None

　　In this paper, we will study the concept of weak convergence and the related topices. First we describe the Riemann-Lebesgue lemma since it is the origin of the weak convergence. In the sequal, in section 2 we dicuss the main properties of weak and weak* convergence in a Banach space. We also details these notions for the particular case of L^p spaces in section 3. Then we will get the weak compactness in L^p spaces. In section 4 we talk a relevant class of periodic oscillating functions, which is the application of weak convergence in L^p spaces. And we will give an example about the weak limit of oscillating periodic functions. In section 5, we introduce some classes of Sobolev spaces and discuss their main properties. Finally, then we will get the compactness result in Sobolev spaces.

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