^{}(M, TM)$ is by using the integral defined by using the volume  form $dV_{g}$ induced by the metric $g$. i.e. $$<X,Y>_{g}=\int_{M}<X_{p},Y_{p}>dv_{g}.$$ 
It is easy to prove that the closure of $(\mathcal{F}^{}(M, TM), \parallel\cdot\parallel_{g})$ is a Hilbert Space usually denoted $$L^{2}(M,TM)$$ i.e. the classical $L^2$ Hilbert spaces in the context of differential geometry. Of course it is possible to define $L^2$ spaces on any tensor bundle $TM^{\otimes k}\otimes T^{\ast}M^{\otimes s}$. Moreover, if the manifold is not compact, then there are many ways to define Hilbert spaces too by imposing regularity conditions. This is very useful to apply functional anaylsis methods to study PDE on manifolds. A nice introduction to these topics can be found in Chapter 10 "Elliptic Equations on Manifolds" of the book Geometry of Manifolds by L. Nicolaescu.  
