In this chapter we study various constructions related to the tensor product of vector spaces. In particular, we introduce symmetric and anti-symmetric tensor algebras, whose Hilbert space versions are called bosonic and fermionic Fock spaces. Fock spaces are fundamental tools used to describe quantum field theories in terms of particles.
We also discuss the notions of determinants, volume forms and Pfaffians, which are closely related to anti-symmetric tensors.
There are several non-equivalent versions of the tensor product of two infinite-dimensional vector spaces. We will introduce two of them, which are especially useful: the algebraic tensor product and the tensor product in the sense ...