In this work we will study the space of functions of bounded total variation, a space of functions with high relevance in various mathematical and applied contexts, such as image processing and the theory of partial differential equations. Following the paper from A. Cohen, W. Dahmen and I. Daubechies, we will characterize the bounded total variation space by studying an atomic decomposition of its functions. To do this, it is necessary to introduce the concept of dyadic cubes and the weighted weak $\ell_p$ sequence spaces. Finally, we will use this atomic decomposition to give a result on the interpolation of Fractional Sobolev spaces.