In this work, first of all, we study the convergence properties of one and multidimensional trigonometric Fourier series. It begins by discussing fundamental concepts such as vec- tor spaces, function spaces, and interpolation theorems. Inspired by Fourier analysis, it explores the feasibility of extending these concepts to infinite-dimensional spaces and or- thonormal systems. The study extensively covers the convergence behavior of Fourier series in the one-dimensional case, including partial sums, Dirichlet kernels, Fourier coefficients, and convergence criteria. It also examines the concept of Fej´er means. In the multidi- mensional context, the thesis explores topics such as partial sums, norm convergence, and summability means. Throughout the thesis, we use and refer to books [1] and [2], as well as the lectures given by my supervisor in his ”Fourier series” lectures [3]. What results are not directly referred to are taken from lectures [3, 4]. No new statements or results have been formulated in the thesis.