Entanglement is a physical property exhibited by a mathematical vector when it cannot be decomposed into tensor products of multiple individual vectors residing in distinct Hilbert spaces. Before examining properties of entanglement, this chapter deals with a preliminary review of basic mathematical concepts of vectors and operators of a Hilbert space. The structure of this chapter develops as follows. Firstly, we provide the definition of a Hilbert space. As a straightforward example, we introduce a quantum bit, or equivalently, a qubit, representing a two-dimensional Hilbert space. Subsequently, the concept of entanglement naturally emerges in the context of two qubits. We then explore existing paradoxes associated with entanglement for identical quantum particles. Lastly, a brief overview of linear operators within a Hilbert space is conducted, incorporating the crucial concept of “mixed states,” laying the groundwork for a more in-depth discussion on entanglement in subsequent chapters.