Further developments of the ideas in Fourier analysis are shown in this chapter. First, the basic definitions and properties of Hilbert spaces are presented. Examples are given with their inner products, e.g., spaces of square integrable functions and square summable sequences. As orthonormal bases in such spaces, the Haar system, the trigonometric system, the sinc system, Legendre polynomials, Hermite and Laguerre functions, and spherical harmonics are considered. As application of the spherical harmonics and the Laguerre functions, an outline of quantum mechanical results for the nonrelativistic hydrogen atom is given, so that the periodic system of elements and the hybridization of a water molecule can be explained by eigenfunctions of the according Schrödinger operator. A final section treats continuous and discrete wavelet transforms. A pointwise reconstruction formula for the continuous wavelet transform is proven, and the algorithm of Mallat for multi-resolutions is shown. For explanations the Haar wavelet is used. Further examples are image compressions and image denoising with Daubechies wavelets. Finally, a spectrogram with an STFT and a wavelet scalogram of an audio piece are computed for comparison and graphically shown. A filterbank with a series of bandpass filters is used for the scalogram.