We introduce tensor products, which are the appropriate tool to construct the state space for systems that can be decomposed into non-interacting simpler components. The boson Fock space, built from such tensor products, is the appropriate state space to describe collections of any number of identical bosons. The annihilation and creation operators act on this Fock space. We then introduce the idea of quantum fields, which are operator-valued distributions, not trying yet to incorporate ideas from special relativity, and we argue that these quantum fields can be viewed as a quantized version of certain spaces of functions.